Keywords: HVDC cable, Analytical calculation, PMLTCT, PMITD.

017 nd International Conference on Applied Mathematics, Simulation and Modelling (AMSM 017) 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 1 State Grid Liaoning Economic Research Institute, Shenyang 110006, China Northeast Electric Power University, Jilin 1301, 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 []-[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 [1], 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, T, 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 6087 give the calculation method of steady-state current capacity of DC cable under 5 KV [8] : M a IIEC RMT1nRM(TT3T 4) (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 IT RMT () 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, T, 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 ILRDC, 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] : RDC R0 [1 ( c 0)] (4) Where R0 is unit length DC resistance of conductor at 0 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

a I RT(10 ) c 1 I RT L 0 L 0 (5) The relationship between temperature θ(r) at any position in insulation layer and conductor temperature θc are as follow [8] : ILRDC r c () r ln( ) r1 (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): r I BT r () r [1I ACln( )] I BCln( ) a L L L r1 1 IL AT r1 Where A=R0, B= (1-0) R0, C=1/λ. The temperature difference of cable insulation layer can be obtained by (7): (7) I BT a L i ( r1) ( r) c i ILAT1 I LBT1 1 IL AT (8) Where T1=Cln(r/r1) In (9), r1and r 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): I E M ( A B) T AT a Where IE, A; θm is PMITD, C. 1 M (9) 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. The boundary conditions of FEA model. In heat transfer theory, common boundary conditions include 3 types [5], as is shown in Figure : (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 Set parameters Calculation parameters /(1/ C) ρ/(ω m) λ/[w/(m K)] 3.93 10-3 1.741 10-8 0.9 r 1/mm r /mm S/m 13.5 31.55 5.516 10-4 θ M/( C) θ M/( C) θ a/( C) 90 0 0 T 1/(K m/w) T/(K m/w) R 0/(Ω/m) 0.48 1.5 3.16 10-5 R M/(Ω/m) A B 3.986 10-5 1.9 10-7.880 10-5 The conductor of cable is made of copper and its, ρ see IEC Standards 6087 [5], λ, r1, r is listed in Table 1. For XLPE insulation, taking PMLTCT θm=90 C [14], PMITD θm=0 C [5]. Besides, assuming that the ambient temperature θa=0 C. Putting the relevant parameters (shown in Table 1) into () and (11), then IT and IE is obtained, and the results are shown in Table. 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 0 C, which caused the insulation temperature difference is less than 0.1 C, the results are shown in table 3, respectively. Table. Current capacity of ±160 kv XLPE DC cable. Constrain condition/( C) analytic method/a FEA method/a θ M=90 I T=1185 I T =1178 θ M=0 I E=105 I E =1050 I DC=105 I DC =1050 As can be seen from Table, 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 0 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. () 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. Reference [1] H. Zhao, J. Jin and X. Lu, "Advantage of HTS DC power transmission," 009 International Conference on Applied Superconductivity and Electromagnetic Devices, Chengdu, 009, pp. 403-406. [] Giovanni M., Massimo M. Extruded Cables for high-voltage direct-current transmission: advances in research and development [M]. Newyork: Wiley-IEEE Press, 013: 58-117. [3] International Electrotechnical Commission. IEC 6087-1-1, Electric cables-calculation of the current rating-part 1-1: Current rating equations (100 % load factor) and calculation of losses [S]. Geneva: HIS, 006. [4] Huang Z.Y., Pilgrim J.A., Lewin P.L., et al. Thermal-electric rating method for massimpregnated paper-insulated HVDC cable circuits [J]. [5] International Electrotechnical Commission. IEC 6087--1, Electric cables-calculation of the current rating-part -1: Thermal resistance-calculation of thermal resistance [S]. Geneva: HIS, 006. [6] IEEE Transactions on Power Delivery, 015, 30(1): 437-444. [7] Boggs S., Damon D.H., Hjerrild J., et al. Effect of insulation properties on the field grading of solid dielectric DC cable [J]. IEEE Transactions on Power Delivery, 001, 16(4): 456-461. 170

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