Electric Power Systems Research

Electric Power Systems Research 78 (2008) 84 88 Contents lists available at ScienceDirect Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr 3D finite-element determination of stray losses in power transformer Livio Susnjic a,, Zijad Haznadar b, Zvonimir Valkovic c a Faculty of Engineering, Vukovarska 58, 5000 Rijeka, Croatia b Faculty of Electrical Engineering and Computing, Unska 3, 0000 Zagreb, Croatia c Polytechnic of Zagreb, Konavoska 2, 0000 Zagreb, Croatia article info abstract Article history: Received 6 March 2007 Received in revised form 3 August 2007 Accepted 0 March 2008 Available online 22 April 2008 Keywords: Stray losses Finite-element analyses Eddy currents Power transformer This paper presents a three-dimensional (3D) finite-element (FE) analysis of eddy current losses generated in the tank walls and yoke clamps of a three-phase 40 MV A power transformer. The time harmonic FE model is used to compute the magnetic leakage field in the case of a short circuit condition of the power transformer. Three cases are analyzed to study the impact of modeling tank walls and yoke clamp plates in FE context in estimation of their losses. The load loss test was carried out on an experimental transformer to validate the simulation. 2008 Elsevier B.V. All rights reserved.. Introduction Accurate predictions of stray losses of a power transformer and their reduction mechanisms are necessary for improving transformer design. The stray losses arise from eddy current and hysteresis effects inside the yoke clamps and tank walls. The main portion of stray losses in carbon steel plates are the eddy current losses [], whereas hysteresis losses comprise between 25 and 30% of total stray losses. Several authors have considered the stray losses of the power transformer, not taking into account hysteresis losses [2,3]. To determine stray losses, a full three-dimensional (3D) finite-element (FE) analysis of the whole structure is required. 3D geometry discretization of a power transformer requires a huge number of nodes and elements because overall transformer dimensions are measured in meters and there exist regions such as tank walls and yoke clamps where the penetration depth of eddy current is measured in millimeters. In order to reduce the number of nodes and elements and to avoid using very demanding computational resources, the computation of eddy current losses by 3D FE is made using a different model of the tank and yoke clamps in the context of the finite element [4 6]. The objective of this research is to investigate the impact of different modeling for the tank walls and yoke clamps on the computed eddy current losses. Corresponding author at: Faculty of Engineering, Department of Electrical Engineering, Vukovarska 58, 5000 Rijeka, Croatia. Tel.: +385 5 65435; fax: +385 5 6546. E-mail address: livio.susnjic@riteh.hr (L. Susnjic). This paper summarizes previously works [7,8] and in addition gives results for simulation by use of the non-linear surface impedance method. First, yoke clamp plates and unshielded tank walls are modeled with skin depth independent shell elements, rather than with the linear surface impedance method, and in the end with the non-linear surface impedance method. The electromagnetic leakage field based on magnetic scalar potential has been calculated for a transformer short circuit condition with a rated current. The magnetic non-linearity of the transformer core material is considered. The influence of regulating coil tapping position on computed losses is also presented. A commercial FE software (Flux 3D V9.3 [9]) was used to perform the simulation shown in this paper. The computed results are discussed and compared with experimental ones. 2. Numerical analysis The transformer FE model is shown in Fig., and the relevant data are given in Table. Fig. 2 shows a sketch of the transformer cross-section. The tank walls and clamps thickness are 0 mm and 25 mm, respectively. B H data of carbon steel material used for the tank and clamps are given in Table 2. The rated ampereturns are prescribed for the appropriate coils of each phase. The ampere-turns balanced equation corresponding to the short circuit condition, in phasor form is Ī H (N H + N R ) + Ī L N L = 0 () Balance of the ampere-turns can be assumed for the coils wound on the same leg. With the exception of the coils region, 0378-7796/$ see front matter 2008 Elsevier B.V. All rights reserved. doi:0.06/j.epsr.2008.03.009

L. Susnjic et al. / Electric Power Systems Research 78 (2008) 84 88 85 Table 2 B H data of carbon steel material used for the tank and clamps H (A/m) B (T) H (ka/m) B (T) 6 0.05.0.250 33 0..26.350 84 0.238.59.436 9 0.324 2.0.504 29 0.527 2.52.55 288 0.642 3.97.63 483 0.89 6.25.720 69.0 7.82.763 788.3 9.80.8 2.. Skin depth independent shell elements Table Transformer data Fig.. The transformer FE model (tank is not shown). Symbol Quantity Value S Rated power 40 MV A f Frequency 50 Hz V H/V L Rated voltages 0 ± 5%/2 kv I H/I L Rated currents 209.9/00 A N L/N H/N R Number of turns 52/677/20+20 Clamp plate and tank conductivity 5 0 6 S/m the sub-regions of the calculation domain are defined with the total scalar magnetic potential formulation. Reduced potential described the coils region. The calculation of the magnetic field from the Biot-Savart s law allows for the exclusion of the coils from the finite element mesh. The transformer is reconectable on the high voltage (HV) side. The current in the coils and the number of turns corresponding to a tapping position are given in Table 3. Estimation methods for computing the eddy current losses are briefly presented as follows. The thin steel plates can be modeled by means of surface region, given frequency, permeability and conductivity of the material. Shell elements independent in terms of skin depth have been used for calculating eddy current losses. The tangential component of the magnetic field in a thin plate of thickness e through the depth of the plate in the z-direction is described analytically by the following expression: [ ( ae ) ( ae )] H t (z) = H t sh sh(ae) 2 + az + H 2t sh 2 az (2) where a =(+j)/ı, H t and H 2t are the field values on both sides of the plate and ı is the skin depth.the volume current density variation has a tangential component only, and is described by J(z) = a sh(ae) [ H t sh ( ae 2 + az ) H 2t sh ( ae 2 az )] Eddy current loss per surface unit in the plate is P = e/2 e/2 J(z) 2 dz (4) 2 where is material conductivity. 2.2. Linear surface impedance method (3) Surface impedance links the component of the magnetic field H tangential on the thin steel surface to the tangential component of the electric field E: nxe = Z s nx( nxh) (5) For linear material it is a ratio of the tangential electric field E s and the tangential magnetic field H s : Z s = E s = + j H s ı The surface current density is defined as (6) K = nxh s (7) The steel plate power loss density (surface density) in W/m 2 is given by P = 0.5Re(Z s ) Hs 2 (8) Table 3 Coils data Tapping position Turns number (LV/HV/RC) Current (A) (LV/HV/RC) Fig. 2. The transformer cross-section. 5% 52/677/0 00/247./0 0 52/677/20 00/209.9/209.9 +5% 52/677/20 + 20 00/82.4/82.4

86 L. Susnjic et al. / Electric Power Systems Research 78 (2008) 84 88 2.3. Non-linear surface impedance method According to [5], the surface impedance for non-linear material (permeability depends on magnetic flux density) over a large range of fields (from low to high fields), and is given by the following formula: Z s = k w (H s )Z sl + ( k w (H s ))Z snl (9) where Z sl and Z snl are the surface impedances for the linear and non-linear material, respectively, and k w is the weighting function. Weighting function k w (H s ) is: k w (H s ) = (0) + k(h s /H k ) H k corresponds to the value of the magnetic field at the knee of the B H curve, and k is the coefficient to be chosen. The magnetic field reaches the tank wall and clamp plates in a mostly normal direction. In this case, the electric field is mostly tangential, remaining unchanged through an interface and is assumed to be mostly sinusoidal. The value of coefficient k equals in this case of sinusoidal electric field [5]. The steel plate power loss density in W/m 2 is calculated by (8). 3. Results Calculations on a three-phase, three-limb transformer rated at 40 MV A and 0/2 kv have been performed. The computed results by skin depth independent shell elements and by linear surface impedance method are shown in Figs. 3 and 4, respectively. These results correspond to the regulating coil tapping position 0. Figs. 3a and 4a shows the relative tank permeability ( rt ) depen- Fig. 4. Permeability dependence of the: (a) tank loss value for both fixed ( rcl =500) and varied relative permeability of the yoke clamps and (b) yoke clamps loss value forbothfixed( rt = 500) and varied relative permeability of the tank (linear surface dence of the tank loss values for both parametrically given clamps permeability ( rcl = 500) and varied relative permeability of the yoke clamps ( rcl = rt ). Figs. 3b and 4b shows the relative clamps permeability ( rcl ) dependence of yoke clamps loss values for both parametrically given ( rt = 500) and varied relative permeability of the tank ( rt = rcl ). The variation of the relative permeability for the tank and yoke clamps is simultaneous in the range from 00 to 000, with an incremental value of 00. The eddy current losses computed by modeling tank walls and yoke clamps with skin depth independent shell elements and the linear surface impedance method are in close agreement. From Figs. 3 and 4 it is obvious that the losses depend on the chosen steel permeability. The clamp plate s loss depends on the permeability of the plate as well as on the permeability of the tank. It has been shown that a higher prescribed permeability of the tank results in a reduced leakage field in the clamp plate area, and as a consequence reduced yoke clamps losses. A similar conclusion applies to the tank loss. The skin depths variation for linear analyses is from 3.8 mm to mm, for the chosen relative permeability of the steel from 00 to 000. The eddy current losses computed by simulation with the non-linear surface impedance method, for different regulating coil tapping positions, are given in Table 4. It could be seen that the losses obtained with non-linear surface impedance analyses are at Table 4 Computed losses Fig. 3. Permeability dependence of the: (a) tank loss value for both fixed ( rcl =500) and varied relative permeability of the yoke clamps and (b) yoke clamps loss value forbothfixed( rt = 500) and varied relative permeability of the tank (skin depth independent shell elements). Tapping position Clamp plates (W) Tank (W) Total losses (W) 5% 3808 0,700 4,508 0 5522 9,680 25,202 +5% 7930 36,476 44,406

L. Susnjic et al. / Electric Power Systems Research 78 (2008) 84 88 87 Fig. 5. Distribution of eddy current density on the tank wall (non-linear surface Fig. 7. Power loss distribution on the tank inner surfaces (non-linear surface Fig. 8. Distribution of magnetic induction (leakage field) on symmetry plain in the coils region (max. value is 0.22 T). Fig. 6. Surface power density on the clamp plate (non-linear surface impedance method). least 30% higher than those with linear surface impedance or independent skin depth shell elements. Fig. 5 shows the distribution of the surface eddy current density on the tank wall. Distribution of power loss density on the clamp plate surface is shown in Fig. 6. Power loss density distribution on the inner tank surface is shown in Fig. 7. Distribution of the magnetic induction or leakage field on the symmetry plain outside the core (in the coils regions) is shown in Fig. 8. Maximum value of the leakage field is 0.22 T. 4. Experimental validation According to IEEE Std. C57.2.90, power transformer load losses should be measured at a load current equal to the rated current Fig. 9. The transformer during its manufacturing (Končar Power Transformers Ltd.).

88 L. Susnjic et al. / Electric Power Systems Research 78 (2008) 84 88 Table 5 Discrepancy between computed eddy current and measured stray losses Tapping position Total losses I 2 R Winding eddy current losses Tank and clamps stray losses Tank and clamps eddy current losses Discrepancy eddy current vs. stray losses 5% 209,500 69,500 9,200 20,800 4,508 30.2% 0 20,00 55,600 2,00 33,400 25,202 24.5% +5% 228,900 46,200 27,600 55,00 44,406 9.4% for the corresponding regulating coils tapping position. The load loss test is accomplished by short-circuiting the secondary winding and applying a reduced voltage to the primary winding, i.e. the voltage necessary to cause a rated load current to flow. A 40 MV A transformer was used to investigate stray losses (Fig. 9). The power absorbed in the short-circuit test consists of the I 2 R and eddy current losses in the winding, and stray losses in constructive steel parts. Stray losses in the constructive steel parts are obtained by subtracting I 2 R losses and eddy current losses in the winding from the power obtained in the load test [0]: P stray = P load P i 2 R P ec () where P load is the load losses (W); P i 2 R the I2 R losses in winding (W) and P ec the winding eddy current losses (W). The stray losses obtained by () are treated as measured losses. The winding eddy current losses are calculated analytically by known distribution of the magnetic leakage field, calculated previously. The magnetic leakage field inside windings are calculated as a 2D axisymetric field []. The method described in Ref. [2] is used to estimate winding eddy current losses. The winding eddy current losses depend on the tapping position of the regulating coils, e.g. fora0tapping position there are 2. kw. Table 5 shows the measurement values of total load losses and I 2 R in windings, calculated winding eddy current losses, stray losses and calculated eddy current losses in the tank walls and clamp plates. Also, stray losses for different tapping positions are compared with the eddy current losses computed by modeling clamp plates and tank walls with the non-linear surface impedance method. The comparison between the computed and the experimental results shows discrepancy. The discrepancy results due to approximation in the modeling clamp plate s geometry and in not taking hysteresis losses into account. For the three methods mentioned above approximation in modeling clamp plates geometry is needed, so brackets (elements for tight the coils) are not included. Made of carbon steel, brackets are high in permeability and liable to invite leakage flux concentration causing eddy current losses. 5. Conclusion In this paper, power transformer eddy current losses in the yoke clamps and unshielded tank walls are computed by 3D FE analyses. The permeability of the tank and clamps has a significant influence on eddy current losses. The losses obtained with non-linear surface impedance analyses are at least 30% higher than those with linear surface impedance or independent skin depth shell elements. Stray losses are a function of many factors including the physical geometry of the cores and coils, the voltage class of the transformer, and the material used in the tank and clamps construction. The computed values of losses by 3D FE analysis do not match closely the test values. The comparison between the computed and the experimental results shows the discrepancy which arise due to approximation in the modeling clamp plate s geometry (where the brackets were not included) and in not taking hysteresis losses into account. The tapping position is shown to have a strong influence on the obtained eddy current losses values. References [] N. Takahashi, Z. Cheng, Team Problem 2, in: Team-Based Benchmark Family (Problem 2/2 a /2 b /2 c ), R&D Center, Reading Tianwei Group Co. LTD., 2005. [2] J.C. Olivares, et al., 2D finite-element determination of tank wall losses in pad transformers Electric Power Research, vol. 7, ELSEVIER, 2004, pp. 79 85. [3] M. Rizzo, et al., Influence of flux collectors on stray losses in transformers, IEEE Trans. Magn. 36 (2000) 95 98. [4] C. Guerin, et al., 3D Eddy current losses calculation in transformer tanks using the finite element method, IEEE Trans. Magn. 29 (993) 49 422. [5] C. Guerin, et al., Surface impedance for 3D non-linear Eddy current problems application to loss computation in transformers, IEEE Trans. Magn. 32 (996) 808 8. [6] C. Guerin, et al., Shell element for computing 3D Eddy currents application to transformers, IEEE Trans. Magn. 3 (995) 360 363. [7] L. Susnjic, et al., Electromagnetic analysis applied to the prediction of stray losses in power transformer, ICEM 2004, Book of Digests, Krakow, 2004, pp. 907 908. [8] L. Susnjic et al., Prediction of Stray Losses in Power Transformer, ISEF 2005, International Symposium on Electromagnetic Fields in Mechatronics, Proceedings, Baiona, CFSA-2.2., 2005. [9] FLUX V.9.3, CAD package for electromagnetic and thermal analysis using finite elements, CEDRAT, 2006. [0] J.C. Olivares, et al., Experimental study to reduce the distribution-transformers stray losses using electromagnetic shields, Electr. Power Syst. Res. 63 (2002) 7. [] M. Christoffel, A. Kuster, Methode zur Berechnung von Kurzschlusskraeften in Transformatorwicklungen mit Hilfe von digitalen Rechenmaschinen, Brown Boveri Mitt. 47 (960) 32 328. [2] W. Dietrich, Berechnung der Wirkverluste von Transformatorenwicklungen unter Beruecksichtigung des tatsaechlichen Streufeldverlaufes, Arch. Elektrotech. 46 (96) 209 222.