Coupled electromagnetic, thermal and stress analysis of large power electrical transformers

Coupled electromagnetic, thermal and stress analysis of large power electrical transformers DANIELA CÂRSTEA High-School Group of Railways, Craiova ROMANIA ALEXANDRU ADRIAN CÂRSTEA University of Craiova ROMANIA ION CÂRSTEA Department of Computer Engineering and Communication University of Craiova Str. Doljului nr. 14, bl. C8c, sc.1, apt.7, Craiova ROMANIA Abstract: Investigating the electromagnetic, thermal and stress fields inside an electrical transformer of large power delivers knowledge for predicting its performance. The mathematical models for the three processes are coupled and allow an accurate analysis on windings level as well as the prediction of the overall performance of the transformer. In this work we survey our research on a model described by a set of equations for the three fields. The interconnection of the three fields is given on the one hand implicitly by the temperature dependence on the material properties, on the other hand explicitly by the heat sources in the thermal model controlled by the Joule's effect in the electromagnetic system. As target example we use a power transformer, important equipment used in transmission and distribution of the energy. We present a numerical magneto-thermal-stress model for an encapsulated three-phase transformer. The model is based on the coupled approach and involves a stress analysis coupled with the computation of the power losses in the coils determined from the electromagnetic field and heat transfer analysis. The numerical models for the analysis of the coupled fields are based on the finite element method for axisymmetric fields. Key Words - Electrical transformers; Coupled fields; Finite element method. 1 Introduction Most electromagnetic systems involve coupled electromagnetic, thermal and mechanical processes. The trend in the design of these devices is characterised by new and performant requirements that involve more accurate mathematical models and computational techniques for accelerated tests in laboratories. Electrical transformers are important devices used to provide an electricity supply for many industrial applications. The performance of the electrical transmission systems is dependent on power distribution transformers. These devices are based on the law of electromagnetic induction. Even though these devices may seem simple devices, the realisation is very complex. In practice the design of these devices is based on simplified models of the transformer neglecting windings eddy currents, the magnetic core nonlinearity etc. In fact, the design of the transformer is dependent on a high number of design variables so that an optimum design involves powerful computers, new numerical algorithms and programming methods. Neural networks and genetic algorithms have been successfully applied to design high-performance transformers. An electrical transformer must satisfy a series of design requirements: efficiency, an imposed percentage of leakage impedance and no-load current. For a design engineer it is desirable to predict the performance of the transformer by numerical simulation. The performance is measured by some parameters: leakage fields, reactances, the flux distribution in the iron core and tank-wall losses. The main features of power transformer operation are its internal temperatures, specially the winding hotspot temperature and the top-oil temperature. These ISBN: 978-960-6766-77-0 29 ISSN 1790-5117

features affect the insulation ageing and, consequently, the life of the device. The internal temperature of the transformer depends on many parameters of the environment as the solar radiation, the wind speed, humidity, the rain/evaporative cooling etc. In other words, the computation of the hot spot depends on a lot of perturbations that are difficult to be measured or estimated. Our work will concentrate on a coupled analysis of the electrical transformers. The analysis of many electromagnetic devices requires the solution of a problem in which the electromagnetic field equations are coupled to other partial differential equations, such as those describing thermal field, fluid flow or stress behaviour. The coupling between the fields is a natural phenomenon and only in a simplified approach the device analysis can be decomposed in independent problems. When a time-varying magnetic field acts on a conducting medium, the eddy-currents are induced. These currents cause heat generation due to the Joule effect. The conducting medium is subjected to Lorenz force as well as the heat supply by eddy current loss. Consequently, two kinds of stress are generated by time-varying magnetic field: A thermal stress caused by eddy current loss A magnetic stress caused by Lorenz force The sum of these stresses represents the solution of the coupled problem. Most of the professional literature in this area consider the electromagnetic field only, an approach equivalent to considering the material properties (electrical conductivity, magnetic permeability) are not dependent on temperature [1]. In fact the electromagnetic and thermal fields are coupled both implicitly and explicitly. Thus, the electrical conductivity for windings is dependent on temperature, and Joule effects of the electrical currents (source and eddy-currents) are heat sources in such devices. The electrical transformers are encapsulated for protection against many undesirable phenomena as the humidity, fire, water etc. As a results of these protections the heat in transformers can have dangerous values so that they must be cooled by natural or forced convection. The heat is mostly generated within transformer windings by Joule Lenz's effect of the electrical current but in an accurate computation the losses from the core can not be neglected. In professional literature the analysis of the transformers is done for sinusoidal excitation currents and low frequencies so that our work is limited to this practical case. In a coupled problem as in our work, we use mathematical models derived from electromagnetics, thermics and elasticity theory. The non-linear mathematical model of the coupled problem prohibits an analytical solution so that we must use numerical methods [5]. An efficient method for numerical simulation of distributedparameter systems is the finite element method (FEM). The mathematical model of the transformer is a set of partial derivative equations of elliptic and/or parabolic type so that we have a typical distributed-parameter system. In this work we present algorithmic skeletons for numerical analysis of the processes in large power transformers using coupled models. In Fig. 1 an axial section in a transformer is shown. Sinusoidal currents cross the phase windings (low and high voltage). 2 Mathematical model of the transformer - coupled approach The mathematical model for the electromagnetic field is defined by Maxwell's equations [2]. This model can have a simplified formulation using the magnetic vector potential A called A-formulation. A transformer has a geometrical and field symmetry so that an axisymmetric model is used to describe the device behaviour. The core and windings have rotational symmetry but other elements (tank, shunts etc) have no symmetry but this aspect can be ignored in design. For axi-symmetric fields in cylindrical coordinates, the A-formulation is a 2D-scalar model in (r-z) plane: σ ( ra) υ ra = J s r t ( ) r (1) where ν represents the reluctivity, ω is the angular frequency, σ is the electric conductivity, A is the complex magnetic vector potential, J s is the source complex current density and r is the radius. Eq. (1) is solved under initial and boundary conditions. For the harmonic-time case, mathematical model is [3]: υ ( ra) υ ( ra) + r r r z r z (2) υ jσω ( ra) = J s r The solution of Eq. (2) gives the θ component of the magnetic vector potential, U= ra θ. With the modified potential U the analogy with the model Oxy is obviously. The temperature distribution within the analysed device can be determined by solving the energy equation [4]: [(cγ )( T ) T ] + [ k( T ) T ] = q (3) t ISBN: 978-960-6766-77-0 30 ISSN 1790-5117

T ( x,0) = T ( x) x Ω (4) 0 where: T(x,t) is the temperature in the spatial point x at the time t; k is the tensor of thermal conductivity; γ is mass density; c is the specific heat that depends on T; q is the density of the heat sources that depends on T; T 0 (x) is the initial temperature. In the coupled problems the internal volumetric heat sources q occurring in Eq. (3) are results of the electromagnetic analysis defined by: A q = σ ( T ) ( ) 2 (5) t with σ the electrical conductivity of the material. The heat sources for the thermal system are generated by the electromagnetic system. Practically there are more sources of power loss in a transformer. Hysteresis and eddy core losses that are due to the magnetisation and no-load current Stray losses that are due to the leakage fluxes in frames and tank The losses from windings: loss due to DC resistance, loss due to induced currents by flux leakage, and circulation loss. Stress analysis problem is the utmost one that imports the temperature field from the heat transfer problem and the magnetic forces from the timeharmonic magnetic problem. The conducting medium is subjected to both temperature change and Lorenz force. Due to this magnetic and thermal loading the device components become deformed. The electrodynamic force is a vector normal to the magnetic induction B according to the formula F = I X B. The force at the end of the winding builds up principally in the axial direction. During a short-circuit the windings turns, turn insulation and spacers are subjected to alternate compression and relaxation. In a stress analysis problem the displacement, strain and stress are of great importance. The physical quantities for stress analysis are: Displacement vector δ Strain vector ε and its principal values Stress vector σ and its principal values Some relevant criteria (Tresca criterion, Drucker-Prager criterion, Mohr-Coulomb criterion, Von Mises stress) In an accurate analysis and synthesis of a transformer we must use a complete mathematical model. The procedure used more and more exclusively in power transformer construction must satisfy all electrical requirements. Yet mechanical behaviour of the device must not be ignored, e.g. the short circuit strength of windings can lead to the explosive destruction of the transformer. The bending strength is of great importance in the case of clamping rings that assure constant pressure on the transformer winding during operation and short circuits. Electrodynamical forces occur primarily during short circuits and have a destructive effect on a large transformer. The radial and axial forces appear between the transformer windings and in windings. Thus, in a transformer with two windings, the current flows in opposite directions in both windings so that both windings repel each other due to the radial forces. These forces pose special problems for the support cylinder adjacent to the core. The axial forces appear in winding and pose problems for insulating material and spacers. The current direction of the current is identical within the same winding so that the forces contract the coils. The consequences of the axial forces are: The heights of the windings are modified The spacers used for cooling between winding conductors are subject to high stresses The axial compression of the winding material (copper) is increased For axisymmetric problems, the displacement field is assumed to be defined by the two components of the displacement vector in direction Or and Oz. Only three components of strain and stress tensors are independent in both plane stress and plane strain cases and four components for the axisymmetric problems due to the radial deformation. The equilibrium equations for axisymmetric problems are [6]: 1 ( rσ r ) τ rz + = f r (6) r r z 1 ( rτ rz ) σ z + = f z (7) r r z where σ r, σ z τ rz are the stress components, and f r, f z are components of the volume force vector. Temperature strain is determined by the coefficients of thermal expansion and temperature difference between strained and strainless states. Components of the thermal strain for axisymmetric problem and orthotropic material are defined by the following equation [6]: α z α r ε 0 = ΔT (8) α θ 0 where α z, α r, α θ are the coefficients of thermal expansion along the corresponding axes for orthotropic material, and ΔT is the temperature difference between strained and strainless states. For linear elasticity, the stresses are related to the strains by the constitutive law (Hooke's law): ISBN: 978-960-6766-77-0 31 ISSN 1790-5117

{ 0 σ} = [ D ]({ ε} { ε }) (9) where [D] is a matrix of elastic constants (Young's modulus, Poisson's ratio, shear modulus), and {ε 0 } is the column vector for the initial thermal strain. 3 Numerical results The nonlinear mathematical model of the problem prohibits the use of the analytical methods. A numerical solution is the single approach for the coupled problem so that a spatial and time discretization must be done. The finite element method (FEM) can handle the solution of coupled problem, which is an elliptic-parabolic problem. The order of the discretization can lead two different numerical models. In our work we do a spatial discretization, firstly. In this way we approximate a distributed-parameter system by a lumped-parameter systems where there is a large professional literature for analysis and synthesis. The system theory was developed for lumped-parameter systems so that we can use the remarkable results from this area. example) or unbalanced. The low voltage windings are usually placed next to the core. The high voltage windings are either mounted separately or wound directly over the low voltage windings (as in our example). There is a gap between the HV and LV windings, which is used for axial cooling of the windings. In many power transformers the electrostatic shields are used. These shields are heat sources because of the eddy voltages generated in accordance with the law of induction. The eddy currents in the electrically conducting shield produce heat losses that must be kept at low values both in normal operation and overcurrents caused by system short circuits. Fig. 2 The meshed domain The transformer under consideration is a 400 MVA, three-phase, wound core, oil immersed, power transformer, shown in the Fig. 1. In Fig. 2 the meshed domain for finite element model is shown with the horizontal axis as rotation axis (Oz). The considered transformer has the voltage ratio 400 kv/24 kv. Windings' physical properties are presented in the Table 1. Fig. 1 Axial section (half) As first example we consider a three-phase transformer in oil where the coils and limbs have rotational symmetry about the axis of the limb, that is, a transformer with cylindrical windings. The windings are wound on former cylinders and a mounted concentrically to the stepped leg of the core. An axisymmetric model is represented in Fig. 1 (an axial section by a surface perpendicular to symmetry plane). Because of the geometrical symmetry, only a half of the window is used for analysis (see Fig. 2). The tank wall is made of mild steel and symmetry axis denoted by 1 is axis Oz in a cylindrical co-ordinate system Orz. The two transformer windings, LV (low voltage) and HV (high voltage), can be balanced (as in our Table 1 Relative permeability 1 1 l Electrical conductivity 3.6e+07 S/m Thermal conductivity 140 W/K.m Young's modulo 6.9e+10 N/m 2 Poisson's ratio 0.33 Shear modulus 2.5e+10 N/m 2 Thermal expansion coefficient 2.33e-05 1/K Specific density 2700 Kg/m 3 We consider the particular case of isotropic elastic material so that in relation (9) D is a symmetric matrix whose entries are functions of only two independent parameters. These parameters are either Young's modulus E and Poisson ratio ν, or the bulk B and shear G moduli. The following relations hold among these parameters: E E B = ; G = 3(1 2υ ) 2(1 + υ) The stress problem is the determination of the displacement vector δ of the strain field ε and stress ISBN: 978-960-6766-77-0 32 ISSN 1790-5117

tensor produced in the device by the magnetic field and temperature field. In Fig. 3 the field lines for the magnetic problem are shown. We considered the case of high permeability for tank and yoke so that a Neumann's boundary condition was considered. The vectors of Lorenz force are plotted in Fig. 4. F = (σ n) ds where σ is the stress tensor and the integral is evaluated over the boundary of the volume. The displacements along some contours can be plotted. In Fig. 8 the displacement at external surface of the high-voltage winding is shown, with starting point at the winding bottom. Fig. 3 Magnetic field lines Fig. 6 Displacement vectors Fig. 4 Lorenz force Fig. 7 Strain tensor The heat developed by the source current and eddy currents in the windings is obtained using the numerical solution of Eq. (3). In finite element models the same geometrical model was used for the three field problems. In Fig. 5 the isotherms are plotted. The solution of the elasticity model lead to the results presented in Fig. 6. The strain tensors are shown in Fig. 7. The absolute value of displacement is computed with the relation: 2 2 δ = δ z + δ r where δ z and δ r are the components along coordinate axes. Fig. 8 Displacement at external surface Fig. 5 Isotherms For stress analysis the total force acting on a particular volume is defined by: 8 Conclusions Transformer lifetime is dependent on a lot of factors that are electrical, thermal and mechanical. Most electrical defects in this apparatus are attributable to the mechanical failure during short-circuits. The use of the modern numerical techniques for analysis of the transformers in laboratories is a strong requirement for the designers. A coupled model is a real approach for the accurate analysis of these devices. ISBN: 978-960-6766-77-0 33 ISSN 1790-5117

We presented some aspects in a coupled analysis for the transformers with cylindrical windings. In a future work we shall present a coupled analysis of the transformers with rectangular windings. The numerical models were obtained by finite element method in formulation with the magnetic vector potential A. In this work we ignored the losses in the core and the tank although they are of a great importance in the transformer operation. These aspects will be included in a future paper. References: [1]. Cârstea, D., Cârstea, I. CAD of the electromagnetic devices. The finite element method. Editor: Sitech, 2000. Romania. [2]. Cârstea, I., Advanced Algorithms for Coupled Problems in Electrical Engineering. In: Mathematical Methods and Computational Techniques in Research and Education. Published by WSEAS Press, 2007. ISSN: 1790-5117; ISBN: 978-960-6766-08-4. Pg. 31-38 [3]. Cârstea, I., Cârstea, D.,, Cârstea, A. A.,. "Numerical simulation of coupled fields in reactors with electromagnetic shield." In: Proceedings of the 8-th International Conference on Applied Electromagnetics, ISBN 978-86- 85195-47-0. PES 2007, Nis, Serbia. [4]. Cârstea, I., Cârstea, D., Cârstea, A. A.,"A domain decomposition approach for coupled fields in induction heating device". In Volume: System science and simulation in engineering. WSEAS Press, 2007. ISSN: 1790-5117; ISBN: 978-960- 6766-14-5. [5]. Marchand, Ch., Foggia, A. 2D finite element program for magnetic induction heating. In: IEEE Transactions on Magnetics, vol. 19, no.6, November 1983. [6]. *** QuickField program, version 5.4. Page web: www.tera-analysis.com. Company: Tera analysis. ISBN: 978-960-6766-77-0 34 ISSN 1790-5117