# Low-Speed Bearing Fault Diagnosis Based on Permutation and Spectral Entropy Measures  
[PERSON]  
1 Ikerlan Technology Research Centre, Basque Research and Technology Alliance (BRTA), P\({}^{\rm o}\). J. M\({}^{\rm a}\). Arizmendiarrieta 2, 20500 Arrasate-Mondragon, Spain; [EMAIL_ADDRESS]
2 Control, Modeling, Identification, and Applications (CoDAlab), Department of Mathematics, Escola d'Enginyeria de Barcelona Est (EEBE), Universitat Politecnica de Catalunya (UPC), Campus Diagonal-Besos (CDB), Eduard Maristany 16, 08019 Barcelona, Spain; [EMAIL_ADDRESS] (F.P. ); [EMAIL_ADDRESS] (Y.V.) 2 May 2020203 July 2020  
in comparison to regular bearings, such as heavy loads (order of kNm), a larger size (diameter of meters), and greater weight (order of tonnes), as well as a clockwise-anticlockwise rotation [8]. Consequently, they have a considerable influence on the functionality and maintenance costs of a wind turbine, and also particular operating conditions [9]. Nevertheless, one particular characteristic of these bearings that is of interest is the operating speed. Low or high operating-speeds can be discriminated by the limit value of 600 revolutions per minute (rpm) [10], which is used to classify bearings. Because the pitch bearings operate under 10 rpm, they are considered very low-speed bearings. In this scheme, CBM and fault diagnosis for bearings use data from multiple sources, such as lubricant condition, acoustic emission (AE) and vibration signals [11]. Although lubricant analysis can be done mainly with spectrometric and ferrographic analysis with positive results [12], AE and vibration signal are the main data source for bearing fault diagnosis [13]. The use of AE for bearings fault diagnosis involves several intrinsic parameters, such as maximum amplitude, duration of emission, counts, number of events, and absolute energy [14]. AE has a large number of studies aimed at regular bearings, which works above 600 rpm. However, the structures and working conditions of low-speed and heavy-load pitch bearings are different enough to use the research conclusions for regular bearings only as a reference. Therefore, a small number of alternatives for indicators and models focused on AE and pitch bearings can be found in the literature. The indicators found in the literature are root mean square (RMS) [15], hyperflatness [16], average signal level [17], and Prony's energy [18]. Because AE signals are paroxysmal and non-linear [19], some nonlinear parameter estimation can be found for general-purpose bearings, but only the largest Lyapunov exponent has been applied effectively for low-speed bearings [20]. Historically, research on CBM and fault diagnosis focused on bearings has been dominated by the analysis of bearing vibration response. In this analysis, information is gathered by feature extraction in both the time and frequency domains. While several indicators are well established for general purpose bearings, similar approaches cannot be used effectively for very low-speed bearing conditions, such as pitch bearings [21]. Therefore, a reduced number of indicators based on statistics can be used, such as covariance and kurtosis [22; 23]. The use of indicators is also feasible in the frequency domain, but the vibration signal must initially be transformed using the fast Fourier transform. When the transformation is done, indicators such as the root mean square frequency and the root variance frequency can be calculated [23]. Due to the particular working conditions at low rotational speed [24], researchers used strategies for noise reduction to subsequently compute these parameters. The methods are mostly based on the decomposition of the signal, as wavelet decomposition and empirical mode decomposition (EMD). Several researchers used wavelet decomposition simultaneously with kurtosis [25], energy spectrum coefficient [26], envelope analysis [27; 28] or harmonic significance index [29]. Regarding EMD, [PERSON] et al. [30] combined EMD with cross-correlation kurtosis to diagnosis a bearing rotating at 15 rpm.