Vibration Analysis and Time Series Prediction for Wind Turbine Gearbox Prognostics

Multiple premature failures of a gearbox in a wind turbine pose a high risk of increasing the operational and maintenance costs and decreasing the profit margins. Prognostics and health management (PHM) techniques are widely used to assess the current health condition of the gearbox and project it in future to predict premature failures. This paper proposes such techniques for predicting gearbox health condition index extracted from the vibration signals. The progression of the monitoring index is predicted using two different prediction techniques, adaptive neuro-fuzzy inference system (ANFIS) and nonlinear autoregressive model with exogenous inputs (NARX). The proposed prediction techniques are evaluated through sun-spot dataset and applied on vibration based health related monitoring index calculated through psychoacoustic phenomenon. A comparison is given for their prediction accuracy. The results are helpful in understanding the relationship of machine conditions, the corresponding indicating features, the level of damage/degradation, and their progression.


INTRODUCTION
There is a growing interest in renewable energy systems with increased concerns over climate change.Wind energy has an attractive share in renewable energy because it diversifies a resource portfolio and improves overall security of the power system.However, the engineering challenge for the wind industry is to design a reliable wind turbine to harness wind energy and turn it into electricity.Despite all technological advancements in wind turbine design and installation, there is a price to pay in maintaining the wind turbine in harsh operating environments and reduced accessibility.According to two large surveys of European wind turbines, conducted over a span of 13 years, gearbox failure is one of the highest risk events in wind turbines (C.C. James, 2011).Hence, there is a need for efficient condition monitoring system for wind turbine gearbox.Condition monitoring is a good tool to assess the damage early in time in order to plan the maintenance activities in a better way.Condition monitoring can be combined with opportunity maintenance to reduce the turbine's unexpected downtimes.
Typically, faults in wind turbine gearbox arise while in operation.Therefore, it is vital to detect, diagnose and analyze these faults as early as possible.The process should be non-destructive in nature to avoid wind turbine's disassembly.This study presents fault detection, features extraction, and prognostics for wind turbine gearbox based on vibration analysis.Vibration analysis is a non-destructive testing (NDT) technique widely used in the industry and in academia.

Classification of Vibration Signals
Vibration signals emanating from the rotating gearbox are analyzed to ascertain the current condition of the gearbox.Vibration signals can be classified into stationary and nonstationary, and based on this classification, the nature of their analysis methods differ.For stationary signals, vibration analysis methods are divided into two domains, namely time and frequency.Time-domain methods include statistical, model based, and signal processing based methods.Frequency-domain methods include spectrum and cepstrum based methods.For non-stationary signals, joint time-frequency vibration analysis methods such as short time Fourier transform (STFT) and wavelet analysis (WA) (Hui Li et al., 2011) are used for fault detection in gearboxes.Other methods of non-stationary analysis for gearboxes include Wigner-Ville distribution, Hilbert-Haung transform, and kurtogram analysis (Jerome Antoni, 2007).

Features Extraction
The features extraction system extracts characteristic signatures from raw vibration signals emanating from gearbox.The extracted features should reflect the changes in the gearbox's health conditions over time.As discussed earlier, the vibration based signal processing analysis is one of the most common non-destructive techniques.Also, with suitable vibration analysis, we can detect many different faults related with gearboxes.We use vibration based features extraction to extract the information that best represents the faulty conditions present in the monitored equipment.Different methods for vibration based features extraction in gearbox fault diagnosis framework have been proposed in research (Halima, E. B. et al, 2008, J. Rafiee et al, 2010).In time-domain, vibration based features such as kurtosis and spectral kurtosis are extensively used (F. Combet et al, 2009).Other studies including statisticalbased and transient-based features detection are performed in the past (Hiram Firpi and George Vachtsevanos, 2008;V. Indira et al, 2010).A comprehensive list of time-domain and frequency-domain features for fault detection and diagnosis of gearboxes is discussed in (Yaguo Lei et al, 2010).

Prognostics
Prognostic plays a very important role in an accurate and reliable decision making.Prognostics can be used effectively in utilization and maintenance of machinery systems.Prognostics uses different machines health related indices including temperature (Jamie Coble et al, 2010), oildebris analysis (Richard Dupuis, 2010), acoustics, andvibration (Eric Bechhoefer et al, 2010).Among these, vibration based prognostics is quite common.In (B.Samanta and C. Nataraj, 2008), researchers have used different health monitoring indices in gearboxes such as gear-wear, gear-chipped, gear-crack, gear-pitting, and shaft misalignment for projecting the gearbox health information in future.They have used neuro-fuzzy approaches for modeling and prediction of gearbox dynamics.A comprehensive review on prognostics is presented in (Andrew K.S. et al, 2006), where decision making process based on diagnostics and prognostics is discussed.Prognostic is performed by estimating the temporal evolution of the features over time (Wang W., 2007).In vibration based prognostics, vibration signals emanating from sensitive components inside the gearbox are recorded, health related features are extracted, and time series prediction techniques are applied to the features trends for prognostics (L.Gelman et al, 2012).Statistical, evolutionary and soft computing approaches are used to estimate the predictors and the use of neural networks and neuro-fuzy methods are very common.This paper proposes a novel method to extract machine's health related vibration features based on psychoacoustic phenomenon along with neural networks and neuro-fuzzy approaches for prognostics.In this study, other than vibration features, we also use sunspot data for measuring the performance of the designed predictors.The rest of the paper is organized as follows.Section 2 is the methodology section where we propose the novel feature extraction technique based on psychoacoustics phenomenon followed by wavelet smoothing and prediction approaches.Section 3 simulates the proposed techniques on sunspot data and realworld vibration data emanating from a planetary gearbox inside a wind turbine.Finally, section 4 concludes the paper.

THE METHODOLOGY
The process of vibration based features extraction and prognostic is shown in Figure 1.Below we discuss each module one by one.
Figure 1.The process of prognostic

Transient Based Features Extraction
The vibration features extraction algorithm presented in this section works on the principle of transient analysis.
Transients are very short and abrupt changes in sound waves due to non-linearity.Non-linearity could be mechanically introduced disturbances in electromechanical systems or unwanted clicks in transmission lines.The transient analysis algorithm calculates a real time estimate of transients caused by non-linearity as perceived by the human ear.The analyzer uses the knowledge about human ear's nature of filtering the signals as presented in (E.Zwicker, and H. Fastl, 2009).The filtering operation ensures that the transients are detected in a way that matches the nature of the cochlea and thereby as perceived by the human ear.Transient analysis gives a much better correlation to the perceived quality of sound than traditional measurements based on frequency analysis.
The vibration features extraction algorithm uses principles of auditory models developed by the auditory physiologists (S. Seneff, 1988).According to psychoacoustic theory, pulses with short rise time or fall time contain a broad spectrum of frequencies.Therefore, it is possible to detect the instantaneous energy in frequency bands in the transient range of the ear.A common method for doing this is to use a filter bank containing a group of band-pass filters covering the frequency interval of interest.The purpose of the band pass filters is to detect the pulses in the frequency band where the pulses have most energy as perceived by the human ear.It will be the filter where the shape of the impulse response of the band-pass filter matches best the shape of the pulses but reverse in time.A gammatone filter bank is used in this study as shown in  The impulse response of a band-pass filter is defined by the following relation (R. D. Patterson et al, 1992): Where R is an arbitrary factor that is typically used to normalize the peak magnitude transfer to unity, N is the filter order, m is a parameter that determines the duration of the impulse response and thus the filter's bandwidth, c F is the filter's center frequency, and  is the phase of the tone.
The magnitude and phase of () . The magnitude () At is the envelope of the signal and is always a positive function.It is expected that pulses with a short rise time will be more annoying to the ear, than pulses with longer rise times.Thus, by considering the pulse envelope, it seems reasonable to focus the measure on the slope in the leading edge.Every transient detected is characterized by the 80% amplitude and the maximum slope in the leading edge.In order to obtain an equal number of detected transients in each frequency bands, the output from the transient analyzer is split into blocks of a pre-specified duration.The duration is found as a trade-off between preserving the complexity of the measurement and having an equal number of detected transients in each channel.
The argument for squaring the maximum steepness in Eq. ( 5) is simply to put the large steepness values in favor.
is the maximum amplitude of the nondifferentiated envelope signal.The vibration features extracted are de-noised through wavelet de-noising techniques and normalized before prediction.

Wavelet Denoising
Features extracted from real signals emanating from complex dynamical systems pose a serious problem of noise.Therefore, it is important to de-noise the extracted features before modeling is performed (Uros Lotric and Andrej Dobnikar, 2005).De-noising the signal is one of the most effective applications of wavelets in signal processing.The wavelet transform-based de-noising methods can produce much higher de-noising quality than conventional methods.Furthermore, the wavelet transform-based methods retain the details of a signal after de-noising (Edmundo G. de Souza e Silva et al, 2010).Wavelets are limited duration, undulatory mathematical functions.The time integral of wavelet functions equals to zero. Figure 5 plots some common wavelets.Similar to Fourier transform, where we use sines and cosines as basis functions, wavelet transform uses wavelets as basis functions.Wavelets are used in many different fields including compression, signal processing, and de-noising (Graps, A, 1995).In Fourier analysis, we approximate a function   fx by sines and cosines functions with different frequencies and amplitudes.Thus, the approximation equation becomes Wavelet analysis is performed in the similar way as Fourier transform but with scaled and translated versions of mother wavelet   x  as basis functions.Mother wavelet can be any one from Figure 5 and the scaling and translation is defined as child wavelets and can be calculated as Where  is a constant, k is wavelet translation and 2 j is scale translation.We can estimate    from the approximation.This procedure is called thresholding and it forms the basis for the wavelet denoising.Wavelet analysis basis functions are finite and limited to one size and this makes wavelet analysis useful technique for detecting local features like discontinuities and spikes in a signal.On the other hand, Fourier analysis basis functions are infinite in nature and an approximation of a specific part of the signal affects the entire signal.Wavelet analysis is joint time-frequency analysis technique as contrast to the Fourier transform that is purely frequency analysis.This feature makes wavelet analysis to detect when in time a particular event took place.Wavelet smoothing or wavelet trend analysis is used to remove high frequency components from the extracted features which can be assumed as noise.

Time Series Prediction
The prediction of time series   Where  is a predictor functions and can be approximated through various conventional, statistical and artificially intelligent techniques like Bayesian, support vector regression, adaptive neuro-fuzzy inference system (ANFIS) and neural networks (NN).This paper uses dynamic neural network called the nonlinear autoregressive model with exogenous inputs (NARX) and ANFIS techniques to approximate the predictor function  .

The NARX
The NARX is a dynamic neural network, used for modeling nonlinear dynamical systems.The NARX can be represented mathematically as Where,  

The adaptive neuro-fuzzy inference system
The basic structure of the ANFIS is shown in Figure 7.The to the corresponding output y .The mapping in the ANFIS system is done through the membership functions (MFs), the rule base and the related parameters that emulate the training dataset.The training phase of the ANFIS uses hybrid learning method.It uses the gradient descent approach for fine tuning the parameters that define the MFs and applies the least squares method to identify the consequent parameters that define the coefficient of each output equation in Sugeno-type fuzzy rule base.The training process continues till the desired stopping criteria is reached, i.e., number of epochs or error tolerance.

SIMULATIONS AND DISCUSSIONS
In this section, the prediction accuracy of both the time series predictors, NARX and ANFIS is compared using standard dataset of sunspot activity for years 1749-2012(RWC Belgium World Data Center, 2012).The sunspot activity data has non-linear, non-Gaussian and nonstationary characteristics and is suitable to test the performance of the predictors.The entire dataset of sunspot activity was normalized between [0 -1] and used for training and testing both NARX and ANFIS predictors.To get a reliable prognosis, the data need to be less sensitive to noise.This requirement can be reached using selected signal processing techniques such as wavelet smoothing or denoising discussed in section 2.2.For wavelet denoising, we use Daubechies wavelet (db4) with nine levels.The threshold technique is set to soft threshold, and the threshold rule used is universal.The universal threshold rule is defined as 2 log( ) L  .Where L is the signal length.We also set the rescaling method as single level where the algorithm considers the noise as white and estimates the standard deviation of the noise from the wavelet coefficient at the first level.We use the normalized Akaike Information Criteria (AIC) for assessing the prediction performance.The AIC can be formulated as (Akaike H. 1974 1 shows the mean absolute error (MAE) and AIC for NARX and ANFIS for sunspot data.The number of membership functions for ANFIS is 3 with 16 fuzzy rules and 104 parameters.The AIC achieved is -4.457.The neuro-fuzzy system proposed in (Wang WQ 2004) with the same datasets produced AIC of 1.527.The ANFIS method gave better prediction performance as compared to (Wang WQ 2004) for the same datasets.The sunspot training and test datasets were also used for assessing the prediction performance of NARX.For NARX, MAE of testing is 0.0034.There are some outliers for both NARX and ANFIS cases and NARX depicts better performance as compared to ANFIS in this case.In the case of NARX, the MAE value is approximately same as ANFIS (sunspot data-set) in Table 1.
In the framework of machine condition monitoring, it is very important to know well in advance the expected behavior of a machine system for its proper operation and maintenance scheduling without any major interruptions.Several approaches are adopted to monitor the system behavior and vibration-based approaches are quite popular for condition monitoring.dotted lines in Figure 6).Similar strategy is adopted in the ANFIS case.
Figure 2. First designed by Patterson and Holdworth (R. D. Patterson et al, 1992), gammatone filter bank is an array of band-pass filters which simulates the response of the human ear's cochlea.At each point along the cochlea, a psychoacoustic measure of the width of the auditory filter is represented by an equivalent rectangular bandwidth (ERB).

Figure 2 .
Figure 2. The gammatone filter bankThe ERB is a psychoacoustic measure of a filter's bandwidth in the filter bank.The bandwidth of a filter increases with an increase in its center frequency.The relationship between ERB and center frequency c F Hz is given by the following equation

Figure 3
Figure 3 shows an impulse response of a gammatone filter with 1000 c F Hz  , 125 m Hz  and 4 N  .To detect the energy in the channels the output signals from the band-pass filter bank are Hilbert transformed.Hilbert transform detects

Figure 3 .
Figure 3. Gammatone filter impulse response After the pulses are extracted through envelope detection or Hilbert transform, feature extraction block follows.In the feature extraction block, amplitudes and slopes of the pulses are calculated as per Eq.(4) and Figure 4.

Figure 4 .
Figure 4. Calculation of amplitude and slope of a pulse Amplitude (maximum magnitude) is a linear detector for pulses in the full frequency area.Slope is very sensitive for catching nonlinear sounds in the high frequency area.Both approximation for the original signal.If the value of this coefficient , jk c is very small and its contribution to the approximation is considered negligible, we can omit the corresponding child wavelet   , jk x

Figure 6
Figure 6 shows the way the NARX is trained and tested.During the training phase, the feedback loops (dotted lines in Figure 6) are not used.During the testing or prediction phase, if multistep-ahead predictions are required, the output values are fed back to both the input regressor   n u and Figure 6.NARX training and testing (Feedback loops are required only during testing)

Figure 8
Figure 8(a) shows the sunspot activity data along with wavelet smoothing.NARX and ANFIS prediction results are shown in Figure 8(b) and 8(c) along with error box-plots in Figure 8(d), respectively.In Figure 8(d), the medians of the box-plots are centered at zero for both NARX and ANFIS predictors and the 2.698  lines are at 0.01  showing about 99.3% of the error observations within the range of 0.01 .Where  is the error's standard deviation.

Figure 9 .
Figure 9. (a) Planetary gearbox (b) GRC Drive train Configuration (Courtesy of National Renewable Energy Laboratory) The test turbine in the field is a stall-controlled, threebladed, upwind turbine with a rated power of 750kW.The turbine generator operates at 1200 RPM and 1800 RPM nominal on two different sets of windings depending on the power.The planetary gearbox has an overall ratio of 1:81.491.It is composed of one low speed (LS) planetary stage and two parallel stages as shown in Figure 9.This study uses data from test case CM_2a with main shaft speed of 14.72 RPM and high speed shaft (HSS) speed of 1200 RPM.The data are collected at a sampling frequency of 40KHz s F  .Figure 10(a) shows an example of a raw

Figure 10 .
Figure 10.(a) Raw vibration signal (one sample) (b) Vibration index trend (350 samples) (c) one-step prediction NARX (d) one-step prediction ANFIS A total of 350 such vibration signals are analyzed in this section.Vibration indices are calculated as proposed in

Figure 11 .
Figure 11.One-step ahead recursive prediction (a) NARX (b) ANFISNARX seems to exhibit more promising results as compared to ANFIS in this case.Also, the MAE error value for NARX is less as shown in Table1(vibration index dataset).We can also use one step recursive prediction for NARX and ANFIS to predict as many future values as we want as shown in Figures11(a) and 11(b).For multi-step recursive prediction, we have to loop the output values back as discussed in section 2.3.1 in the NARX case (feedback are used for predicting the wind turbine gearbox health related vibration based index trend.The prediction performance of the predictors, NARX and ANFIS, is illustrated on two data-sets, sunspot activity and vibration index.Both the NARX and ANFIS predictors perform quit well in this study.Results show the effectiveness of the predictors in estimating the variations of the monitoring indices.In this work, one-step-ahead and recursive multi-step ahead prediction is considered for both the NARX and ANFIS cases.Although, the k step ahead prediction is not performed in this research, it can be done easily by predicting the target ()y n k  with the previous values () xn , () x n k  , ( 2 ) x n k  ,( 3 )x n k  and so on.The potential application of these techniques for the development of on-line prognostic and estimation of remaining useful life (RUL) for machine condition is under consideration for further work.
In each block, only the transient with the maximum slope is gathered.The maximum slope is found by differentiating the signal and finding maximum amplitude of the differentiated signal.The logarithmic index is calculated as follows BN is the total number of sub-blocks.The maximum steepness in band pass filter i and sub-block ).
u d  .Table