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NUMERICAL SIMULATION OF COUPLED FIELDS IN LARGE POWER CABLES Daniela CÂRSTEA1, Ion CÂRSTEA2 Industrial Group CFR, Craiova. Str. Brâncuşi nr. 9, ROMANIA 2 University of Craiova, Craiova. Str. Decebal nr. 5, ROMANIA Abstract. This paper deals with the heat generated by ohmic losses in different parts of the electromagnetic devices. The problem is described by a coupled thermal-electric set of equations. The coupling between the two fields is the thermal effect of the electrical current or a material property as the electrical conductivity. A computation algorithm is presented for coupled problems in two dimensions. A numerical algorithm based on the finite element method (FEM) is presented describing the solution of two-dimensional systems. In our example we consider only steady-state regime although many transient regimes appear in the behavior of the electromagnetic devices. A CAD product was developed for coupled problems. The algorithm has been implemented in our CAD product and several examples from electrical engineering were solved with good results for the engineers. Keywords: Finite element method; CAD; coupled problems. 1 Introduction In some electromagnetic devices with strong current densities, it is important to compute the currents and the temperature caused by Joule-Lenz effects. There are a lot of industrial applications in this area. Many works in the professional literature present coupled models for the electromagnetic devices and this work is toward this direction with emphasis on the development of efficient algorithms in numerical computation of the coupled models. The examples are many. It is the case the contacts of the HV switcher equipment submitted to high current densities. The contacts of the circuit breakers at high temperature generated by high currents can melt the contacts. Another case is the dielectric heating in HV cables with polluted or imperfect insulation layers. An accurate computation of the temperature distribution is useful for designers of these equipments. In our work we use as target example a highvoltage (HV) and large-power cable in direct current. The problems that appear at this object are complex and depend on the cable regime. At a step voltage the electric field has a variation in time and the solution at this problem involves an iterative approach. After the stabilization of the electric transient regime, the cable is heating. This is a transient regime and the heating constant is more large than the electric constants. The problem is complex because of the imperfect insulation, that is the solid dielectric has impurities that destroy the high resistivity of these. These situations can be solved numerically and these aspects are presented in this work. 2 A 2D model for coupled electric and thermal fields A coupled model for electrical current and its thermal effect involves development of mathematical models for the two distinct physical fields but the solving of these models must be done simultaneously. Our models are based on Maxwell equations and heat conduction [1]. The static field distribution can be modeled by the following equations: E 0; E J with: - the material resistivity, E - the electric strength and J – the current density. A 2D-field model was developed for a resistive distribution of the electric field. An electric vector potential P was introduced by the relation [2, 4]: J P Laplace’s equation describes distribution (for anisotropic materials): the field x (x P x ) y ( y P y )0 (1) A finite element model for the above Laplace's equation can be obtained by Galerkin's procedure. With this model a resistive distribution of the electric field is computed and the heat source (JouleLenz’s effect) can be calculated as being (for isotropic materials): q J 2 P 2 P 2 2 2 ( J x J y ) [( ) ( ) ] (2) x y Mathematical model for the thermal field is the conduction equation [1]: x (kx T x )+ y (ky T y )+ q = c T t (3) with: T (x, y, t) - temperature in the point with coordinates (x, y) at the time t; kx , ky – thermal conductivities; -specific mass; c – specific heating; q – heating source. It is obviously that there is a natural coupling between electrical and thermal fields. Thus, the resistivity in equation (1) is a function of T, and the heating source q in (2) depend on J. Numerical models for the two field problems can be obtained by the finite element method. An iterative procedure was used for the temperature distribution. The algorithm in pseudo-code has the following structure: 1. Choose the initial value of the temperature 2. Repeat {Computations for electrical field} Compute the resistivity ρ Solve the numerical model for electric potential {Computations for thermal field} Compute the heating source q by (2) Solve the numerical model for the temperature Until the convergence_test is TRUE The convergence test is the final time of the physical process but in the internal loop of the cycle repeat-until we have an iterative process because the electric conductivity depends highly on temperature T and electric field E. In a first approximation we neglect the dependency of the resistivity by the electric field strength. In insulation the resistivity decreases with the temperature so that we can use the relation: (T ) (T0 )[1 (T T0 )] ρ (T0) is the resistance at ambient temperature T0 (usually +20 0C) and ρ (T) is the resistivity at the instantaneous temperature T. 3. The case of a large-power cable Our example is a high-voltage direct-current (HVDC) cable with two insulation layers. The leakage current in dielectric is caused by the finite resistivity of the dielectric insulation. At the application of a high voltage the field has a capacitive distribution initially. This distribution is for a short time so that it is not interset for the temperature distribution. Finally the field has a resistive distribution. Between these limits there is an intermediate field that can be computed by an iterative procedure. We limit our discussion at the final distribution of the electric field (resistive case). Generally speaking there is no perfect dielectric insulation so that a leakage current exists. Ohmic losses cause the dielectric heating. A parallel-plane model can be used to compute the electric and thermal fields. Fig.1 –Analysis domain and mesh The physical properties of this example (relative permitivity, electric resistivity, and thermal conductivity) are given below for a cable with the following geometrical dimensions: r1=15 [mm]; r2=30 [mm] Electrical properties are: =1.1014 [. m]; U=10 000 [V] Thermal properties are: kx=ky=0.15 [W/m. K] convective coefficient h=12 [W/m.K]; medium temperature: 40 0C. c=1800 [J/kg.K ];γ=1300 [kg/m3] The time domain for the transient regime is 1800[s]. The analysis domain is the insulation space. The symmetry of the problem can reduce the analysis domain to a quarter (Figure 1). In Figure 2 the map of the density current and with Pcond - the ohmic losses per cable meter in the inner conductor as Joule-Lenz’s effect. Thus, Neumann’s condition is: T p n C1 with C1 – the interface of the cable conductor and insulation. At the interface insulation-environment we consider a convective condition by the form: T h(T T ) n C 2 with h – the convective coefficient, T∞ - the ambient temperature and C2 – the boundary of the cable and the external medium. In Figure 3 the map and the vectors of the heat transfer are plotted at the final time of the time domain. Fig.2 – Map and vectors of the current density vectors of the current densities are plotted. The highest current density is near the conductor surface. Initially, the field strength has the highest strength near the conductor surface but this diustribution can be changed by the thermal field. It is possible to have the highest field strength at the boundary of insulation with external medium. The second stage of the numerical algorithm is the simulation of the thermal field. The heat source is the thermal effect of the current in the dielectric insulation and the load current of the cable. It is obviously that the ohmic losses in the cable conductor is the most important heat source. In our model we consider that there is a constant heat flux on the interface conductor-insulation. The source of this flux is the Joule-Lenz’s effect of the load in the cable. The mathematical model for the heat transfer is the conduction equation (3). The boundary conditions are: Neuman’s condition at the interface conductor-insulation, and convective condition at the boundary insulation-environment. The Fig.4 Neumann's condition can and be computed –Analysis domain mesh by the conductor losses in the case the cable was loaded before switching of the step voltage, that is the current in the cable has been raised long before and the temperature distribution in the cable is stable. In this case the value of the heat flux is computed with the relation: p Pcond 2r0 Fig.3 –Map and vectors of the heat flux The variation of the temperatures versus time at the interfaces of the insulation with the conductor and environment are plotted in Figure 4. The highest temperature is near the conductor and this fact leads to important changes in the electric field distribution in the insulation. The solution plotted in the Figure 3 is obtained in hypothesis that the resistivity of the insulation is constant on the time domain and space. Practically, the heat sources depends on the temperature because the electrical properties (resistivity of insulation and conductor) vary with the temperature and space variables. The highest temperature is near the conductor and the resistivity of the insulation decreases in this zone much more than at zone near the environment.The field distribution is a hyperbolic function if there is no temperature drop in insulation. With no temperature drop in insulation, the maximum value of the electric field T (K) 80 70 60 50 40 30 20 10 0 0 300 600 900 1200 1500 1800 Time (s) Fig. 4 - Temperatures vs. time at conductor (green) and environment (red) surfaces strength is near the conductor.For large loads that lead to high temperatures in conductor, the field near the environment may become higher than the highest field strength near the conductor. An analytical relation is proposed in the work [5]. In figure 4 the variations in time of the temperature are plotted at two points of interest : the conductor surface and environment surface. 4 Conclusions In our work we presented some examples that illustrate the real coupling between the electric and thermal fields. The mathematical models are solved using a finite element formulation. A CAD product for 2D or axi-symmetric problems was developed by the authors [1,4]. The validation of the simulation for our examples was achieved by comparing the results obtained from our program with results from open literature or obtained with similar software as program Quickfield [6]. In the models used in the work we have both explicit and implicit couplage between the two phenomena. The couplage is achieved both by the right term of the conduction equation (heat source) and electric conductivity of the material (in Laplace equation of the electric field). At each time step, the physical properties are computed using the temperature distribution computed by a distributedparameter model. In another variant of our software product we use a lumped-parameter model for temperature distribution in the case of HVDC cables. References: [1]. Cârstea, D. Numerical simulation of magnetothermal processes using automatic computation systems. Ph.D. Thesis, 1999. Romania. [2]. Bastos, J.P.A., Sadowski, N., Carlson, R. “A modelling approach of a coupled problem between electrical current and its thermal effects”. In: IEEE Transactions on Magnetics, vol.26, No.2, March 1990. [3]. Cârstea, D., Cârstea. I. “A CAD product for inverse thermal problems”. AMSE Press, Modelling, Measurement & Control, B, 2002Vol.71, N0 3,4. ISSN 0761-2516, pg. 61-70. [4]. Cârstea, D., Cârstea. I., CAD in electrical engineering. The finite element method. Publisher: Sitech, Craiova, 2000. Romania. [5]. Jeroense, M.J.P., Morshuis, P.H.F. Electric Fields in HVDC Paper-Insulated Cables. In: IEEE Transactions on Dielectrics and Electrical Insulation. Vol.5, No.2, April 1998. [6]. Quickfield package. Tera Analysis Co.