# Six Sigma-Based Frequency Response Analysis for Power Transformer Winding Deformation
## 2 Materials and Methods
### Proposed FRA6\(\sigma\) Approach--Define Phase
#### 2.1.4 Step 4: Sigma Level and Process Capability Determination  
The fourth step in the proposed novel FRA6\(\sigma\) approach focuses on assessing the stability of the transformer's response using process capability indices (\(C_{p}\) and \(C_{pk}\)). These indices quantify how well the \(X_{latest}\) measurement remains within acceptable variation limits. A high \(C_{p}\) and \(C_{pk}\) value indicates a stable transformer, while lower values suggest potential mechanical changes or faults. \begin{table}
\begin{tabular}{c c c} \hline \hline
**k Value** & **Confidence Interval (\%)** & **Sensitivity Level** \\ \hline
3 & 99.73\% & More strict \\
2 & 95.45\% & Moderate \\
1.5 & 86.64\% & Sensitive \\ \hline \hline \end{tabular}
\end{table} 
Table 3: \(k\) Factor determination for confidence interval determination. Process Capability Index (\(C_{p}\)) Computation  
The \(C_{p}\) evaluates how well the transformer's response fits within the Six Sigma limits. It is defined as follows in Equation (6). \[C_{p}=\frac{UCL-LCL}{6\sigma} \tag{6}\]  
If \(C_{p}\geq 1.33\), the transformer FRA response is considered stable. If \(C_{p}<1.33\), significant variations exist, indicating mechanical changes. Process Capability Index Relative to the Mean (\(C_{pk}\)) Computation  
While \(C_{p}\) measures the overall spread of the response; it does not indicate whether the FRA response is shifting toward one boundary (UCL or LCL). This is where \(C_{pk}\) comes in. It detects if the FRA response is skewed towards one limit, signifying early-stage transformer drift. \(C_{pk}\) is calculated as follows in Equation (7). \[C_{pk}=min\bigg{(}\frac{UCL-X_{latest}}{3\sigma},\frac{X_{latest}-LCL}{3\sigma} \bigg{)} \tag{7}\]  
where  
\(X_{latest}\) represents the latest FRA response value at each frequency. If \(C_{pk}<1\)--the transformer shows abnormal drift. If \(C_{pk}>1.33\)--the process is well within control. Once \(C_{p}\) and \(C_{pk}\) values are calculated, the transformer's process stability is assessed based on Six Sigma levels as proposed in Table 4. The following table classifies the transformer's condition based on the calculated \(C_{p}\) values, ensuring structured decision-making for fault detection and maintenance. The 6-Sigma rule defines process capability in terms of \(\sigma\) and can be expressed mathematically as follows in Equation (8) or Equation (9). \[\sigma=\bigg{(}\frac{UCL-LCL}{6\sigma}\bigg{)} \tag{8}\]  
or  
\[\sigma=3\times min\bigg{(}\frac{UCL-X_{latest}}{3\sigma},\frac{X_{latest}-LCL} {3\sigma}\bigg{)} \tag{9}\]  
While the \(\overline{X}\)-chart monitors the mean transformer response, a \(\overline{R}\)-chart has also been proposed which is crucial for tracking variability in measurements. The \(D_{4}\) constant is applied to establish upper control limits (\(UCL_{R}\)) for the range of FRA values. While \(\overline{X}\)-charts track deviations in mean FRA responses, they do not capture variations in process stability. To account for this, \(\overline{R}\)-charts are introduced, which monitor changes in measurement range over time. \begin{table}
\begin{tabular}{c c c c} \hline
**Cp** & **Sigma Level (\(\sigma\)-Level)** & **Process Stability** & **Action Required** \\ \hline
2.00 & \(\geq\)6\(\sigma\) & Highly stable & No action required \\
1.75 & 4\(\sigma\)–5\(\sigma\) & Acceptable but monitor trends & Routine monitoring \\
1.50 & 4\(\sigma\)–5\(\sigma\) & Acceptable but monitor trends & Routine monitoring \\
1.25 & 3\(\sigma\)–4\(\sigma\) & Marginal stability & Investigate for early-stage faults \\
1.00 & 3\(\sigma\)–4\(\sigma\) & Marginal stability & Investigate for early-stage faults \\
0.75 & \(<\)3\(\sigma\) & Process out of control & High probability of transformer failure \\
0.50 & \(<\)3\(\sigma\) & Process out of control & High probability of transformer failure \\ \hline \end{tabular}
\end{table} 
Table 4: Proposed FRA6 \(\sigma\) level analysis.