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1 11-E-TRN-1315 Determining Intensity of Radial Deformation and Axial Displacement of Transformer Winding Using Angular Proximity Index K.Pourhossein Tabriz Branch, Islamic Azad University, Tabriz, Iran G.B.Gharehpetian Electrical Engineering Department, Amirkabir University of Technology, Tehran, Iran 1 E.Rahimpour ABB AG, Power Products Division, Transformers, R&D Department, Bad Honnef, Germany Keywords: Power transformer, Transfer function (TF), Radial deformation (RD), Axial displacement (AD), Vector space model, Arc-cosine similarity measure (ACSM) Abstract Transfer function method is a widely-used technique to detect mechanical defects of transformer winding. Although TF is emerging as a powerful diagnostics technique, there is still no general guideline for interpreting measured TFs. The present paper is an attempt to diagnose intensity of radial deformation (RD) and axial displacement (AD) of transformer winding through a vector space perspective, using transformer s transfer function. To determine the intensity of winding deformation or displacement, the angular proximity index is used. The measurement results show the effectiveness of this index. The angular proximity index constructs a nearly linear mapping between any TF curve and its defect intensity. The results are verified using experiments and compared with the other well-known index, i.e., correlation coefficient. 1. INTRODUCTION Power transformers are of the most important elements in power systems and their sudden outage is a very costly event [1]. The monitoring and diagnostic techniques, which can evaluate the integrity of transformer, are essential to increase reliability of the equipment. When a transformer is subjected to high fault currents, the mechanical structure and windings are subjected to intense mechanical stresses causing winding axial displacement and radial deformation (Fig.1). The displacement may also occur due to loss of clamping pressure. Winding displacements and deformations in power transformers are difficult to be detected by conventional methods of diagnostic tests. They result in relative changes to the internal inductances and capacitances of the winding. These changes can be detected externally by Frequency Response Analysis (FRA) method [1]. FRA method, sometimes called Transfer Function (TF) method, is a powerful diagnostic test technique. It is based on measuring the TF of power transformer windings over a wide range of frequencies and comparing the results of these measurements with a reference set.

2 Differences may indicate damage to the transformer, which can be investigated further using other techniques or by an internal examination [2]. The comparison of results is usually made by plotting a graph of the amplitude of the TF against frequency for sets of measurements. An experienced observer then examines the two curves for any significant differences. The main problem with this method of comparison is that the experts opinion may lack both transparency and objectivity [2]. In recent years, several works have been developed for the evaluation of transformer TFs. A good survey in this topic has been presented in [2], where merits and demerits of major works on TF interpretation have been discussed [3]. In this paper a vector-based approach has been utilized to diagnose intensity of RD and AD of power transformer winding. Here, the transformer TF is treated as a vector in a highdimensional vector space. The angle between an unknown TF vector of the transformer and its intact TF vector is used for intensity determination of RD and AD of transformer winding. 2. Test Objects and Measurements The test object for RD studies is a high voltage winding with 30 double inverted disks, 11 turns in each disk and a one-layer low voltage winding with 23 turns. The deformation has been performed on the double disk winding in four stages, as shown in Fig.2-a and described in Table 1. A high voltage winding with 31 double inverted disks, 6 turns in each disk, and a four-layer concentric low voltage winding, 99 turns in each layer are used as a test object to study AD of the low voltage winding. These windings have been manufactured for special experiments and have the construction of transformer windings with a rated voltage of approximately 10 kv. This special prototype can be axially moved in 2 cm steps. The movement is the shift of the internal layer winding with respect to the outer winding as shown in Fig.2-b. The AD has been done in four stages. Each stage is equal to 2 cm axial 2 movement. The test object is 83 cm high, so a 2 cm axial displacement is equal to 2.4 % of its total height. Fig.1: Radial deformation and axial displacement in transformer winding Intensity of RD Stage 1 Stage 2 Stage 3 Stage 4 Table 1: Different intensities of RD Description (Deformation is about 7% of disc radius.) deformed on one side. deformed on two opposite sides. deformed on three sides with 90º shift with respect to each another. deformed on four sides with 90º shift with respect to each another.

3 (a) RD (b) AD Fig.2: Test objects for RD and AD Fig.3 shows the terminal connection for the analysis of RD and AD. The measured TF is defined, as follows: (1) Figs 4 and 5 show the measured TFs related to RD and AD, respectively. All measured TFs show the high frequency behavior of test objects with several dominant resonances in the studied frequency range. Before comparing two TFs, they should be transformed into more compact and computationally appropriate form. 3. RD and AD Intensity Determination Vector spaces are commonly used for information organization and retrieval purposes [4]. Vector space model is an algebraic model for representing objects, as vectors of identifiers. Using vector space model, any TF Fig.4: Measured TFs related to RD U HV LV IN Fig.3: Terminal connection for TF measurements on test objects Fig.5: Measured TFs related to AD curve can be represented as a vector in an appropriate vector space. Then, any TF curve can be efficiently demonstrated by its amplitude in an appropriate number of frequency points. The number of these points, n, usually differs from some hundred to some thousand points. On the other hand, it is well known that sinusoid functions with different 3

4 frequencies are orthogonal functions [5]. Thus, any TF curve can be represented by a 1 vector and its elements determine the amplitude of TF curve on its orthogonal basis vectors, hence:,,,... (2) where, is projection of on (j-th basis vector) for j=1, 2,, n. In other words, any TF vector is spanned by n basis vectors in the n-dimensional Euclidean space. Any change in TF curve (because of RD or AD) changes the location of n discrete points on the amplitude axis and consequently modifies the projection of TF vector on its orthogonal bases and leads to change the direction of the resultant TF vector in the n- dimensional Euclidean space. Intense defects cause more deviation in the direction of TF vectors. As a result, it can be said that the intensity of RD and AD for different TF curves can be estimated using the angle between the defected TF vector and the intact one. The angle between two vectors of and, can be calculated using arccosine similarity measure (ACSM), as follows [6]: (3) where, stands for the dot product of two vectors and. indicate the length of a vector. Calculated values of ACSM for the test objects and for four intensities of RD and AD are presented in Tables 2 and 3, respectively. Considering Tables 2 and 3, it can be obviously seen that computed ACSMs have regular variations with respect to the defect intensity. Therefore severe defects cause more deviation in the direction of TF vectors (Fig.6). As a result, it can be said that the extent determination of the RD and AD for different TF curves can be accurately estimated using the ACSM. In this stage, the ACSM is compared with the well-known correlation coefficient that usually used for TF comparison. 4 Table 2: ACSM for different stages of RD RD intensity Without RD 0 Stage Stage Stage Stage Table 3: ACSM for different stages of AD AD intensity Without AD 0 Stage Stage Stage Stage Fig.6: Deviation in direction of intact TF vector ( ) in presence of slight defect ( ) and sever defect ( ) Correlation Coefficient (CC) is a measure for the similarity of two transfer function curves. For and the correlation coefficient can be calculated, as follows [2]:,.. (4) The computed values of the correlation coefficient and ACSM for TFs of Figs 4 and 5 are presented in Fig.7. Considering Fig.7, it can be said that ACSM acts as a mapping from the space of the TF to the space of the defect intensity. Angular proximity between TF vectors in their vector space is an efficient index to estimate the intensity of RD and AD in transformer winding. Due to regular and nearly linear behavior of angular proximity index (ACSM),

5 in comparison to correlation coefficient, it is a very good alternative for correlation coefficient in transformer diagnosis. Correlation Coefficient RD AD Defet Extent RD AD Fig.7: Comparison of (a) ACSM and (b) correlation coefficient, for different intensities of RD and AD 4. Conclusion The TF curve of a transformer can be represented by a vector in a high-dimensional vector space. The radial deformation (RD) and axial displacement (AD) of transformer winding divert the TF vector from its initial direction. Thus the intensity of RD and AD can be estimated using the angular proximity of the defected and intact TF vectors. Angular proximity index acts as a mapping from the space of the TF to the space of the defect extent. Due to regular and nearly linear behavior of this index, in comparison to correlation coefficient, and in presence of its value for a predetermined defect extent for a specific transformer, the intensity of the mechanical defect can be determined for any other stage of defect and for any other transformer with the similar design (sister transformer). (a) (b) Defet Extent References [1] Christian J., Feser K. Procedures for detecting winding displacements in power transformers by the transfer function method IEEE Transactions on Power Delivery 2004; 19: [2] Secue J.R., Mombello E. Sweep frequency response analysis (SFRA) for the assessment of winding displacements and deformation in power transformers Electric Power Systems Research 2008; 78: [3] Rahimpour E., Jabbari M., Tenbohlen S. Mathematical comparison methods to assess transfer functions of transformers to detect different types of mechanical faults IEEE Transactions on Power Delivery 2010; 25: [4] Rijsbergen C.J.V. Information retrieval; 2nd edition Butterworth- Heinemann, [5] Kreyszig E. Advanced engineering mathematics; 10-th edition Wiley, [6] Zhu S., Wu J., Xiong H., Xia G. Scaling up top-k cosine similarity search Data & Knowledge Engineering 2011; 70: