Short Circuit and Open Circuit Natural Frequencies of 3-Φ Transformers: Derived Analytical Expressions and its Applications
Frequency Response Analysis (FRA) method is perhaps the most sensitive tool that can detect even the slightest of winding/core movements. High sensitivity, non-invasiveness, non-destructiveness, and on-site capability are some salient features − making it an ideal monitoring and detection tool. The existence of Standards (IEEE, IEC, and CIGRE) is ample testimony of its global acceptance and superior detection capabilities. The detection principle is based on observing a deviation between two measured FRAs which implies a possible fault. Naturally, the next logical step is to analyze these deviations to determine the type of fault, estimate the extent of damage and its severity, and as a bonus, predict its location, if possible. However, even after three decades, arriving at these inferences is still at the research level. Even though there is a consensus among all the standards on FRA test/measurement procedures, best-suited terminal connections, cable layout, grounding practices, etc., they remain largely silent regarding interpretation and diagnostics. A detailed analysis of literature compiled in Chapter 1 reveals that lack of a mathematical foundation might be one reason for the present plight of FRA. So, developing a generic mathematical-based approach for interpretation and location of incipient mechanical winding damages in actual 3-Φ transformer windings using measured FRA, is imperative. Development of such a generic method necessitates derivation of closedform expressions, which can directly link measured FRA quantities to the electrical parameters of the winding. For assessing damage severity, the challenge is to identify a quantity which is not only extractable from measured FRA, but also be sensitive, monotonic, and traceable to the fault. Driven by this philosophy, this thesis aims to address the following - 1. Propose a unified and general approach to derive closed-form analytical expressions (for each multiphase winding) to link the measured open and short circuit natural frequencies to electrical parameters of the winding and valid for any condition of the neutral 2. Define a quantity calculable from the measured FRA’s peak/trough frequencies which is physically related to mechanical damage in the winding and capable of yielding some physical insight about damage 3. Develop novel methods using the derived analytical expressions to identify an incipient, discrete and localized axial and/or radial displacement in any multiphase winding and applicable for any condition of the neutral In the second chapter, a generic and unified analytical method is developed (applicable to any 1-Φ or 3-Φ winding) starting from the basic mutually coupled lossless ladder network model to derive equations which relate the harmonic sum of squares of short circuit natural frequencies (SCNF) and open circuit natural frequencies (OCNF) to the elemental winding inductances and capacitances. Complete derivation details are discussed, and all the derived formulae were cross-verified by extensive numerical circuit simulations. Each of these derived expressions has a strikingly similar structure and possesses a unique property, viz., the contribution of series capacitances and ground capacitances are decoupled. This important property paves the way for estimating a physical quantity directly responsible for the winding resonances, viz., the effective air-core inductance (Leff). This estimation requires multiple FRA measurements. Chapter 3 presents complete details of the concept, its derivation, measurements, and experimental results for all 1-Φ and 3-Φ windings. Loss of clamping pressure in a winding is not directly identifiable by any means other than an FRA measurement. But, this damage cannot be judged by merely comparing two FRAs. So, a clamping pressure measurement experiment was carried out on a single isolated winding to ascertain the sensitivity and monotonicity afforded by the quantity, Leff, to a change in clamping pressure. Driven by the promising results, author proceeds to build a method based on Leff to find the location of a discrete and localized axial displacement (AD) in any 3-Φ winding configuration. Details of this method, experimental results, and measurement steps are presented in Chapter 4. Proceeding further, Chapter 5 discusses concept of a new method, measurement steps and experimental results to identify presence of a Radial Displacement (RD) in a 3-Φ star winding with neutral-open, as well as, in a delta connected winding. Driven by success, the concept was extended to identify the simultaneous occurrence of a discrete and localized AD and RD in one phase of a 3-Φ star winding, with neutral-open. Preliminary experimental results proved the method could successfully identify faulted phases that contained AD and RD. All experiments reported in the thesis were carried out on transformer windings rated at 33 kV, 3.5 MVA. The results are encouraging, and the author believes that true potential of the proposed methods can be judged when implemented on actual transformers. In summary, this thesis presents, perhaps for the first time, a mathematical basis for identifying and diagnosing axial and radial displacements in 1-Φ and 3-Φ windings using the peak/trough frequency data from the measured FRA. The author believes this is a small step forward in advancing FRA as a diagnostic tool.