Eddy Current Losses in the Tank Wall of Power Transformers

Eddy Current Losses in the Tank Wall of Power Transformers Erich Schmidt Institute of Electrical Drives and Machines, Vienna University of Technology A 14 Vienna, Austria, Gusshausstrasse 25 29 Phone: +43-1-5881-37221, Email: erich.schmidt@tuwien.ac.at Introduction To maintain quality, performance and competiveness, the finite element method is increasingly utilized not only for the verification of contractual values of existing transformers, but also for the initial design and for the design optimization of transformers. The detailed geometry of windings, core, core clamping construction, tank wall and shielding requires 3D numerical calculation methods in order to obtain high accurate values for the material utilization as well as the iron and power losses. With a modern design of power transformers, there are two key points for the initial design. First, an optimal utilization of the insulation material with regard to the specification is of great interest for the costs as well as the long time quality. Secondly, there are a lot of degrees of freedom with the arrangement of the tank wall shielding for reducing the eddy current losses in the tank wall. This report presents 3D finite element analyses with regard to the eddy current losses in the tank wall. In this case, the calculations serve as the reference for simplified calculation approaches which can be included with the initial design, the design review or the design optimization. The eddy current losses in the tank wall caused by the leakage magnetic flux strongly depend on the geometry and material properties of the core and the steel tank wall. Usually, a tank wall shielding by means of sheets of high electric conductivity and high magnetic permeability will be used to reduce the additional tank losses. In those cases, the arrangement of the shielding has a lot of degrees of freedom. Thus, optimization techniques with numerical field calculations are used together as an approach towards an optimal design. Representation of Eddy Currents in Ferromagnetic Material The calculation of eddy current losses in magnetic steel is a problem difficult to solve. Due to the nonlinear magnetic material properties, numerical methods are considerable difficult to apply because they are chiefly developed for non-magnetic conductive materials [1]. Several finite element formulations are known for nonferromagnetic and even ferromagnetic eddy current carrying sheets [1, 2]. Mainly, they describe the required formulations for rather thin sheets as well as laminations and neglect the influence of saturation in the ferromagnetic material. When considering the eddy current losses in the nonlinear tank wall, the relationship between the magnetic excitation H and the magnetic flux density B within the material can be assumed to be a step function as shown in Fig. 1 [3, 4]. This means that in principal, the flux density in the material can only be the saturation flux density B S, either positive or negative. Consequently, the magnetic flux in the material keeps staying at the surface. This surface flux generates a saturated layer with the thickness only depending on the peak value of the total amount of magnetic

B B(H) = B S sign (H) H Fig. 1: Simplified magnetic characteristic of the ferromagnetic material flux entering the material. A variation in time of the flux takes place by the movement of a surface with the velocity as given by v S = ω 2 cos ωt kπ 2, < ωt kπ < π, k =, ±1, ±2,.... (1) This surface separates regions with positive saturation flux density from regions with negative saturation flux density. The movement always starts at the material surface and is directed towards the inside of the material as depicted in Fig. 2. The eddy currents are flowing also only in this saturated layer of thickness, parallel to the surface and perpendicular to the direction of the flux density. ωt = ωt = π/4 ωt = π/2 x x x ωt = 3π/4 ωt = π ωt = 5π/4 x x x Fig. 2: Magnetic flux density profile in saturated layer at various points in time during a power frequency cycle The behaviour described above can be used to generate comparable simple finite elements which nevertheless represent the actual behaviour quite accurate. The important properties are represented with surface elements which deal with the fundamental harmonics of flux and current density. linear case: J S = 2 H S step-function: J S = 8 5 3π H S sin ( ωt + ϕ ), (2) sin ( ωt + ϕ ). (3)

Contrarily to linear eddy currents, now there is a phase shift between the fundamental harmonics of tan ϕ =.5 instead of tan ϕ = 1.. Moreover, the magnitude of the current density is approximately 35% higher in comparison to linear models [3]. Finite Element Modelling Table 1 lists the main data of the investigated power transformer. The coil windings of this three-limb transformer consist of a two layer low voltage winding and a two layer high voltage winding with two tapping windings. Table 1: Main characteristics of the power transformer Rated power 16 MVA Rated high voltage 165 kv Rated low voltage 67 kv Rated frequency 5 Hz Short circuit impedance.12 Cooling system ONAN/ONAF Contrarily to 2D cartesian and axial-symmetric analyses as proposed in the literature, full 3D approaches are used throughout all calculations. They are done with the intent on detailed investigations of unsymmetry effects in this three-limb transformer and their influence on the eddy current losses in particular in the tank wall. The 3D solver uses a nodal formulation of the magnetic vector potential A and an incorporated Coulomb Gauge [5, 6]. Fig. 3 shows the finite element model of the above power transformer. Due to an assumed symmetry of the core and tank wall arrangement, the model includes one quarter of the transformer. Table 2 lists the date of the complete model. Table 2: Data of the transformer model Number of Elements 1225 Number of Nodes 1162 Number of Equations 33 CPU time [s] 876 Due to symmetry and uniqueness, the following boundary conditions are applied with finite element calculations: A Neumann boundary condition B n = is used at the horizontal xy mid-plane. A Dirichlet boundary condition B n = is used at the vertical xz mid-plane. Dirichlet boundary conditions B n = are modelled at all outer boundary planes of the tank wall. In the subsequent analyses, the eddy current carrying tank wall is represented with surface elements as proposed in [1]. With these surface elements, the suggestions mentioned above are introduced additionally. Moreover, an anisotropic permeability of the non-conducting core and anisotropic conductivities of the coil windings are utilized with the finite element model as described in [3, 7].

Fig. 3: Active parts of the finite element model of the power transformer containing core, low and high voltage windings, core clamping plate and steel tank wall (partly shown) Due to the current excitation with impressed current densities and negligible eddy currents inside the coil windings, any anisotropic coil conductivity is given as [ σ ] = σ C (4) with regard to a local cylindrical coordinate system with each of the three limbs. Regarding to Fig. 3, the anisotropic core permeability can be written as [ µ ] = µ xx µ yy. (5) µ zz The different magnetization directions of limbs and yokes are considered with appropriate functions µ xx and µ zz in dependence of the local saturation. The core lamination is represented by an effective permeability obtained from the stacking factor k F as where an averaged value µ F obtained from µ xx and µ zz is used. µ yy = µ µ F µ F (1 k F ) + µ k F, (6) As proposed in [8], the anisotropic nonlinear core permeabilities and the isotropic nonlinear permeability of the core clamping plates are modified by using the averaged magnetic coenergy as given by ( ) 4 µ co B = max H(t) 4 2 T t T /4 H(t) dt B dh. (7) H()

Numerical Results Table 3 shows the eddy current losses in the tank wall obtained from an excitation with opposite rated MMFs within the LV and HV windings including both tapping windings. Thereby, region 1 is along the winding height, region 2 means the height between windings and yoke, regions 3&4 are corresponding to the height of the yoke, regions 5&6 represent the upper chamfer of the tank wall and finally region 7 is above the yoke. Fig. 7 depicts the power loss distribution at the inner tank wall planes with regard to this current excitation. The larger values of the left part of phase 1 and the right part of phase 3 are due to the much greater tank wall planes in these regions. The unsymmetries of the listed values between the regions are caused by the three-limb construction because the finite element mesh is reflected and shifted in the centered regions of the three phases. These calculated values are in good accordance to the total eddy current losses of 19 kw obtained from measurements. Table 3: Eddy Current Losses [W] in Various Tank Wall Regions Phase 1 Phase 2 Phase 3 left right left right left right Region 1 3398 189 11 197 177 342 Region 2 254 17 135 124 128 294 Region 3 12 125 121 118 127 95 Region 4 423 238 234 25 235 397 Region 5 25 142 166 158 142 193 Region 6 95 575 621 614 571 94 Region 7 219 287 297 296 263 23 Summary 4696 2563 2674 2612 2543 4678 Fig. 4 and Fig. 5 show the distribution of the magnetic flux density at the inner tank wall planes for rated current excitation of the HV and LV windings without and with tank wall shielding, respectively. Fig. 6 and Fig. 7 show the corresponding distributions of the eddy current density and the power loss density at the inner tank wall planes. Concluding Remarks With the aim of an improvement of initial design and design optimization calculation methods, the eddy current losses in the steel tank wall of power transformers in the range of 1 MVA to 2 MVA are investigated with 3D finite element analyses. These calculations are carried out with the intent on detailed investigations of unsymmetry effects in the three-limb transformers and their influence on the eddy current losses in particular in the tank wall. The several global and local results will further be used to investigate simplified calculation approaches for an inclusion in the initial design and the design optimization. References [1] Biro O., Bardi I., Preis K., Renhart W., Richter K.R.: A Finite Element Formulation for Eddy Current Carrying Ferromagnetic Thin Sheets. IEEE Transactions on Magnetics, Vol. 33, No. 2, March 1997.

[2] Brauer J.R., Cendes Z.J., Beihoff B.C., Phillips K.P.: Laminated Steel Eddy-Current Loss versus Frequency Computed Using Finite Elements. IEEE Transactions on Magnetics, Vol. 36, No. 4, July 2. [3] Schmidt E., Hamberger P., Seitlinger W.: Calculation of Eddy Current Losses in Metal Parts of Power Transformers. Proceedings of the 15th International Conference on Electrical Machines, ICEM, Brugge (Belgium), 22. [4] Schmidt E., Hamberger P.: 3D Finite Element Analysis of the Winding Support and the Core Clamping System of Power Transformers. Accepted for publication in Proceedings of the 13th International Symposium on High Voltage Engineering, ISH, Delft (Netherlands), 23. [5] Biro O., Preis K.: On the Use of the Magnetic Vector Potential in the Finite Element Analysis of Three-Dimensional Eddy Currents. IEEE Transactions on Magnetics, Vol. 25, No. 4, July 1989. [6] Biro O., Preis K., Richter K.R: Various FEM Formulations for the Calculation of Transient 3D Eddy Currents in Nonlinear Media. IEEE Transactions on Magnetics, Vol. 31, No. 3, July 1995. [7] Schmidt E.: Representation of Laminated and Slotted Configurations in the Finite Element Analysis of Electrical Machines and Transformers. Accepted for publication in Proceedings of the 14th Conference on the Computation of Electromagnetic Fields, COMPUMAG, Saratoga Springs (NY, USA), 23. [8] Biro O., Paoli G., Buchgraber G.: Complex Representation in Nonlinear Time Harmonic Eddy Current Problems. IEEE Transactions on Magnetics, Vol. 34, No. 5, March 1998.

. -.4 -.8-1.2-1.6-2. - 2.4-2.8-3.2-3.6.4.8 1.2 1.6 2. 2.4 2.8 3.2 3.6 4. MagneticFluxDensity FullVector TypeOfData Magnitude Frequency 5Hz Fig. 4: Magnetic flux density at the inner tank wall planes for rated current excitation of the HV and LV windings without shielding. -.4 -.8 -.12 -.16 -.2 -.24 -.28 -.32 -.36.4.8.12.16.2.24.28.32.36.4 MagneticFluxDensity FullVector TypeOfData Magnitude Frequency 5Hz Fig. 5: Magnetic flux density at the inner tank wall planes for rated current excitation of the HV and LV windings with shielding

. - 1.2e+5-2.4e+5-3.6e+5-4.8e+5-6.e+5-7.2e+5-8.4e+5-9.6e+5-1.1e+6 1.2e+5 2.4e+5 3.6e+5 4.8e+5 6.e+5 7.2e+5 8.4e+5 9.6e+5 1.1e+6 1.2e+6 EddyCurrentDensity FullVector TypeOfData Magnitude Frequency 5Hz Fig. 6: Eddy current density at the inner tank wall planes for rated current excitation of the HV and LV windings with shielding 1. - 31.623-1. - 316.228-1. - 3162.3-1. - 31622.8-1.e+5-3.2e+5 31.623 1. 316.228 1. 3162.3 1. 31622.8 1.e+5 3.2e+5 1.e+6 PowerLossDensity TypeOfData Real Frequency 5Hz Fig. 7: Power loss density at the inner tank wall planes for rated current excitation of the HV and LV windings with shielding