# An Application of Instantaneous Spectral Entropy for the Condition Monitoring of Wind Turbines
## 5 Results
### Signal #1
#### 5.1.1 Sensitivity Analysis for the GWT Parameters  
Since the \(\text{ISE}(t)\) is a synthetic feature derived from the TF transform of the recorded signal, better time and frequency resolution will return more reliable results. Thus, it is essential to optimise the CWT settings. In the case investigated here, as mentioned in Section 3, the particular shape of the Generalised Morse (mother) Wavelet depends exclusively on the values considered for the doublet of parameters \((\beta,\gamma)\). Therefore, to improve the capabilities of the proposed DSF, fine-tuning these two parameters becomes the most critical aspect of the whole procedure. For this reason, a dedicated sensitivity analysis has been performed. The following cases were considered:  
1. Symmetry equal to \(\gamma=1,\ 1.5,\ 2,\ 2.5,\ 3,\ 3.5,\ \text{or}\ 4\). 2. \(\beta\) varying from 2 to 40 in steps of 2. Hence, a total of 140 combinations were analysed. The range of \(\beta\) was defined to not exceed the suggested \(\frac{\beta}{\gamma}\leq 40\) ratio [49]. Accordingly, \(P^{2}\) (as the product of the lowest values of both \(\beta\) and \(\gamma\)) ranges from a minimum of 2 to a maximum of 160. The aim of this optimisation is dual. For the training dataset, the data should behave as homogeneously as possible, to clearly define a NOCs model. This implies the signal stationarity (that is, constant mean \(\mu\) and standard deviation \(\sigma\)) and low variability (i.e., low sigma values). For a constant gamma (in the previous example of Figure 4, \(\gamma=3\)) increasing \(\beta\) decreased the absolute value of both \(\mu(\text{ISE})\) and \(\sigma(\text{ISE})\). This latter point resulted in a narrower interval of confidence. In turn, this increased the sensitivity to damage as the threshold was lowered. The same trend was observed for all the other values of \(\gamma\) investigated here as well. Regarding the testing part (second and third tracts of the concatenated signal), it is also noticeable how larger \(\beta\) values increased the detectability of the fault condition, making a more marked transition from the \"pre\" to \"post\" damage insertion conditions. Again, this finding was verified for all values of \(\gamma\in[1,4]\). Figure 4: Resulting ISE (signal #1). Values for \(\gamma=3\) and \(\beta\) equal to (**a**) 2, (**b**) 14, (**c**) 28, and (**d**) 40. As anticipated, this derives from the detail of the TF representation. Figure 5 shows this point for the examples reported in Figure 4. Please note that to avoid any potential aliasing issue and considering the very high sampling frequency, the TF was truncated at \(f_{s}/4\) (3200 Hz). The instant corresponding to the damage event is marked by the red dashed line. One can see that the TF representation for \(\beta=2\) and \(\gamma=3\) is clearly unreliable. The TF representations become more and more refined over the frequency axis as both (i) the length of the time support and (ii) the number of oscillations under the GWT envelope (and therefore the instantaneous frequency resolution) increase with \(P^{2}\) (Figure 6). Since \(P^{2}=\beta\gamma\), this latter effect can be achieved by independently increasing \(\beta\) or \(\gamma\). Figure 5: Time–frequency distributions corresponding to the results shown in Figure 4 (\(\gamma=3\) and \(\beta\) equal to (**a**) 2, (**b**) 14, (**c**) 28, and (**d**) 40). Figure 6: Mother wavelets (in the time domain) corresponding to the results shown in Figure 4 (\(\gamma=3\) and \(\beta\) equal to (**a**) 2, (**b**) 14, (**c**) 28, and (**d**) 40). On the other hand, the effects of varying \(\gamma\) for constant \(\beta\) can be summarised as follows. For any value of the symmetry value, lower values of \(\beta\) (\(\beta=2\) in the example of Figure 7) have too coarse a frequency resolution and are thus almost unusable for the intended purposes, as it can be seen from the ISE values barely changing when moving from pre- to post-damage conditions.