El mundo de la vida como urdimbre y el cuerpo como tejedor de tramas

This subsection proposes two new indices for evaluating deviation in the low frequency sub-bands. The purpose of these two indices is to effectively support the identification of DRC, SCTFM and OCFM.

Area Ratio Index (ARI)

The common deviation in sub-band LF1 is vertical displacement. This kind of characteristic deviation usually occurs when the winding under test is in d- ifferent remanence condition, or when some kind of electrical faults appears, i.e short-circuit between turns and open circuit in a winding. CC is not sen- sitive to vertical changes. Although σ can detect this kind of deviation, the calculated value of σ cannot detect the degree of deviation explicitly. In other words, by calculating σ for this case above (LF1), it is difficult to define a threshold value for the purpose of the classification. Besides, the calculation- s of σ and CC do not have the ability of differentiating whether the curve is shifting upwards or downwards with reference to the reference curve. The shifting direction (upwards or downwards) in sub-band LF1 also contains vital

information to classification. In consequence, to cover these deficiencies ofCC and σ in sub-band LF1, a new index, Area Ratio Index (ARI), is designed to differentiate the changing directions of curves and reflecting the degree of vertical deviation in quantitative manner. This index is specially designed for sub-band LF1 where the curves are generally linear or smooth.

The definition of Area Ratio Index (ARI) is:

ARI(X, Y) = N ∑ i=1 (yi −xi) N ∑ i=1 (M AX−min(xi, yi)) , (5.4.3)

M AX = max{max(X), max(Y)}, (5.4.4) where X = {x1, x2, . . . , xN} and Y = {y1, y2, . . . , yN} denote vertical pixel positions (py) of two frequency responses (in (px,py) coordinate system) and N is the number of pixels in horizontal axis that corresponds to the maximum frequency point.

The physical understanding of (5.4.3) is illustrated in Fig. 5.11. It can be seen that the absolute value of ARI is equivalent to the area ratio of the deviation area in LF1 (yellow (dark) area in Fig. 5.11) to the accumulated maximum area (doted area). In most of the FRA curves of the power trans- former in usual use, the slope angle is similar and does not change much (eg. typically 20 dB/decade). Therefore, the area ratio can reflect the degree of de- viation. The interval ofARI is [-1, 1]. The sign of ARI represents the shifting direction of the suspected curve compared to the reference curve. Take X as reference, then the positive sign means that the suspected curveY moves below X (shifting downwards), and the negative sign means that the suspected curve Y moves above X (shifting upwards). Therefore, both of direction of change and degree of vertical deviation in sub-band LF1 are described by ARI.

ARI can accurately used to extract the information in LF1, but for some LV windings or test results under connection of end-to-end shirt-circuit (see Chap- ter 2), the determination of the threshold values for this index has limitations due to the flat slope angle at the beginning of the curves (much smaller than 20 dB/decade). This can be solved by adding a weighted parameter according

px(pixel)

py(pi

xe

l)

Figure 5.11: Explanation of Area Ratio Index (ARI)

to the type of the input FRA data, but it is not considered in this thesis. In future study, when the data sets can cover more types of transformer, it should be considered when determining the threshold values for ARI.

Angle Difference of the anti-resonance in LF2 (AD)

The apparent anti-resonance (minimum, Zm mentioned in Subsection 5.3.2 ) in sub-band LF2 contains essential information and can be easily detected in automatic processing of data. The shift of this point effectively reflects the change of Lm orCg or both of them. A new index, Angle Difference (AD), is designed to extract the information of the changing angle of this point, and its definition is:

AD=atan2d((b1−b2),(a2−a1)), (5.4.5) whereatan2d(B, A) represents a function of four-quadrant inverse tangent in degrees as explained in Fig. 5.12. The minimum anti-resonance of baseline in LF2 is chosen as the origin of plane, and the range of AD is (−180◦, −180◦]. X(a1, b1) represents the pixel positions of Zm of the reference in LF2, and Y(a2, b2) represents the pixel positions ofZm of the suspected curve.

The index AD is purposefully designed for coherent usage with ARI. It is only used to calculate the changing angle rather than the severe degree of

px(pixel)

py(pi

xe

l)

Figure 5.12: Explanation of Angle Difference (AD)

curve shift. The index ARI can reflect the change of electrical parameters which could be Lm or winding resistance R. AD can indicate the change of parameters, which could be Lm or Cg. Whereas using them together, it is able to differentiate the changed electrical parameter(s) from each other. The novelty of these two proposed indices is that as a part of the hybrid indices, they substantially increase the sensitivity of the hybrid indices at low frequencies. A combined use of ARI and AD is able to distinguish the key parameter that causes deviation at low frequency sub-bands. It consequently supports the identification of DRC, SCTFM and OCFM. The implementation of these indices will be given in the case studies.

5.5

Hierarchical Dimension Reduction Classi-