## 2 Research Methodology
### Modeling Methods Improvements  
Classical approaches, such as analytical methods, finite element modeling (FEM), or experimental techniques, can be too costly, complex, or time-consuming in cases where the system has nonlinear effects, complex structures, or composite flaws. The direct application of classical approaches in these cases is not always accurate, whereas their hybridization or optimization significantly increases the analysis efficiency. Modern methods, including neural networks, genetic algorithms, optimization techniques, and machine learning, enable adaptive improvements of the classical approaches, enhancing diagnostic precision with reduced computational cost. In order to conquer the challenge of time-varying harmonic analysis in power systems, an adaptive frequency-shift filtering approach for time-varying harmonic analysis is proposed in the paper [87]. An infinite impulse response (IIR) filter bank is initially applied to separate each harmonic component from the original power signal. Next, considering the time-varying nature of practical harmonics, an improved Teager energy operator (ITEO) was used to detect and segment steady-state intervals of the targeted harmonic. Then, in the detected steady-state intervals, an adaptive frequency-shift filtering process was conducted to estimate the frequency, amplitude, and phase values of the time-varying harmonic. The contribution of this paper can be stated as follows in terms. Unlike conventional frequency domain techniques like FFT, which suffer from spectrum leakage and fence effects, particularly under asynchronous sampling conditions, the method is performed solely in the time domain. This enables more precise harmonic analysis without the typical drawbacks of frequency domain conversions. The technique uses an infinite impulse response (IIR) filter bank to analyze harmonics and an enhanced Teager energy operator (ITEO) to detect steady-state intervals within the harmonics. The secondary frequency-shift filtering process yields the amplitude estimation formula for the amplitude  
\[A_{h_{s}}=\frac{2|y_{K_{2}}(n_{3})|}{\left|G\left(e^{-\beta h_{s}\Delta\omega _{K_{2}}}\right)\right|^{K_{2}}} \tag{42}\]  
and phase  
\[\varphi=h_{s}\Delta\omega_{K_{2}}\left[n_{3}+\frac{K_{2}(M_{c}-1)}{2}\right] -arg\left[y_{K_{2}}(n_{3})\right]+\frac{\pi}{2} \tag{43}\]  
for the specific harmonic, where \(K_{2}\) represents the \(K_{2}\)-stage MAF with a length of \(M_{c}\); \(n_{3}\) denotes an arbitrary point in the secondary frequency-shift filtering signal \(y_{K_{2}}(n)\); \(\Delta\omega_{K_{2}}\) is the deviation in angular frequency of the fundamental frequency after the secondary frequency-shift filtering. The hybridization is novel and improves the precision of time-varying harmonic parameter estimation. The technique is intended to tackle time-varying harmonic dynamics, which are becoming more common in contemporary power systems because of the incorporation of renewable energy resources. Classical methods fail at these time-varying features, whereas the approach overcomes this challenge. However, this method has some severe limitations. Foremost among them is its high sensitivity to several system parameters, such as, but not limited to, a frequency shift, filter order, and window length. No standard method of selecting these parameters is available at present, so their choice is made exclusively experimentally; therefore, their incorrect selection can undermine the validity of the results. In addition, the use of IIR filters can introduce time delays in detecting stable signal intervals. The authors note that the detection of the harmonic steady-state intervals can be subjected to a fixed delay, which should be compensated for precise parameter estimation. This could render the algorithm less straightforward to use in practice. The article by [35] examines a fractional-order differential model for the dynamics and stability of a single-mass rotor as an advancement of standard rotor dynamics models.