1. PROGRAM FOR COMPUTATION OF OPERATION PARAMETERS OF LARGE TRANSFORMERS Ivo DOLEŽEL CZECH TECHNICAL UNIVERSITY, PRAHA, CZECH REPUBLIC Pavel KARBAN UNIVERSITY OF WEST BOHEMIA, PLZEŇ, CZECH REPUBLIC

2. ABSTRACT About fifteen years ago Prof. Doležel started developing a relatively large program package for solution of a wide spectrum of tasks associated with operation of large power transformers. The program ordered by one of the principal Czech manufacturers – ŠKODA Plzeň – was written in FORTRAN 77 for environment MS-DOS supplemented with memory extender DBOS and contained the following modules: Solution of the 2D magnetic field distribution in the window. It could be calculated either analytically (provided that the window contains no ferromagnetic subdomains such as shunts) or numerically, using the finite element method. Determination of radial and axial forces in particular windings, their self- and mutual inductances and short-circuits voltage. Determination of additional losses in the windings. Building of the equivalent circuit of the transformer in the form of a complete polygon (that respects both resistances and inductances of particular windings). Investigation of steady-state operation regimes using the equivalent circuit. Solution of the 2D electric field distribution in the window. FEM solution of the time- dependent 2D magnetic field distribution in the magnetic cores (eventual transversal magnetic flux is neglected) and corresponding magnetization losses. Computation of the time-dependent magnetization currents in particular phases.

3. ABSTRACT But even when the results mostly very well corresponded with measured values and the package was successfully used for more than 10 years, it was not able to carry out some fine computations due to the maximum permissible number of nodes in discretization mesh (under DOS this number was only several thousand). That is why were asked to completely rewrite the code for the WINDOWS environment, improve (as far as it is possible) the algorithms and build in it some more modules containing: Computation of all transversal capacitances between particular windings and between the individual windings and tank. Computation of the longitudinal capacitances of the windings. The time dependence of distribution of voltages and currents in the windings for a step voltage or voltage wave brought to one of them (for specified initial conditions). Export of results to special html sheets suitable for handling via Internet. The paper presents and describes the basic structure of the program written in Borland Delphi, algorithms used for the computations, way of visualisation of the results and possibilities of their storing and next processing. Particular steps of the solution are illustrated on a typical example.

4. STRUCTURE OF THE PROGRAM - PREPROCESSOR The program written in Borland Delphi consists of three basic modules. Preprocessor containing input of the complete set of geometrical data (dimensions of the cores, window, coils and other elements such as shunts, information about particular coils (arrangements, structures and numbers of partial turns in the coils, voltages of currents), relevant material data (for example permittivities or permeabilities). Main window of the preprocessor

5. STRUCTURE OF THE PROGRAM - PREPROCESSOR Properties of the transformer The sheet for input the corresponding coil data

6. STRUCTURE OF THE PROGRAM PROCESSOR AND POSTPROCESSOR Processor that allows (March 2005) assembling data for generation of the mesh, computing magnetic field and electric field, determining distribution of forces and total losses in particular coils (the total losses consist of the basic and additional losses), determining parameters of the equivalent circuit with respecting resistances of coils, evaluating (from the viewpoints of delivered power and voltage drops) any operation regime defined by connection of terminals of particular coils and values of loads, Postprocessor that permits visualising distribution of various quantities (coloured maps of distributions of magnetic vector potential and magnetic flux density in magnetic fields, electric potential and electric field strength in electric fields, distribution of losses and forces in particular coils and writes into files other integral quantities such as total energy of magnetic and electric fields, inductances, capacitances or short-circuit voltage), exporting all calculated data in the form of tables into Excel and MatLab for the purpose of their eventual further processing, exporting the input and output data to special sheets for their further handling via Internet.

7. STRUCTURE OF THE PROGRAM Main window of the postprocessor Processor list of computations

8. STRUCTURE OF THE PROGRAM Distribution of the additional losses in a winding Distribution of magnetic field along axis x

9. DESCRIPTION OF THE ALGORITHMS USED FOR COMPUTATIONS Magnetic field Magnetic field in the window is supposed to be described by equation (in 2D cylindrical coordinate system) for magnetic vector potential. This equation reads (the vector potential as well as current density in the windings has only one nonzero component in the circumferential equation that are denoted as A  and J , respectively) The boundary condition along the window are of the Neumann type, so that it is necessary to prescribe its value at an arbitrary point. While equation on the rectangular area of the window can be solved analytically (the results are given by an infinite sum of the Bessel, Struve and other higher functions), equation must be solved numerically, using the finite element techniques. Electric field Distribution of electric field is described by the Laplace equation for scalar potential in the form with the Dirichlet boundary condition along the window that sounds (the magnetic circuit is supposed to be grounded). It is, moreover, necessary to describe distribution of voltage along the present windings that depends on the operation regime solved and input the relative permittivity of all elements. Equation must be solved numerically.

10. DESCRIPTION OF THE ALGORITHMS USED FOR COMPUTATIONS Forces and losses in the windings The force acting on a turn of given dimensions that carries current in magnetic field is given by integral which in components gives Here denotes the volume of the turn. As the skin-effect at industrial frequencies is rather low, computations of forces are performed on the assumption that is constant over its cross-section (this, however, is not the case of the additional losses!!). For a ring of rectangular cross- section we obtain On the other hand, the losses the same turn is calculated from relation where denotes the electrical conductivity. Now, of course, the nonuniform distribution of the current density must be respected, which is realized using (relative complicated, but very accurate) analytical expressions. In fact, computations are carried out not for the turns, but for partial conductors in them (which is more accurate).

11. EXPORT OF RESULTS A typical sheet with basic information about the transformer

12. CONCLUSION The results are automatically stored in corresponding files. But, at request, they may be exported to Excel or Matlab for further numerical processing. Moreover, they can be printed (in common with various input data) into special sheets (for the purpose of Internet handling, presentations etc.). The program is in the stage of permanent development. The results of each module were carefully tested by comparison with experiments or values calculated by some other method. Nowadays, a new module is being completed containing computations of the time-dependent distribution of voltages and currents along the windings when one of them carries a wave of given shape. This necessitates additional calculation of the transversal and longitudinal capacitances of all windings, evaluation of their conductances and solution of a system of partial differential equations providing the required distribution.