Upper Frequency Bound on Circuit-Based Models for Transformer Windings

Abstract

The power generation, transmission and utilisation are necessarily being carried out at different voltage levels, and require transformers for performing the voltage level conversions. As a result, transformers form one of the most critical elements of the power system. Incidentally, they are also one of the costliest equipments in any electric power stations, with cost ranging up to millions of dollars. Their repair work also proves to be quite expensive and time consuming. Moreover, the revenue loss due to the consequential line outages can be intolerable. The electrical insulation in the transformers age under electrical, thermal, mechanical and synergy of these stresses. The electrical stresses are due to the continuous operating voltage, temporary overvoltages and the transient overvoltages. Classically, the surges generated by switching operation and natural lightning formed mainly the transient overvoltages. An adequate design of the transformer insulation requires a detailed knowledge on the electrical stress distribution all along the winding. Unlike that in simple airgaps found in the transmission lines, the transformer winding complicates the stress distribution by modulating its spatio-temporal distribution. This necessitated a detailed modelling of the winding, well beyond the normal two-port network model employed in power system studies. Both distributed and ladder network models have been proposed in the earlier literature to accurately depict the response of the simplified winding models for fast rising lightning and switching surges. Depending on the adopted model, varieties of theoretical approaches ranging from travelling and standing wave theory-based approaches to finite-difference-equation based approaches, have been proposed. With the advent of digital computers, ladder network models assumed priority and non-uniform winding could be modelled. There was also another experimental based approach in which the frequency or time domain response of the winding at its terminals (and taps if made available) were measured and various system identi fication approaches were attempted to either describe the terminal response for different surges or use it for possible identification of the physical (geometric) changes in the winding structure. However, as this approach cannot be employed for the winding that are yet to be fabricated and further cannot provide any insight into various interior stresses, they will not be considered hereafter. With the increase in power rating of the transformers, the size of the winding also became bigger. Then the adequacy of the above said modelling approaches for analysing the stress under the chopped lightning impulse was questioned. Meanwhile, the propagation of the Partial Discharge (PD) pulse, which can have rise time of the order of few nanoseconds, could not be e ectively analysed by the classical approaches. With the advent of Gas Insulated Substation (GIS), another overvoltage called the Very Fast Transient Overvoltage (VFTO), caused by the operation of mainly the disconnector switches became a matter of concern. These overvoltages have frequency spectrum ranging up to tens to hundreds of MHz. The higher frequency content of the above said entities have led to serious concern over the validity of the circuit-based modelling. To overcome the problem, transmission line modelling for the turns/coils of the winding were proposed and commonly employed. In this approach, both Single Transmission Line model (STL) and Multi-Conductor Transmission Line Model (MTL) were adopted to evaluate the surge distribution along the winding. The same was also employed for the modelling of the propagation of PD pulses. However, the transmission line modelling requires the existence of Transverse Electro Magnetic (TEM) mode of wave propagation, which is rather di cult to realise for the initial critical part of VFTO and for the entire PD waveforms. Incidentally, the laboratory validation provided in some of the literature were plagued by the electromagnetic scaling issues, which render the validation provided quite inadequate. In other words, it has become highly essential to trace the underlying dynamic electromagnetic fields, rather than resorting to convenient simplified modelling approaches. The present work was taken up to address this basic problem. Its scope is identified as: (i) Find a suitable numerical electromagnetic field calculation approach for the problem in hand, and (ii) Noting that the circuit-based modelling is the language of electrical engineers, provide an upper frequency bound to such modelling approaches for the transformer windings. Simplifications which are routinely made in evaluating the surge response of the windings like neglecting role of bushing, tank and other phases, are also made in this work. At the same time, it is worth noting here that the present work can be considered as a first step in finding the full-wave response of windings