In this paper, resonance effects in transformer windings are thoroughly investigated and analyzed. The resonance isdeterminedbymakinguse of anaccurate approachbasedonthe applicationofthe impedance matrix of a transformer winding. The method is validated by a test coil and the numerical results are verified by an ATP-EMTP model. Further analysis is applied on a transformer winding for which the inductance and the capacitance matrix as well as the winding losses are previously determined. By having determined the amplification factor, it can be found the location where the most severe transients may occur. It is also shown that maximum resonance overvoltage depends on the duration of the excitation and its resonance frequency.

Introduction

Transformers are important devices which are inevitable for the existence and the operation of power systems. The study of transient behavior of transformer voltages and currents is important for transformer designers and system planners, in order to know the interaction between the transformer and the system during different disturbances. Transformers may normally possess more resonance (natural) frequencies, which exist because transformer windings and coils can be seen as a number of series inductances and shunt capacitances. When a transformer is excited by a voltage that oscillates with a frequency equal to some of the resonance frequencies, a resonance occurs. During this process, the total winding impedance is determinedby the copper losses ofthe transformer winding.Hence, the resonance is a phenomenon in which the terminal transformer impedance is fully resistive and the imaginary impedance part is equal to zero. In this case, the total impedance becomes either minimum (series resonance) or maximum (parallel or anti-resonance). In case of a series resonance, the transformer is exposed to high overvoltages voltages, and the voltage distribution in the winding is non-linear since winding capacitances cannot be ignored. The evaluation of this distribution is important in order to know, which of the windings experience the highest stresses and under which conditions;lightning or switching. One important parameter that provides insight about voltage amplitudes along the winding is the amplification factor. This parameter was studied in Ref. [1]. During non-standard waves, resonance overvoltages may take different values. The analysis is performed to a single transformer even though the procedure is valid for multi-transformer windings as long as the impedance matrices, the elements of which are frequency dependent, are accurately determined. Nowadays different types of models are applied to study transformer transients. The powerful vector fitting model, which is very accurate belongs to the group of blackbox modeling [2]. Its application depends on the measured admittance matrices within broad frequency range. A model based on 2 port network representation by making use of a Frequency Response Analysis (FRA) is another example of an efficient black box approach [3]. Another type of models are white box models. These are numerical models that make use of inductance-, capacitance- and resistance matrices. The advantage of the white box models is that transient analysis is performed within broad frequency range. However, the disadvantage is that the accuracy strongly depends on the accuracy of computed parameters, particularly inductance and capacitance matrix as well as losses, which are frequency dependent [4–۶]. Finally, the last type of models are gray box models. These models are built in EMTP-based software packages, and some transformer parameters previously determined by white box model can be tuned to the measured values (black-box). In this way, inaccuracies of white box model can be eliminated.