## 2 Research on Winding Looseness Tests of Power Transformers
### Construction of Frequency-Domain Feature Map Based on Gram Angle and Field  
The spectrum distribution of transformer windings under different looseness fault states changes obviously, and the change in the fundamental frequency amplitude under  
Figure 5: Two-dimensional feature map generation flowchart.  
Figure 6: Generated two-dimensional color feature map.  
each looseness state is positively correlated with the current. Therefore, the construction of frequency-domain feature maps under different load current conditions can more accurately reflect the characteristic information of different looseness faults of windings. The two-dimensional image generated by the Gram-angle field transformation [31] is capable of more effectively reflecting the characteristic information of the frequency-domain signal.  
Before constructing the frequency-domain feature map, the original one-dimensional vibration signal needs to be transformed by Fourier transform to obtain the spectrum curve, and then the amplitude information corresponding to the frequency is saved in matrix form by Gram-angle field transform. Finally, the frequency-domain feature map is obtained by pseudo-color processing. The detailed procedures are presented below:  
(1) Complex modulation refines the spectrum:  
The vibration signal frequency spectrum of the power transformer winding is predominantly concentrated in the range of 0-1000 Hz. In this experiment, the vibration signal acquisition rate is 100 Hz. After Fourier transform, the characteristic information contained in the low-frequency component cannot be clearly displayed in the spectrum. Therefore, it is necessary to perform complex modulation refinement operations to filter out the high-frequency part, so that the generated spectrum curve is concentrated within 0-1000 Hz [32]. The complex modulation refinement spectrum flow is shown in Figure 7.  
As shown in Figure 7, the time series of the original vibration signal of the transformer \(x_{in(n)}\) is transformed by Hilbert transform [33], and the complex analytical sequence \(\hat{x}_{n}\) is constructed, which is \(\hat{x}_{n}=\{a_{1}+jb_{1},a_{2}+jb_{2},\cdots,a_{n}+jb_{n}\}\). Then, the complex analytic signal is modulated by the complex exponential signal to achieve the spectrum shift. The modulated signal undergoes low-pass filtering. This process removes the high-frequency components of the signal, leaving only the frequency signals within the 0-1000 Hz range. The filtered signal is resampled, and the sampling frequency is \(f_{s}/D\). The resampled signal undergoes a discrete Fourier transform, followed by frequency adjustment and refinement before being output.  
(2) Data normalization processing:  
The spectrum curve data after complex modulation refinement are normalized to make them range between [\(-\)1, 1], which can be directly used as the input of the cosine function. The detailed formula for normalized calculation is presented as follows:  
\[\widetilde{x}_{i}=\frac{2(x_{i}-x_{\text{min}})}{x_{\text{max}}-x_{\text{min} }}-1 \tag{8}\]  
Figure 7: Complex modulation refinement spectrum flowchart.  
(3) Gram matrix generation:  
The complex sequence after complex modulation refinement \(\hat{x}_{n}\) contains the amplitude and angle information of different frequency bands, and it does not need polar coordinate conversion. The obtained Gram-Angle Summation Field (GASF) matrix is as follows:  
\[GASF=\begin{pmatrix}\cos(\theta_{1}+\theta_{1})&\dots&\cos(\theta_{1}+\theta_{n })\\ \dots&\cos(\theta_{i}+\theta_{i})&\dots\\ \cos(\theta_{n}+\theta_{1})&\dots&\cos(\theta_{n}+\theta_{n})\end{pmatrix} \tag{9}\]  
In this formula, \(\theta_{i}\) is the angle value of the \(i\)-th sequence.  
Carry out pseudo-color processing on the generated Gram matrix to obtain a two-dimensional characteristic graph in the frequency domain, and then compare it with the two-dimensional map generated without complex modulation refinement, as shown in Figure 8 below.