High frequency transformer model derived from limited information about the transformer geometry
Introduction
Numerous transformer models have been developed during the past decades, some of which are capable of representing the electromagnetic behaviour of a transformer at high frequencies. Fast front transients related to lightning strikes or switching of vacuum circuit breakers are sources of the high frequency electromagnetic waves, which are studied in the paper. High frequency transformer models should consider resonances which may occur inside the transformer when stimulated by a high frequency overvoltage wave. Unfortunately, these models are often too complex to be of practical use or they require confidential information on a transformer geometry.
High frequency transformer models can be classified into three different groups, depending on the data needed for their construction: Black Box, White Box and Grey Box [1], [2], [3], [4], [5], [6]. Due to the limitations of the Black Box models [5], [6] (they require measurement data which is not always available) and the White Box models [1] (they require a detailed knowledge of the transformer’s design), the Grey Box models have been introduced [2], [3], [4]. Grey Box is the common name for the models which range between the Black Box and the White Box models’ approaches. The aim of the Grey Box models is to obtain a physical and accurate model of transformers from the data which is usually provided by transformer manufacturers such as nameplate data, basic geometry of the transformer window and measurement results. These models can be used both for the calculation of the voltage distribution along the windings of a transformer and for the calculation of the transferred overvoltages between its sides.
There are two different approaches to construct all these models. The first one (Grey Box or White Box) is to construct a network with lumped or distributed parameters values of which are calculated from the geometry of the transformer window and adjusted based on measurement results if necessary [2], [3], [4]. The complexity of these models depends on the level of precision with which the transformer’s inner geometry is taken into account. The second approach (for Black Box) is to construct a model from the frequency response analyser (FRA) measurements, made on the transformer terminals. The model response can be fitted with rational functions [7] or with a generic circuit model the parameters of which represent electromagnetic relations inside the transformer [8], [9], [10], [11], [12]. [13] proposes to use artificial neural network methods to determine the Grey Box model parameters from the FRA measurements.
Recently, in addition to the classical Grey Box modelling approach, some studies were carried out in the area of practical determination of the Black Box transformer model derived from the White Box model [14], [15]. It is possible to directly transform the complex White Box models to the state space equations which describe the transformer at the terminals of interest. By using this approach, the White Box models can easily be used in an EMTP-type software program. Another advantage of this approach is that the transformer manufacturer is able to provide utilities with accurate transformer models without divulging any confidential information related to the transformer’s inner design.
The second section of the paper describes a detailed procedure for the calculation of the frequency dependent nodal admittance matrix of a Grey Box transformer model which is based on finite element method (FEM) calculations and derived from limited information about the transformer geometry. It also presents some approximations based on the concepts of complex permeability and homogenization, which have been made in order to reduce the model size. The inclusion of the model in EMTP-like software using rational approximation, passivity enforcement and state space equations is explained in the third section. Some elements of validation based on lightning impulse test measurements conducted on a 64 MVA, 24/6,8/6,8 kV, YNd11d11 power transformer are given in the fourth. Section five is the conclusions.
Section snippets
Grey Box transformer model principle
In this section a model which can be classified as a Grey Box transformer model is described. It is based on finite element method (FEM) calculations and its parameters are derived from limited information about the transformer geometry.
First the concept of the Grey Box models implemented as segmented lumped RLCG networks is presented. In what follows, the method for deriving the RLCG parameters from the geometry of the transformer window and its windings is described in detail. Both constant
Inclusion of the model in EMTP-RV
To include the calculated frequency dependent nodal admittance matrix of the Grey Box model, Ynodal_reduced(f) inside the EMTP-RV software program, the procedure based on fitting the admittance matrix coefficients using the rational approximation and passivity enforcement is applied, as shown in Fig. 13. Such an approach is widely used when it comes to representing multiple-input, multiple-output systems (MIMO) such as power transformers [35], [36], [37], [38], [39].
The fitting of the
Model’s verification based on lightning impulse tests measurements
This section is devoted to the validation of the developed Grey Box transformer model whose responses are compared with the field measurements made on a 64 MVA, 24/6.8/6.8 kV, YNd11d11 power transformer.
Conclusions
In this paper a state of the art Grey Box transformer model is presented. The model is capable of representing power transformer behaviour in the case when it is energized with a high frequency electromagnetic wave. The parameters of the model are calculated with an electromagnetic field calculation software program from limited information on the transformer’s inner design.
To represent the transformer’s behaviour accurately for a wide frequency range, it is necessary to take into account the
Acknowledgment
The authors express their thanks to Siemens Končar Power Transformers for providing the measurement results which were used for the development and validation of the models presented in this paper.
This work has been supported in part by the Croatian Science Foundation under the project “Development of advanced high voltage systems by application of new information and communication technologies” (DAHVAT).
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