# Six Sigma-Based Frequency Response Analysis for Power Transformer Winding Deformation
## 2 Materials and Methods
### Proposed FRA6\(\sigma\) Approach--Define Phase
#### 2.1.2 Step 2: Computing Statistical Control Limits--Analyze Phase  
To establish a reliable control chart for FRA, it is essential to compute key statistical parameters, including the mean (\(\overline{X}(f)\)), standard deviation (\(\sigma(f)\)), and the control limits (UCL and LCL). These parameters define the expected variation range in the transformer's response under normal conditions. If deviations beyond these limits are observed, they may indicate mechanical or electrical faults.  
1. Mean Response (\(\overline{X}(f)\)) Calculation  
The mean response at each frequency point is obtained by averaging multiple baseline FRA measurements. This provides a reference curve that represents the transformer's normal operating state. The mean response is calculated as follows in Equation (2).  
\[\overline{X}(f)=\frac{1}{n}\sum_{i=1}^{n}X_{i}(f) \tag{2}\]  
where  
\(Xi(f)\)--FRA magnitude for different baseline measurements (if multiple fingerprint measurements exist)  
\(n\)--number of baseline measurements.  
NB: if multiple baseline measurements are available, they help reduce the effect of measurement noise and improve accuracy in determining the normal response of the transformer.  
2. Standard Deviation (\(\sigma(f)\)) Calculation  
To measure the dispersion of the baseline FRA data, the standard deviation at each frequency point is calculated as follows in Equation (3). The standard deviation quantifies how much individual baseline responses deviate from the mean response.  
\[\sigma(f)=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}\left(X_{i}(f)-\overline{X}(f) \right)^{2}} \tag{3}\]  
where  
\(X_{i}(f)\)--represents each individual baseline measurement.  
\(\overline{X}(f)\)--is the mean response.  
\(n\)--the number of baseline measurements.  
NB: if only a single fingerprint measurement is available, the standard deviation cannot be computed directly. In such cases, a predefined threshold must be used to approximate the expected variability.  
3. Control Limits Calculation  
Once the mean and standard deviation are calculated, the control limits are defined to establish acceptable variation thresholds. The upper control limit (\(UCL\)) and lower control limit (\(LCL\)) determine the range within which \(X_{latest}\) responses should remain under normal operating conditions. The \(UCL\) is expressed as follows in Equation (4).  
\[UCL(f)=\overline{X}(f)+k\sigma(f) \tag{4}\]  
The corresponding \(LCL\) is expressed as follows in Equation (5).  
\[LCL(f)=\overline{X}(f)-k\sigma(f) \tag{5}\]By default, a factor of 6\(\sigma\) is used in Six Sigma analysis to ensure that 99.99966% of normal FRA variations remain within these control limits. This minimizes false alarms while ensuring that significant deviations indicate actual transformer defects. However, the choice of \(k\) can be adjusted based on the required confidence interval as tabulated in Table 3.  
Using these control limits, the \(X_{latest}\) measurement can be evaluated. If the response remains within the UCL and LCL than the transformer is considered stable. However, if deviations beyond these limits are observed, they indicate potential faults such as winding deformation, core displacement, or insulation failure. This proposed novel statistical approach ensures a robust and objective method for monitoring transformer health and enabling early detection of mechanical and electrical defects.