HIGH FREQUENCY TRANSFORMER MODEL FOR COMPUTATION OF SECTIONAL WINDING TRANSFER FUNCTIONS USED FOR PARTIAL DISCHARGE LOCALISATION

In this contribution a research work is illustrated that has been carried out to develop a model for the computation of sectional transfer functions of transformer coils in the high frequency range with the aim of partial discharge localisation. The mathematical descriptions of the model in time and frequency domain are given and the results of using standard identification and optimisation methods as well as their disadvantages are discussed. Finally it is shown that genetic algorithms can be used for parameter estimation of the model for achieving satisfactory results. INTRODUCTION Transfer functions (TF) of power transformer at the high frequency range are commonly used for different applications in power engineering such as transient analysis, insulation coordination and design procedures. In recent years TF applications have been extended for diagnostic purposes thus to the detection of turn to turn short circuits or insulation defects [1]. Determination of mechanical deformations of the transformer active parts as well as studies on surge overvoltages along the winding have been carried out. For this reason extensive efforts had been performed to develop an appropriate model for transformer coils accurate enough in high frequency ranges. New investigations have been performed at the ScheringInstitute showing that TF of transformer coils can also be used for the evaluation and localisation of partial discharges (PD) [2]. At the high frequency range the capacitance of the transformer winding becomes increasingly valid. Because the partial discharge pulses have a wide frequency spectrum, while they being transmitted along the windings they are exposed to attenuation, distortion and reflection. Knowledge of the frequency transfer characteristic between measuring point and PD source can allow a localisation of the PD origin and a precise evaluation of the apparent charge. For this purpose the knowledge of the sectional winding transfer functions (SWTF) is required. Because there is no access to the sections of the winding, after the transformer insulation has been impregnated an appropriate modelling of the electrical property of the coil is needed. There are two main ideas for transformer modelling namely the black-box or terminal model and the detailed or internal model. Black-box models are especially useful for the insulation coordination of HV systems, because in this case no knowledge of the internal behaviour of the system is required. With the detailed model the physical geometry and material characteristics of the windings are converted into lumped RLC units in order to simulate the behaviour of the whole transformer winding. In addition to these two modelling ideas a hybrid category for winding modelling may be used which does not consider each turn of the winding as a RLC unit but each coil, thus simplifying the modelling [3]. For PD localisation and evaluation the knowledge about the SWTFs of the winding is essential because the basic idea of this method is calculating measured signals at the terminals to different possible origins along the winding using SWTFs. Direct measurements of SWTFs on a distribution transformer with accessible points along the winding and the use of these SWTFs for the localisation of PDs had shown excellent results [4]. These results are primarily related to the accuracy of the SWTFs which had been measured using a network analyser. The objective of this work is to develop a method to determine the parameters of winding models of actual transformers while accessing only the terminals of the winding. The necessity of such a procedure is provoked by PD measurements on transformer in operation where the coils are already immersed in the insulating liquid and therefore no longer accessible. Although some researchers have posted that an estimation of the winding parameters using common analytical identification methods is impossible without using the sectional winding voltages [5], this contribution shows, that it is possible to overcome this problem by means of new soft computing methods such as genetic algorithms. MODEL DESCRIPTION AND FORMULA For PD evaluation a model is required which describes the physical dimensions of the windings as precise as possible within the interesting frequency range. Therefore the detailed model as shown in Figure 1 has been used for interpretation of the high frequency behaviour of transformer coils. The required number of RLC units to describe the correct frequency behaviour depends on the considered frequency range and the ability to estimate the RLC parameters accurately. For PD localisation and evaluation applications it is usually sufficient to locate the disk unit of the winding where the PD has occurred. Therefore the number of the RLC units has been chosen equivalent to the number of coil sections, thus each winding section is considered as a black-box represented by a RLC unit. The leakage inductances and losses are modelled by L and R. Cs and Cg represent the coil to coil and coil to ground capacitance respectively. Rs represent the losses due to the insulation between adjacent winding sections and Rg shows it for each section to ground. The mutual inductances M between each winding section and the other ones are modelled by a current controlled voltage source as it is depicted in Figure 1. There are different methods to determine the transfer function of the winding. The most common methods are impulse response and frequency sweep analysis. ) ( 0 t i ) ( ) ( 1 1 t v t v N− ) (t N ) ( 1 t i ) (t N Figure 1: Model of a transformer winding grounded through a pure resistive impedance for calculation purposes The former method is performed by a steep impulse voltage applied to the winding as input signal while simultaneously the voltage or current at the other terminal is measured as output. The stimulating frequencies generated with this method are directly related to the steepness of the impulse. If the input impulse involves enough frequency components to excite all desired oscillatory modes of the winding, the frequency behaviour or transfer function can be calculated. The second method for calculating the transfer function works by applying a sinusoidal test signal with variable frequency as input and measuring the amplitude and phase shift of the output signal for different input frequencies. With this approach, which is used e.g. by network analysers, the results are more precise. In order to compute the SWTFs as they are used for PD localisation, the measured TF or any other input-output data of actual winding can be used for estimating the model parameters of Figure 1. Subsequently the SWTFs can be obtained by a computation of sectional voltages caused by an injection of current into different nodes of the model. For this purpose the equations of the model in time and frequency domain are necessary allowing an estimation procedure and the SWTFs calculation. MODEL DESCRIPTION IN TIME DOMAIN For a description of the model in time domain it is more convenient for computer calculations to write the equations in state space. The common state space equations are: