# Six Sigma-Based Frequency Response Analysis for Power Transformer Winding Deformation
## 2 Materials and Methods
### Outlier Detection and Treatment in FRA-Six Sigma  
Outlier detection is a crucial component of the proposed FRA6\(\sigma\) framework, as it directly affects the accuracy of statistical control charts and process capability assessment. In FRA, outliers may occur due to transient environmental disturbances, instrumentation noise, or sporadic anomalies in measurement conditions. To ensure diagnostic robustness, the FRA6\(\sigma\) methodology incorporates a structured approach to identifying and treating outliers during both the baseline construction phase and the runtime assessment of transformer condition. During the establishment of the baseline fingerprint, multiple FRA measurements are taken under known healthy conditions. For each frequency point, the mean (\(\overline{X}\)) and standard deviation (\(\sigma\)) are calculated. Any data point lying beyond \(\pm 3\sigma\) from the mean is flagged as a potential outlier. These flagged points are examined to determine their origin. If the outliers are determined to be the result of external measurement noise or equipment instability--such as poor grounding, temperature drift, or electromagnetic interference--they are excluded from the statistical baseline. However, if similar deviations appear consistently across multiple tests, they are retained as legitimate indicators of potential internal mechanical stress, rather than random fluctuations. This approach ensures that genuine early-warning signs are not mistakenly discarded as statistical noise. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Ref.** & **Frequency Range (Hz)** & **Category/Region** & **Likely Transformer Faults** & **Physical Interpretation** \\ \hline
[46, 47, 56, 57, 58, 59] & 10,000–100,000 & Mid frequency & Axial and radial winding & (1) Influenced by inter-winding and intra-winding capacitance. \\  & & & variation, insulation & (2) Significant for radial and axial \\  & & & compression failure & deformation in the windings. \\  & & & insulation breakdown & (1) Small, localized defects in \\
[46, 47, 60, 61, 62] & 100,000–1,000,000 & High frequency & between winding turns, & conductor insulation. \\  & & & tap changer defects, & (2) Sensitive to changes in grounding \\  & & & grounding issues & and inter-turn short circuits. \\
[46, 47, 63, 64] & 1,000,000–2,000,000 & Upper high & Partial discharges, floating & (1) Dominated by parasitic capacitance \\  & & frequency & metal parts, loose clamps & and localized insulation breakdown. \\  & & & and connections & (2) Detects floating conductive parts, \\  & & & loose clamps, and insulation failure. \\ \hline \hline \end{tabular}
\end{table} 
Table 7: _Cont._ \begin{table}
\begin{tabular}{c l l} \hline \hline
**Condition** & **Observation** & **Conclusion** \\ \hline Case 1: Transformer is stable & FRA data remains within 6\(\sigma\) control & No significant mechanical \\  & limits. C\({}_{\text{p}}\) and C\({}_{\text{pk}}\) \textgreater{} 1.33. & changes detected. Transformer \\ Case 2: Gradual degradation & FRA response approaches UCL/LCL & is in good condition. \\  & but remains within limits. & Early signs of mechanical drift. \\  & C\({}_{\text{p}}\) and C\({}_{\text{pk}}\) = 1.0–1.33. & Requires monitoring. \\ Case 3: Potential defects & FRA data crosses UCL/LCL in specific & Possible transformer defects (e.g., \\  & frequency bands. C\({}_{\text{p}}\) or C\({}_{\text{pk}}\) \textless{} 1. & winding deformation, insulation failure). \\  & Immediate action required. \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Transformer case study classification. During runtime evaluation, where a new frequency response (\(X_{latest}\)) is compared to the established baseline, any deviation that exceeds the Six Sigma control limits (\(\pm 6\sigma\)) is not treated as a statistical outlier but rather as a significant signal, potentially indicative of a fault. Unlike traditional quality control processes that may discard such points, the FRA6\(\sigma\) framework interprets them as indicators of structural or dielectric changes within the transformer.