2017 2nd International Conference on Applied Mathematics, Simulation and Modelling (AMSM 2017) ISBN: 978-1-60595-480-6 Analytical Calculation Method for Steady-state Current Capacity of HVDC Cables Chun-sheng WANG 1 , Yong-qiang MU 1 , Yong WANG 1 , Yan LIU 1 , He-yan ZHU 1 and Tao HAN 2 1 State Grid Liaoning Economic Research Institute, Shenyang 110006, China 2 Northeast Electric Power University, Jilin 132012, China Keywords: HVDC cable, Analytical calculation, PMLTCT, PMITD. Abstract. The accurate calculation of the steady-state current capacity of HVDC cable is of great significance for the full utilization of its transmission capacity. Firstly, the method of calculating the steady-state current capacity of the HVDC cable is proposed. The method also considers permissible maximum long-term conductor temperature (PMLTCT) and permissible maximum insulation temperature difference (PMITD). Secondly, the steady-state carrying capacity of ±160 kV cross-linked polyethylene DC cable is calculated by this method, and verified by the finite element method. Finally, the influence of ambient temperature, PMLTCT and PMITD on the steady-state current capacity of DC cable are studied. It is found that the current carrying capacity curves of DC cable which consider PMLTCT and PMITD may intersect with ambient temperature change. If the ambient temperature is higher than intersection temperature, the steady-state current capacity is determined by PMLTCT, and otherwise, the steady-state current capacity is determined by PMITD. Introduction With the change of energy utilization and the development of power grid, the role of DC transmission is becoming more and more important in the transmission and distribution of power [1] . HVDC cable is an important part of DC transmission, which undertakes the task of DC power transmission in underground and submarine. Steady-state current capacity is an important indicator of cable transmission capacity. Accurate calculation of cable steady-state current capacity is of great significance for the full utilization of cable transmission capacity [2]-[3] . There are many researches on calculation current capacity of AC cable, among them, the equivalent thermal model is widely accepted [4]-[5] . Compared to AC cable, to calculate DC cable steady-state current capacity need take PMLTCT in account [6] . Besides, it also need consider electric field distribution of insulation layer [7]-[8] . Under the DC state, electric field distribution of insulation depends on the conductivity of cable, while conductivity affected by temperature and electric field strength [9]-[11] . For the calculation of electric field distribution in insulation layer, which involved in the steady-state current capacity calculation of HVDC cable, the analytical method proposed in [12], which requires temperature and electric field intensity at one point in the insulation layer as a reference. In [13], the iterative method is used to solve electric field, which proves that temperature difference of insulation layer is closely related to electric field distribution. In this paper, an analytical method for steady-state current calculation of HVDC cable is proposed, which adopts equivalent thermal circuit model of DC cable, ignores leakage current that caused by insulation temperature rise, meanwhile, considers PMLTCT and PMITD. Finally, this analytical method’s validity is verified by finite element method, and the influence of ambient temperature, PMLTCT and PMITD to the steady-state current capacity of DC cable are studied. 166
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Analytical Calculation Method for Steady-state Current Capacity of
HVDC Cables
Chun-sheng WANG1, Yong-qiang MU1, Yong WANG1, Yan LIU1, He-yan ZHU1
and Tao HAN2
1State Grid Liaoning Economic Research Institute, Shenyang 110006,
China 2Northeast Electric Power University, Jilin 132012,
China
Keywords: HVDC cable, Analytical calculation, PMLTCT, PMITD.
Abstract. The accurate calculation of the steady-state current
capacity of HVDC cable is of great significance for the full
utilization of its transmission capacity. Firstly, the method of
calculating the steady-state current capacity of the HVDC cable is
proposed. The method also considers permissible maximum long-term
conductor temperature (PMLTCT) and permissible maximum insulation
temperature difference (PMITD). Secondly, the steady-state carrying
capacity of ±160 kV cross-linked polyethylene DC cable is
calculated by this method, and verified by the finite element
method. Finally, the influence of ambient temperature, PMLTCT and
PMITD on the steady-state current capacity of DC cable are studied.
It is found that the current carrying capacity curves of DC cable
which consider PMLTCT and PMITD may intersect with ambient
temperature change. If the ambient temperature is higher than
intersection temperature, the steady-state current capacity is
determined by PMLTCT, and otherwise, the steady-state current
capacity is determined by PMITD.
Introduction
With the change of energy utilization and the development of power
grid, the role of DC transmission is becoming more and more
important in the transmission and distribution of power [1]. HVDC
cable is an important part of DC transmission, which undertakes the
task of DC power transmission in underground and submarine.
Steady-state current capacity is an important indicator of cable
transmission capacity. Accurate calculation of cable steady-state
current capacity is of great significance for the full utilization
of cable transmission capacity [2]-[3].
There are many researches on calculation current capacity of AC
cable, among them, the equivalent thermal model is widely accepted
[4]-[5]. Compared to AC cable, to calculate DC cable steady-state
current capacity need take PMLTCT in account [6]. Besides, it also
need consider electric field distribution of insulation layer
[7]-[8]. Under the DC state, electric field distribution of
insulation depends on the conductivity of cable, while conductivity
affected by temperature and electric field strength [9]-[11]. For
the calculation of electric field distribution in insulation layer,
which involved in the steady-state current capacity calculation of
HVDC cable, the analytical method proposed in [12], which requires
temperature and electric field intensity at one point in the
insulation layer as a reference. In [13], the iterative method is
used to solve electric field, which proves that temperature
difference of insulation layer is closely related to electric field
distribution.
In this paper, an analytical method for steady-state current
calculation of HVDC cable is proposed, which adopts equivalent
thermal circuit model of DC cable, ignores leakage current that
caused by insulation temperature rise, meanwhile, considers PMLTCT
and PMITD. Finally, this analytical method’s validity is verified
by finite element method, and the influence of ambient temperature,
PMLTCT and PMITD to the steady-state current capacity of DC cable
are studied.
166
Equivalent Thermal Circuit Model
According to heat transfer theory, under DC operation state, the
heat transfer process of cable body and surrounding medium can be
equivalent to a thermal circuit model, as shown in Figure 1. In
Figure 1, Φ denote its unit length loss of conductor, W/m; T1, T2,
T3 and T4 denote its insulation layer, water-resistant layer, outer
protection layer and the ambient circumstance thermal resistance,
K·m/W, respectively; θc, θi, θb, θs and θa denote its conductor,
interface between insulation layer and water-resistant layer,
interface between water-resistant layer and outer protective layer,
cable surface and ambient temperature, °C, respectively.
Figure 1. Equivalent thermal circuit of DC cable.
The Principle and Method of Analytical Calculation
Considering PMLTCT
IEC standards 60287 give the calculation method of steady-state
current capacity of DC cable under 5 KV [8]:
1 2 3 4(T T ) M a
IEC M M
(1) Where IIEC is steady-state current capacity, A; θM is PMLTCT,
°C; RM is unit length DC resistance
of conductor at permissible maximum long-term temperature, Ω/m; n
is the number of conductors. Generally, the HVDC cable is
single-core cable, and n is 1in equation (1), so:
M a T
(2) Where IT is steady-state current capacity when consider PMLTCT,
A; T is the total thermal
resistance of cable and ambient environment, equal to the sum of
T1, T2, T3 and T4, K·m/W.
Considering PMITD
As shown in Fig.1, according to heat transfer theory, the
difference in temperature between cable core conductor and
surrounding environment medium is equal to the product of core
conductor loss and total thermal resistance [8], that is:
c a T (3) Where 2
L DCI R , IL is cable current carrying, A; RDC is the unit length
resistance, Ω/m.
Considering the linear variation of DC resistance of the conductor
with temperature, so RDC can be obtained by the following formula
[8]:
0[1 ( 20)]DC cR R (4) Where R0 is unit length DC resistance of
conductor at 20°C, Ω/m; is the temperature coefficient
of conductor’s DC resistance, 1/°C. The core conductor temperature
θc is obtained by formula (3) (4):
167
(5) The relationship between temperature θ(r) at any position in
insulation layer and conductor
temperature θc are as follow [8]:
2
1
(6) Where θ(r) is the distribution of insulation temperature, °C; r
is the radius of any position in
insulation layer, mm; λ is the thermal conductivity of insulation
layer material, W/ (m·K). θ (r) can be obtained by (4), (5),
(6):
2 2 2
2 1 1
r I AT r
(7) Where A= R0, B= (1-20 ) R0, C=1/2λ. The temperature difference
of cable insulation layer can be obtained by (7):
2 2 2
I BT r r I AT I BT
I AT
(8) Where T1=Cln(r2/r1) In (9), r1and r2 denote its inner and outer
radius of insulation layer, respectively, its unit is mm. The
steady-state current carrying IE of high voltage when consider
PMITD, can be obtained by
(8):
Finite Element (FEA) Model
In order to verify the validity of this analytical method for
calculate steady-state current capacity of HVDC cable in this
paper, a 3D FEA model, which is a ±160 KV XLPE DC cable used for
analytical calculation in one project, was established in COMSOL
Multiphysics software, and the results of those two methods were
compared.
Figure 2. The boundary conditions of FEA model.
In heat transfer theory, common boundary conditions include 3 types
[5], as is shown in Figure 2: (i) the temperature value of boundary
is provided, which is called the first kind of boundary condition;
(ii) the heat flux density of boundary is provided, which is called
the second kind of boundary condition; (iii) the surface heat
transfer coefficient between boundary object and
168
surrounding fluid, as well as temperature of surrounding fluid are
provided, which is called the third kind of boundary condition. The
finite element simulation of current capacity is carried out by 3D
FEA model, current-carrying can be applied on cable directly, and
it is not necessary to equalize the core conductor as a heat
source. Meanwhile, DC resistance of the conductor can change
linearly with temperature, which consistent with analytical
method.
Results and Analysis
Steady-state Current
In this paper, ±160 KV XLPE cable is taken as an example to
calculate steady-state current capacity of DC cable, and the
relevant parameters used for the calculation are shown in
Table1.
Table 1. Calculation parameters of steady-state current capacity of
±160 kV XLPE DC cable.
Known parameters
/(1/°C) ρ/(Ω·m) λ/[W/(m·K)] 3.93×10-3 1.724 1×10-8 0.29
r1/mm r2/mm S/m2 13.25 31.55 5.516×10-4
Set parameters θM/(°C) θM/(°C) θa/(°C)
90 20 20
T1/(K·m/W) T/(K·m/W) R0/(Ω/m) 0.48 1.25 3.126×10-5
RM/(Ω/m) A B 3.986×10-5 1.229×10-7 2.880×10-5
The conductor of cable is made of copper and its , ρ see IEC
Standards 60287[5], λ, r1, r2 is listed in Table 1. For XLPE
insulation, taking PMLTCT θM=90°C [14], PMITD θM=20°C [5]. Besides,
assuming that the ambient temperature θa=20°C.
Putting the relevant parameters (shown in Table 1) into (2) and
(11), then IT and IE is obtained, and the results are shown in
Table 2. With IT, IE as the initial load current value,
respectively, the temperature distribution of cable is calculated
by FEA method. And use secant method to fine-tune load current
values. The load current value is recorded as IT’, with compared to
the permissible maximum long-term temperature 90°C, which caused
the conductor temperature is less than 0.1°C, and another load
current value IE’, with compared to permissible maximum value 20°C,
which caused the insulation temperature difference is less than
0.1°C, the results are shown in table 3, respectively.
Table 2. Current capacity of ±160 kV XLPE DC cable.
Constrain condition/(°C) analytic method/A FEA method/A θM=90
IT=1185 IT’=1178
θM=20 IE=1052 IE’=1050 IDC=1052 IDC
’=1050
As can be seen from Table 2, a difference between those two type
methods for calculating steady-state current capacity is less than
0.6%.
Analysis Influence Factors
The value of ambient temperature θa is taken from 0°C to 50°C, θM
is taken 20°C and 30°C, respectively, θM is taken 70°C and 90°C,
respectively. Then the relationship curves of IT, IE and θa
can be obtained by proposed analytic method, respectively, as shown
in Figure 3.
169
Figure 3. Relationship between current capacity and ambient
temperature of HVDC cable.
As can be seen from Figure 3, IT and IE showed a downward trend
with rise of θa. But the decline rate of IT is much faster than IE.
Besides, it is also found that the curves IT and IE intersect at L,
M and N. When the ambient temperature θa is lower than the
intersection temperature, curves IE is located below the curves IT,
so the steady-state current capacity is determined by IE. When the
ambient temperature θa is higher than the intersection temperature,
curves IT is located below the curves IE, so the steady-state
current capacity is determined by IT.
Summary
In this paper, the analytical calculation method of steady-state
current capacity of HVDC cable is derived. Taking one ±160 KV XLPE
cable as an example, and calculate this example by proposed
analytical method. Compared with FEA method, the conclusions can be
obtained as follow:
(1) An analytical method for calculate the steady-state current
capacity of HVDC cables is proposed, and the method mentioned here
takes both account of PMLTCT and PMITD. Then, taking the ±160 KV
XLPE cable as an example, the validity of this method is
verified.
(2) The curves of steady-state current capacity which consider two
constraint conditions may have intersection with the change of
ambient temperature. When ambient temperature θa is lower than
intersection temperature, the steady-state current capacity is
determined by IE. And when ambient temperature θa is higher than
intersection temperature, current capacity is determined by
IT.
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