Advanced 3D Numerical Methods for Power Transformer Analysis and Design

Transcription

1 Energy Conversion, MEDPOWER 2002, Athens, Greece, November 4-6, 2002 Advanced 3D Numerical Methods for Power Transformer Analysis and Design M Tsili 1, C Dikaiakos 1, A Kladas 1, P Georgilakis 2, A Souflaris 2, C Pitsilis 2, J Bakopoulos 2, D Paparigas 2 * 1 2 Electric Power Division Schneider Electric SA, Department of Electrical & Computer Engineering Elvim Plant National Technical University of Athens PO Box 59 9, Iroon Polytechneiou Street, Athens, Greece Inofyta Viotia, Greece (*) Tel: (+3) , Fax: (+3) , dimitris_paparigas@mailschneiderfr ABSTRACT: Power transformer analysis and design focussing on the equivalent circuit parameter evaluation by magnetic field numerical calculation is presented The proposed method adopts a particular reduced scalar potential formulation enabling 3D magnetostatic problem solution This method, necessitating no source field calculation, in conjunction with a mixed finite element boundary element technique, results in a very efficient 3D numerical model for power transformer design office use Such a methodology is very promising for investigation concerning losses and short circuit voltage variations with the main geometrical parameters Keywords: Boundary element methods, design methodology, finite element methods, leakage field, theory of images, shell type transformers, short circuit voltage I INTRODUCTION Numerical modeling techniques are now-a-days well established for power transformer analysis and enable representation of all important features of these devices [1]- [3] More particularly, techniques based on finite elements present interesting advantages for nonlinear characteristics simulation The leakage inductance evaluation has been extensively analyzed [14], as well as eddy current loss in transformer tank walls [10,11,13], iron lamination characteristics [12,15] and design considerations [9] Moreover, the combination of boundary and finite elements is widely used for electromagnetic problems since the electromagnetic field is not only confined to the conductors but it expands over extensive parts of air, where the use of a boundary element representation can significantly decrease the computational effort [6]-[8] In the present paper, a reduced scalar potential formulation is adopted, necessitating no prior source field calculation This technique is applied for the analysis of the leakage field distribution and the parameters calculation of a 630 kva, 20-15kV/400V three phase shell type power transformer Its results are compared to those of a classical three dimensional finite element method The method is also applied to a simplified magnetostatic problem, approximating the transformer geometry, which is also treated with the theory of images, in order to evaluate its validity Furthermore, the results of the application of a mixed finite element - boundary element method in three dimensional scalar potential problems are discussed and conclusions are drawn over its applicability to power transformer analysis II MAGNETIC FIELD MODELING TECHNIQUES In the followings, formulations used in conjunction with various numerical techniques developed for three dimensional transformer analysis will be considered in detail, namely: the finite element method, techniques based on the theory of images, the boundary element method as well as mixed methods A Finite Element Method (FEM) The active part of the shell type distribution transformer considered (Fig 1) can be modeled by a 3D magnetostatic field analysis Many scalar potential formulations have been developed for 3D magnetostatics, but they usually necessitate a prior source field calculation by using Biot-Savart s Law This presents the drawback of considerable computational effort We have developed a particular scalar potential formulation enabling efficient modeling of laminated iron cores with or without air-gaps and needing no prior source field calculation [1] Fig 1 Active part configuration of the three phase shell type distribution transformer considered According to our method the magnetic field strength H is conveniently partitioned to a rotational and an irrotational part as follows : H=K- Φ (1) where Φ is a scalar potential extended all over the solution domain, while K is a vector quantity (fictitious field distribution) defined in a simply connected subdomain

2 Energy Conversion, MEDPOWER 2002, Athens, Greece, November 4-6, 2002 comprising the windings that satisfies Ampere s law and is perpendicular on the subdomain boundary Iron Core Part Low Voltage Winding High Voltage Winding A typical geometry used for the winding leakage field evaluation is shown in Fig 4a This is similar to a winding part out of the core window It consists of a circular winding with rectangular cross-section near a planar wall of infinitely permeable iron According to the theory of images, the magnetic field in the air (leakage field), can be calculated by considering the image of the coil with respect to air-iron surface, carrying an opposite current of the same magnitude, as shown in Fig 4b This assumption approximates the boundary condition (H t =0) along the boundary between the air and the infinitely permeable iron The magnetic field calculation in such a case is of almost closed form [5] y(mm) Fig 2 Perspective view of one phase transformer part modeled x(mm) Fig 2 shows the perspective view of the one-phase transformer part, comprising the iron core as well as the low and high voltage windings Due to the symmetries of the problem, the solution domain was reduced to one fourth of the device These symmetries were taken into account by the imposition appropriate boundary conditions that is natural Dirichlet boundary condition (Φ=0) along xy-plane and Φ natural Neumann boundary condition ( =0) along the n other outer three faces corresponding to the transformer casing tank walls In such a case the distribution of K is straightforward As an example such a distribution corresponding to the low voltage winding is shown in Fig 3 Such a formulation is well suited for finite element discretization [1,4] Kz μ --> a y(mm) J x(mm) -J y ZLVmax YLVmin YLVmax b Fig 4 Geometry of the simplified problem studied by the theory of images a: circular winding near an infinitely permeable iron wall b: equivalent representation for the field in the air according to the theory of images C Boundary Element Method (BEM) XLVmin XLVmax Fig 3 Fictitious field distribution corresponding to Low Voltage winding B Theory of Images Frequently, in order to evaluate the accuracy of a proposed field analysis method, field calculations for simplified configurations that approximate the transformer geometry can be conducted with the use of the theory of images [4] x The boundary element method is derived through discretization of an integral equation that is mathematically equivalent to the original partial differential equation The boundary integral equation corresponding to Laplace equation is of the form: G( s, s) Φ( s ) c ( s) Φ( s) + Φ( s) G( s, s) ds = 0 (2) n n Γ

3 Energy Conversion, MEDPOWER 2002, Athens, Greece, November 4-6, 2002 where s is the observation point, s' is the boundary Γ coordinate, n' is the unit normal and G the fundamental solution of Laplace equation in free space The re-formulation of the PDE that underlies the BEM consists of an integral equation that is defined on the boundary of the domain and an integral that relates the boundary solution to the solution at the points in the domain Therefore, while in the finite element method an entire domain mesh is required, in the BEM formulation a mesh of the boundary only is required, resulting to the significant reduction of the problem size As the solution domain, in case of transformer simulation, comprises extensive parts of air, the adoption of a boundary element technique to represent the respective subdomains, improves the model performance D Mixed FEM-BEM Method Let us consider a coupled finite element/boundary element solution domain, comprising m FEM nodes, n BEM nodes and r common nodes in the interface boundary The matrix form of equations corresponding to the BEM region can be written as follows: [H] [U] = [G] [Q] (3) where [U] is a vector of the BEM nodal potential values, [Q] Φ(s ) a vector of the BEM nodal values, while H ij and G ij n G correspond to the equation (2) integrals i (s,s) ds n and G i (s,s)ds respectively, [6] Γj Γj 1 r r+1 m m+1 m+l m+l+1 m+l+r m+l+r+1 m+2l+r 1 r r+1 m m+1 m+l Sij Hij m+n+1 m+l+r -Tij -Gij m+l+r+1 m+2l+r Φ1 Φr Φr+1 Φm Φm+1 Φm+n θφr+1 θφm θφm+1 θφm+n (5) where T ij are the terms used to link the finite element region to the boundary element region (involving the potential and normal derivative values of the FEM-BEM interface boundary nodes) [7,8] III RESULTS AND DISCUSSION In order to compare the precision of the proposed reduced scalar potential formulation with respect to the classical one evaluating the source field through Biot-Savart's law, they have been applied in 3D finite element analysis of a transformer under short circuit The case of the one phase part of a 630 kva, 20-15kV/400V three phase shell type power transformer shown in Fig 2 has been considered The magnetic field distribution is shown in Fig 5 while Fig 6 compares the local magnetic field results (along the contour AB shown in Fig 5) obtained by the two models, which are in = F O Hmag (A/m) Classical Proposed A Fig5 Distribution of magnetic induction magnitude during short-circuit test The matrices of the FEM region can be written as follows: [S] [U] = [F] (4) B x (mm) Fig 6 Comparison of magnetic field density magnitude along AB line where [S] is the stiffness matrix and [F] the source vector The global system matrix is of the form:

4 Energy Conversion, MEDPOWER 2002, Athens, Greece, November 4-6, ,04 0,03 0,02 Bz (T) Bz (T) 0,01 0,00-0,01-0,02-0,03 0,04 0,03 0,02 0,01-0,02-0,03 0,00 0,20 0,40 0,60 0,80 1,00 1,20 1,40 1,60 0 a y(m) 0,00 0,20 0,40 0,60 0,80 1,00 1,20 1,40 1,60-0,01 y (m) b Fig 7 Calculated Bz induction component variation along y axis a: Solution using the theory of images b: 3D FEM model very good agreement The 3D finite element method has also been compared to the theory of images technique in the case shown in Fig 4 The variations of Bz component along y-axis calculated by the two methods are shown in Figs 7a and 7b respectively These figures illustrate the good accuracy obtained for the leakage field distribution of a winding in the vicinity of iron by the 3D finite element method with respect to a solution of almost closed form Fig 8 Finite element mesh cross-section showing the winding discretization Table 1 COMPARISON OF MEASURED AND CALCULATED FLUX DENSITIES Distance (m) Measured (mt) Calculated (mt) Error (%) The described model has also been applied for the analysis of the leakage field distribution and short circuit voltage of a 1250 kva, 20 kv/400 V, shell type three phase transformer In a first step, although there is a slight disymmetry of the windings, due to the terminal connections in one side (Fig 1), the problem has been considered symmetric In such a way the solution domain is reduced to one fourth of the device as shown in Fig 9 The mesh adopted for the transformer analysis comprises nodes A cross section of the mesh adopted at the region of the windings is shown in Fig 8 Fig 10 gives the normal flux density of leakage field distribution along the transformer symmetry plane under short circuit conditions in one phase This case is used in order to calculate the short circuit voltage of the transformer as well as the respective forces on the windings The leakage field distribution near the core for the same test is shown in Fig 11 The low field values observed can be explained by the extended paths of the leakage flux in the air Table 1 compares the flux density computed along the plane Iron Core Parts Low Voltage Winding High Voltage Winding Fig 9 Geometry of the transformer considered (only one fourth of the transformer is modeled due to the symmetries)

5 Energy Conversion, MEDPOWER 2002, Athens, Greece, November 4-6, 2002 Fig 10 Normal flux density of leakage field distribution along the transformer symmetry plane (short circuit conditions in one phase) of symmetry of the winding with the distance from the high voltage winding to the measured values by a hall effect probe in case of the short circuit test The precision is good in the vicinity of the windings The measured short circuit voltage deduced by this test is 595% while the value computed by the proposed method is 605% The local force distribution in the high voltage winding under short-circuit is shown in Fig 12 This distribution corresponds to the winding cross-section shown in Fig 11 The paper examined various numerical methods for power transformer analysis and design involving 3D magnetostatics A formulation based on a particular reduced scalar potential technique necessitating no source field calculation, has been adopted A mixed finite element - boundary element method presents important advantages for this class of problems This methodology has been checked in short circuit cases and is very promising for design office applications IV CONCLUSION Fig 12 Distribution of the axial force component in high voltage winding Fig 11 Flux density distribution near the core under short circuit

6 Energy Conversion, MEDPOWER 2002, Athens, Greece, November 4-6, 2002 V REFERENCES [1] "A new scalar potential formulation for 3D magnetostatics necessitating no source field calculation", A Kladas, J Tegopoulos, IEEE Trans on Magnetics, Vol 28, Nr 2, 1992, pp [2] "AI Helps Reduce Transformer Iron Losses", P Georgilakis, N Hatziargyriou, D Paparigas, IEEE Computer Applications in Power, Vol 12, Nr 4, 1999, pp [3] "A Novel Iron Loss Reduction Technique for Distribution Transformers Based on a Combined Genetic Algorithm - Neural Network Approach", PS Georgilakis, ND Doulamis, AD Doulamis, ND Hatziargyriou, SD Kollias, IEEE Trans on Systems, Man, and Cybernetics, Part C, Applications and Reviews, Vol 31, Nr 1, Feb 2001, pp [4] "Leakage Flux and Force Calculation on Power Transformer Windings under short-circuit: 2D and 3D Models Based on the Theory of Images and the Finite Element Method Compared to Measurements", AG Kladas, MP Papadopoulos, JA Tegopoulos, IEEE Trans on Magnetics, Vol 30, Nr 5/2, Sept 1994, pp [5] "The treatment of singularities in calculation of magnetic field by using integral method", ZX Feng, IEEE Trans on Magnetics, Vol 21, Nr 6, Nov 1985, pp [6] The boundary element method for engineers, London, Pentech Press, 1984, pp [7] "Hybrid Finite Element Boundary Element Solutions for three dimensional scalar potential problems", G Meunier, J L Coulomb, S J Salon, L Krahenbul, IEEE Trans on Magnetics, Vol 22, Nr 5, Sept 1986, pp [8] "Applications of the hybrid finite element - boundary element method in electromagnetics", S J Salon, J D Angelo, IEEE Trans on Magnetics, Vol 24, Nr 1, Jan 1988, pp [9] Dikaiakos, C, Mofori, K, Kladas, A, Georgilakis, P, Souflaris, A, Paparigas, D, "Power Transformer Analysis and Design based on mixed Finite Element-Boundary Element Technique» 13th COMPUMAG Conference, Evian, France, July 2001, pp [10] "Comparison of transformer loss prediction from computed and measured flux density distribution", A J Moses, IEEE Trans on Magnetics, Vol 34, Nr 4, 1998, pp [11] "Losses calculation in transformer tie plate using the finite element method", C Lin, C Xiang, Z Yanlu, C Zhingwang, Z Guoqiang, Z Yinhan, IEEE Trans on Magnetics, Vol 34, Nr 5, 1998, pp [12] "Iron lamination efficient representation in power transformers", J Proussalidis, ND Hatziargyriou, AG Kladas, International Journal of Materials Processing Technology, Elsevier, Vol 108 (2), 2001, pp [13] "Three dimensional flux calculation on a three-phase transformer", ΙL Nahas, B Szabados, RD Findlay, M Poloujadoff, S Lee, P Burke, D Perco, IEEE Trans on Power Delivery, Vol 1, Nr 3, 1986, pp [14] "Magnetic field analysis and leakage inductance calculation in current transformer by means of 3D integral method", K Zakrzewski, B Tomczuk, IEEE Trans on Magnetics, Vol 32, Nr 3, 1996, pp [15] "Three phase transformer modeling for fast electromagnetic transient studies", BC Papadias, ND Hatziargyriou, JA Bakopoulos, JM Proussalidis, IEEE Trans on Power Delivery, Vol 9, Nr 2, April 1994, pp VI BIOGRAPHIES Production and Management of Energy from the National Technical University of Athens in 2001, where he follows post-graduate studies His research interests include transformer and electric machine modeling as well as analysis of generating units by renewable energy sources Antonios G Kladas ( kladasel@centralntuagr) was born in Greece, in 1959 He received the Diploma in Electrical Engineering from the Aristotle University of Thessaloniki, Greece in 1982 and the DEA and PhD degrees in 1983 and 1987 respectively from the University of Pierre and Marie Curie (Paris 6), France He served as Associate Assistant in the University of Pierre and Marie Curie from During the period he joined the Public Power Corporation of Greece, where he was engaged in the System Studies Department Since 1996 he joined the Department of Electrical and Computer Engineering of the National Technical University of Athens (NTUA), where he is now Associate Professor His research interests include transformer and electric machine modeling and design as well as analysis of generating units by renewable energy sources and industrial drives Pavlos S Georgilakis (S 1998, M 2001) ( pavlos_georgilakis@mailschneiderfr) was born in Chania, Greece in 1967 He received the Diploma in Electrical and Computer Engineering and the PhD degree from the National Technical University of Athens, Greece in 1990 and 2000, respectively In 1994 he joined Schneider Electric AE, Greece He has worked in the Development and also the Quality Control Departments of the Industrial Division of the company From 1999 to 2001 he was the R&D Manager of Schneider Electric AE At present he is Business and Activity Manager in the Marketing Division He is member of IEEE, CIGRE, and the Technical Chamber of Greece Athanasios T Souflaris ( thanassis_souflaris@mailschneiderfr) was born in Athens, Greece in 1956 He received the Diploma in Electrical Engineering from the Technical University of Pireaus, Greece in 1981 He joined Schneider Electric AE in 1985 as Transformer Design Engineer and from 1988 he is the Transformer Design Manager of Schneider Electric AE Costas P Pitsilis ( costas_pitsilis@mailschneiderfr) was born in Athens, Greece in 1958 He received the Diploma in Electrical and Mechanical Engineering from the Technical University of Greece in 1984 He joined Schneider Electric AE in 1988 as Transformer Design Engineer and from 1997 he is Manager of Technical and Industrial Production Methods of Schneider Electric AE and also he is Manager of several Technical Projects John A Bakopoulos ( yiannis_bakopoulos@mailschneiderfr) was born in Athens, Greece in 1966 He received the Diploma in Electrical and Computer Engineering from the National Technical University of Athens (NTUA), Greece in 1988 He joined Schneider Electric AE in 1989 and has been engaged in different positions such as Technical, Commercial, Development and Projects Departments From 1994 to 1997 he was Quality Assurance Manager and from 1998 to 2001 he was Production Manager of Elvim factory, producing Distribution Transformers, LV and MV panels At present, he is Development and Customer Service Manager in Sales Division Dimitrios G Paparigas ( dimitris_paparigas@mailschneiderfr) was born in Komotini, Greece in 1945 He received the Diploma in Electric Machines and Instruments from the Moscow Energy Institute, USSR in 1972 From 1974 to 1976 he worked in the Engineering Department of Masina Xrisolouris, Greece Since 1976, he has been with Schneider Electric AE (former Elvim), Greece He has been engaged in several positions, such as Quality Control Manager, Production Manager, Transformer Design Manager and Quality Assurance Director At present, he is Technical Division Director and Industrial Division Director Marina A Tsili ( el95136@centralntuagr) was born in Greece, in 1976 She received the Diploma in Electrical and Computer Engineering from the National Technical University of Athens in 2001 where she follows post-graduate studies Her research interests include transformer and electric machine modeling as well as analysis of generating units by renewable energy sources Christos N Dikaiakos was born in Greece, in 1975 He received the Diploma in Electrical and Computer Engineering from the University of Patras in 1999 and the interdisciplinary post-graduate specialization in