model random coefficient approach, time-dependent ARMA models, linear parameter varying ARMA models, wind turbines.

Transcription

1 Damage/Fault Diagnosis in an Operating Wind Turbine Under Uncertainty via a Vibration Response Gaussian Mixture Random Coefficient Model Based Framework Luis David Avendaño-Valencia and Spilios D. Fassois,. Stochastic Mechanical Systems and Automation (SMSA) Laboratory Department of Mechanical Engineering & Aeronautics University of Patras, GR 65 Patras, Greece fassois@mech.upatras.gr Internet: Abstract The study focuses on vibration response based health monitoring for an operating wind turbine, which features time-dependent dynamics under environmental and operational uncertainty. A Gaussian Mixture Model Random Coefficient (GMM RC) model based Structural Health Monitoring framework postulated in a companion paper is adopted and assessed. The assessment is based on vibration response signals obtained from a simulated offshore 5 MW wind turbine. The non stationarity in the vibration signals originates from the continually evolving, due to blade rotation, inertial properties, as well as the wind characteristics, while uncertainty is introduced by random variations of the wind speed within the range of to m/s. Monte Carlo simulations are performed using six distinct structural states, including the healthy state and five types of damage/fault in the tower, the blades, and the transmission, with each one of them characterized by four distinct levels. Random vibration response modeling and damage diagnosis are illustrated, along with pertinent comparisons with state of the art diagnosis methods. The results demonstrate consistently good performance of the GMM RC model based framework, offering significant performance improvements over state of the art methods. Most damage types and levels are shown to be properly diagnosed using a single vibration sensor. Keywords: non-stationary identification, vibration-based damage diagnosis, structural health monitoring, uncertainties, Gaussian mixture model random coefficient approach, time-dependent ARMA models, linear parameter varying ARMA models, wind turbines. Contents Introduction The Wind Turbine, the Damage/Fault Scenarios and the Sensors. Wind turbine description and simulation The damage/fault scenarios The sensors and the vibration response signals Brief Overview of the GMM RC Damage/Fault Diagnosis Framework and Methods 7. The elementary models Construction of the GMM RC representation Damage diagnosis based on GMM RC representations Modeling and Analysis of the Vibration Response Signals. LPV-AR and FS-TAR based modeling Model based analysis of the dynamics Copyright c 6 by L.D. Avendaño-Valencia and S.D. Fassois. All rights reserved. Corresponding author. fassois@mech.upatras.gr Tel/fax: (++ ) (direct); (central). Preprint submitted to Elsevier August 5, 6

2 5 Damage and Fault Diagnosis Results 5. Description of the optimization of the damage diagnosis methods via cross-validation GMM-RC-ML based methods Detection results Identification results GMM-RC-KL based methods Detection results Identification results Comparison with a Model Parameter Based Method Detection results Identification and level estimation results Discussion Concluding Remarks 6 References 6 Appendix A The spectral correlation and the Melard Tjøstseim PSD of a TARMA model B Brief overview of the model parameter based damage diagnosis method Important Conventions Bold-face upper/lower case symbols designate matrix/column-vector quantities, respectively. Matrix transposition is indicated by the superscript T. A functional argument in parentheses designates function of a real variable; for instance x(t) is a function of analog time t R. A functional argument in brackets designates function of an integer variable; for instance x[t] is a function of normalized discrete time (t =,,...). The conversion from discrete normalized time to analog time is based upon (t )T s, with T s designating the sampling period. A hat designates estimator/estimate of the indicated quantity; for instance θ is an estimator/estimate of θ. Main Acronyms AUC : Area Under the ROC curve ARMA : AutoRegressive Moving Average BIC : Bayesian Information Criterion : False Positive Rate FS : Functional Series GMM : Gaussian Mixture Model LPV : Linear Parameter Varying NID : Normally Independently Distributed PDF : Probability Density Function RC : Random Coefficient ROC : Receiver Operating Characteristic RSS : Residual Sum of Squares SSS : Series Sum of Squares TARMA : Time-dependent ARMA TNR : True Negative Rate TPR : True Positive Rate TV-PSD : Time-Varying Power Spectral Density Main symbols y= [ y[] y[] y[n] ] T R N : N-sample length observation (random vibration response) vector θ = [ θ θ ] T θ n R n : Parameter vector v={o,a,b,c,...} : Structural state (class) (o: Healthy; a: Damage type A; b: Damage type B;...) m : Model of the vibration response M : GMM representation L : Dimensionality of the GMM representation p(x) : Probability Density Function (PDF) associated with the corresponding random vector x P(a) : Probability of the random event a p(x, y) : Joint PDF of the random variables x and y p(y x) : Conditional PDF of y given x

3 . Introduction Vibration response based Structural Health Monitoring (SHM) aims at damage diagnosis for structures based on measured vibration response signals [, ]. The application of this technology is of particular importance for wind turbines, since these are costly, remotely located, structures for which maintenance and repair costs are considerable [, ]. SHM is thus important for avoiding structural damage, which may itself lead to downtimes and even catastrophic events, while keeping maintenance and repair costs low. Nonetheless, the design of vibration response based SHM systems for operating wind turbines poses certain challenges, including the following: In operation diagnosis. Uninterrupted operation of wind turbine facilities is necessary for maximizing power production and economic revenue. Therefore, it is most desirable to perform SHM during normal operation. Appropriate SHM methods must thus have the capability to deal with the dynamics characterizing an operating wind turbine, which feature cyclo-stationary and, in a broader sense, non-stationary behavior [5, 6, 7]. Moreover, since the actual exciting forces are not measurable, the methods must be solely based on random vibration response signals. Changing environmental and operational conditions. Wind turbines operate in a constantly changing environment determined by varying winds and weather conditions. Besides, they are set to operate at different regimes in response to the varying power demands. As a consequence, the characteristics of the random vibration response may change considerably with changes in the environmental and operational conditions [7]. Thus, the SHM system must be capable of coping with such changes and distinguish them from those due to damage [, 8]. Complex models of the dynamics. Physics based models of the wind turbine dynamics under varying excitation and operational conditions are generally quite complex for use in practical SHM systems. As a consequence, data based models derived from random vibration response signals, obtained at specific locations, need to be preferably employed. Such models may also lead to more practical and potentially more effective SHM [, 9]. Most of the currently available vibration based SHM methodologies for wind turbines utilize characteristic quantities (features) derived from frequency domain representations or modal properties of the structure. However, the extraction of frequency domain or modal characteristics from random vibration responses of operating wind turbines requires specialized techniques to cope with non stationarity. For this purpose advanced Operational Modal Analysis (OMA) techniques for cyclo-stationary vibration response may be used [5,,, ]. Although the case of color wind excitation may be dealt via certain methods [,, 5], the vibration response is not just cyclo stationary, but more generally non stationary. This is due to various reasons, such as the nature itself of the wind excitation, variations on the rotor speed in response to power demand and wind speed [6, 7, 7], and so forth. In order to account for the above and achieve robustness and high diagnostic performance, SHM may be performed based on time frequency or time scale representations of the non parametric [8, 9, ] or parametric [,, ] types. In this context empirical mode decomposition, Hilbert-Huang transform, and multi-resolution analysis via wavelet decomposition have been employed for the representation of non stationary vibration signals [, 5, 6]. Yet, the use of non parametric time frequency representations for vibration based SHM is characterized by a number of potential drawbacks, including the requirement for large volumes of data and inferior accuracy. Pattern recognition methods, including a feature extraction and selection stage, may be used to cope with the first one [8, 9]. Yet, this involves increased processing requirements during the baseline and inspection phases of the diagnosis operation. On the other hand, parametric methods, using Time dependent AutoRegressive Moving Average (TARMA) and Linear Parameter Varying ARMA (LPV ARMA) representations, are quite appealing for vibration based SHM, as they potentially offer superior accuracy and compact representations of the underlying dynamics [7, 8]. Previous studies by the present authors have confirmed the advantages of TARMA modeling for the analysis of the vibration response of operating wind turbines [6]. Besides, effective damage diagnosis techniques are already available for the case of FS-TARMA and FS-TARX (X designates the presence of a measurable exogenous excitation) models [9, 9, ]. Nonetheless, these methods are generally not well suited for cases characterized by significant levels of uncertainty. For cases with significant uncertainty Random Coefficient (RC) stochastic models potentially making use of Multiple Model (MM) type representations may be used [,, ]. An alternative approach would be to explicitly include cause and effect expressions for various uncertainties in the dynamics []. Yet, this requires measurable uncertainty sources, which is not always the case.

4 The problem of vibration based Structural Health Monitoring (SHM) for structures with time dependent dynamics under significant environmental and operational uncertainty has been addressed in our companion paper [], in which a proper diagnosis framework has been postulated. This is based on Gaussian Mixture Model Random Coefficient (GMM RC) representations of the non stationary random vibration response. The GMM representation may be thought of a collection of proper models, and is characterized by high flexibility and effectiveness in representing uncertainty, as well as simplicity of construction (estimation) under limited data records. The aim of the present study thus is an illustration and systematic, thorough, assessment of the GMM RC model based SHM framework for the case of an operating wind turbine. Earlier studies, reporting preliminary results, may be found in our conference papers [5, 6, 7]. The adopted GMM RC model based SHM framework includes three main entities: (i) Non stationary time dependent parametric models (either of the Functional Series FS or the Linear Parameter Varying LPV types) for representing the non stationary (time dependent) dynamics, (ii) a GMM RC representation for modeling an individual health state of the structure (also referred to as class) under uncertainty, and (iii) proper decision making. The rationale behind this formulation stems from the expectation that a GMM RC representation may best capture uncertainty from a limited set of baseline vibration responses in a simple and effective way. The illustration and assessment of the damage diagnosis framework is performed on an offshore 5-MW baseline wind turbine simulated via the FAST aeroelastic and structural dynamics simulation code [8], for which the average wind speed at each analysis interval is the external variable inducing uncertainty. Five types of damage, each one of various distinct levels, are considered. Damage diagnosis is based on a single vibration acceleration response signal measured at either the tower top (fore-aft and lateral directions) or on the blade (at 5% of the blade length from the root; flapwise and edgewise directions). In a first step the identification and analysis of the non stationary dynamics embedded in the vibration response of the wind turbine is illustrated through the use of FS TAR and LPV AR models. In a second step GMM RC model based damage diagnosis methods are formulated, optimized, and applied. Comparisons with a single model parameter based method introduced in [] and a Time Frequency Representation with Principal Component Analysis based method [, 9] are also performed. The rest of this paper is organized as follows: The specific problem of vibration based SHM for operating wind turbines is presented in section, including a brief presentation of the FAST simulation package and the damage/fault scenarios utilized. A brief overview of the GMM RC model based damage diagnosis framework is provided in section. Modeling and analysis of the non stationary random vibration response for distinct structural states via FS TAR and LPV AR models is provided in section. The results of damage diagnosis using the GMM RC model based framework using FS TAR and LPV AR models are provided in section 5, along with comparisons with state of the art methods. The conclusions from this study are finally summarized in section 6.. The Wind Turbine, the Damage/Fault Scenarios and the Sensors.. Wind turbine description and simulation The SHM problem considered in the study features an NREL (National Renewable Energy Laboratory) offshore 5-MW baseline wind turbine [], which is standard for component design, and aerodynamic, aeroelastic, structural and control system simulation and assessment. The NREL 5-MW wind turbine is a conventional three-bladed upwind variable-speed and variable blade-pitch-to-feather-controlled turbine. Its main properties are summarized in Table. Table : Main properties of the NREL 5-MW baseline wind turbine as shown in [] Property Value Rating 5 MW Rotor orientation, configuration Upwind, blades Control Variable speed, collective pitch Drive train High speed, multiple-stage gearbox Rotor, hub diameter, hub height {6,, 9} m Cut-in, rated, cut-out wind speed {,., 5} m/s Cut-in, rated rotor speed {6.9,.} rpm Rated tip speed 8 m/s Rotor, nacelle, tower mass {,,7.6} kg

5 The random vibration response of the wind turbine is obtained via simulation using the FAST (Fatigue, Aerodynamics, Structures and Turbulence) simulation code, which is an aeroelastic simulator capable of predicting the loads of two and three bladed horizontal axis wind turbines [8]. FAST uses up to DOFs to describe the wind turbine structure, including [8] - DOFs for the blades ( flapwise, - edgewise), up to DOFs for the rotor shaft ( for torsion, for the hinges before the first bearing, and for pure rotation), DOF to describe the tilt stiffness of the nacelle, and DOFs to describe the torsion of the tower and its displacements in the fore-aft and lateral directions. Although in actual operational conditions there are several variables inducing uncertainty in the wind turbine dynamics, for the present study the minute average wind speed is selected as the uncertainty inducing variable. More specifically, the value of the average wind speed used on a single simulation is drawn from a Gaussian distribution with mean 5 m/s and standard deviation m/s. Moreover, the input wind excitation is simulated using a Kaimal turbulence model with a power law wind profile type [7]. The turbulence model is adjusted so that an expected turbulence intensity value of % at a wind speed of 5 m/s is complied. The wind turbine response is simulated under the rated rotor speed of. rpm. Then, for each simulation, the wind turbine starts at the rated rotor speed with all the control systems on-line. The simulated period is of s and the vibration signals are sampled at 5 Hz. For each simulation, the portion from 5 s to 5 s of the obtained vibration signals, corresponding to 5 samples, is used for further analysis... The damage/fault scenarios Five types of damage (damage scenarios) are simulated, each one for four distinct levels. The simulated damages correspond to typical damages or malfunctions in wind turbines, as discussed in references such as [,,, ]. Damage A Increased mass at the tip of the third blade. Increased mass on the last % of the length of the third blade to simulate the effect of water filtration into the blade structure, or to partly simulate the effects of ice growing over the tip of the blade (see Figure (b)). Four levels of damage are simulated by increasing %, %, 6% and 8% of the blade mass density on the last % of the length of the third blade. Each damage level is equivalent to an increase of.7%,.7%,.% and.7% of the total mass of the blade. Damage B Stiffness reduction at the root of the third blade. Reduction of the stiffness of the root of the third blade in the edgewise direction, which simulates the effect of fatigue damage in the blade root, being one of the most common types of damage in blades (see Figure (c)). Four damage levels are simulated by decreasing 5%, %, 5% and 6% the blade edgewise stiffness from the blade root up to % of the distance to the tip of the blade. Damage C Stiffness reduction at the tower base. Reduction of the stiffness in the lateral direction at the base of the tower to simulate fatigue damage in the welding joints or screws in the base of the tower, where it is exposed to the highest loads. (see Figure (d)). Four damage levels are simulated by decreasing 5%, %, 5% and % the tower stiffness in the lateral direction, from the base up to % of the tower height. Damage D Yaw error. Error in the yawing mechanism. Four damage levels are simulated by inducing,, 6 and 8 degree errors from the upwind position in the yawing system. This type of problem derives from a malfunction in the yawing control system and rapidly increases the fatigue loads in the whole structure. However, this is not a structural damage, but instead is a failure that may lead to rapid fatigue damage in other structural components. Damage E Reduction of the damping coefficient in the rotor low speed shaft. Reduction of the damping coefficient in the low speed shaft (LSS) to simulate fatigue damage in the transmission system. Four damage levels are simulated by decreasing 5%, %, 5% and 6% the damping coefficient of the LSS... The sensors and the vibration response signals For each damage type and level, as well as for the healthy state, realizations of the wind turbine response are simulated using distinct seeds for the generation of the turbulence time series. Measurements from four virtual accelerometers are considered. These are located as depicted in Figure (a), that is two at the tower top in the fore-aft and lateral directions, and two more at 5% of the distance from the root to the tip of the third blade in the edgewise and flapwise directions. Besides, the instantaneous angular position of the rotor (azimuth) is also measured. 5

6 Sensor location Increased mass - blade tip Reduced stiffness Tower base Tower top Lateral Blade Edgewise Tower top Fore-aft Blade Flapwise % blade length (b) Reduced stiffness - blade root % blade length (a) (c) % tower height (d) Figure : Location of the virtual sensors on the wind turbine structure and depiction of the types of damage used in the experiment: (a) Four virtual sensors are located on the structure: two at the tower top in the fore-aft and lateral directions, and other two about 5% of the length from the root of the rd blade in the flapwise and edgewise directions; (b) Damage type A: increased mass in the last % of the length of the rd blade; (c) Damage type B: the stiffness at the root of the rd blade in the edgewise direction is reduced; (d) Damage type C: the stiffness at the base of the tower is reduced. In Figure Welch estimates of the Power Spectral Density (PSD) of the vibration response signals obtained for simulated realizations of the wind turbine for the healthy state and different sensor locations are shown. The Welch spectral estimates are obtained based on Gaussian windows of 96 samples, 89 samples overlap based on 8 samples signal length. The plots also show the main natural frequencies derived from periodic linearization analysis for the rated rotor speed on the same wind turbine model by means of FAST, as shown in [] (see also Table ). As may be seen from the estimated spectra, the vibration response is characterized by complex dynamics, where the presence of other frequency modes, apart from those predicted form the aforementioned analysis, is observed. As shall become evident via the non-stationary analysis in the sequel, most of these modes are related to frequency and amplitude modulations stemming from the time-varying dynamics. Furthermore, it is observed that the PSDs feature large variability among different realizations, which is a consequence of wind speed variations and turbulence. Table : Natural frequencies of the NREL Offshore 5-MW Baseline Wind Turbine obtained via periodic linearization analysis []. Mode Description Frequency [Hz] st Tower Fore-Aft. st Tower Lateral. st Drive Train Torsion 5 st Blade Asymmetric Flapwise Yaw 66 5 st Blade Asymmetric Flapwise Pitch st Blade Collective Flap 99 7 st Blade Asymmetric Edgewise Pitch.79 8 st Blade Asymmetric Edgewise Yaw nd Blade Asymmetric Flapwise Yaw.97 nd Blade Asymmetric Flapwise Pitch.9 nd Blade Collective Flap.5 nd Tower Fore-Aft.9 nd Tower Lateral.96 6

7 Figure : Comparison of the Welch spectral estimates on simulated vibration response signals from the healthy state at different sensor positions. The thick lines depict the sample average PSD, whereas the dotted lines indicate individual realizations. Red lines indicate the main natural frequencies of the wind turbine as provided by [].. Brief Overview of the GMM RC Damage/Fault Diagnosis Framework and Methods Let y[t] represent the vibration response of the structure, with t =,,...,N being the discrete time and T s the sampling period. The structure may operate in one of several health states v={o,a,b,c,...}, where o stands for the healthy state, while a, b, c and so on represent various types of damages or faults. Damage (or fault) diagnosis refers to the problem of determining the current state of the structure given a newly acquired vibration response signal (the test signal) y u = [ y u [] y u [] y u [N] ] T and a model to represent the vibration response at each state v. In the Gaussian Mixture Model Random Coefficient (GMM RC) approach, each one of these models correspond to a proper GMM representation for each structural state. A GMM is constructed as a set of conventional models M={m v,l }, with m={θ v,l,s} and l =,...,L, all of them corresponding to the structural state v []. In this section the definition and construction of GMM RC representations using LPV AR elementary models, as well as related damage diagnosis tests based on the marginal likelihood and the Kullback Leibler divergence are briefly presented... The elementary models In the case of non-stationary vibration response, the type of Linear Parameter Varying AR (LPV AR) models may be used as the elementary model in the GMM RC representation, whenever the non stationary dynamics are governed by a (measurable) scheduling variable β[t] R. An LPV AR(n a ) [pa,p s ], with n a representing the AR order, and p a and p s representing the dimensionality of the AR and innovations variance functional subspaces is defined as follows [7, ]: y[t]= a i (β[t]) p a j= σ w (β[t]) p s j= n a i= a i (β[t]) y[t i]+w[t] w[t] NID (,σ w(β[t]) ) (a) a i, j G ba( j) (β[t]) F AR = { } G ba() (β[t]),...,g ba(pa) (β[t]) { } s j G bs( j) (β[t]) F σ w = G bs() (β[t]),...,g bs(ps) (β[t]) (b) (c) 7

8 where a i (β[t]) are the AR parameters, w[t] a zero mean NID innovations with variance σw (β[t]), F stands for a functional subspace of the respective quantity, b a( j) ( j =,..., p a ), b s( j) ( j =,..., p s ) are the indices of the specific basis functions that are included in each functional subspace, while a i, j and s j stand for the coefficients of projection of the AR parameters and innovations variance. Functional Series Time-dependent AR (FS TAR) models may be thought of as a special case when β[t] t. The LPV AR modelm={θ,s} is fully determined by the parameter vector θ and the structures, each defined as: θ = [ a, a, a na,p a s s s ps ] T n S={n a, b a, b s } () where n=n a p a + p s and b a = [ ] T b a() b a(pa ) p a, b s = [ ] T b s() b s(ps ) p s... Construction of the GMM RC representation The identification of single FS TAR models by means of maximum likelihood methods is broadly discussed in [7, 8]. Nonetheless, for the purpose of constructing GMM RC representations, it is necessary to obtain a common model structure that would effectively represent a large set of vibration responses. Therefore, the estimation of the parameter vector and the posterior evaluation of the model performance are carried out in different non-intersecting training and validation segments, denoted as: [ ] y (tr) m y m = y m (val) [ ] β (tr) β m = m β (val), m=,,...,m m where M is the total number of vibration response signals available for each structural state on the baseline phase, y m is each one of the available vibration response vectors, while y (tr) m and y m (val) are the segments used for parameter estimation (training) and performance evaluation (validation), respectively. Then, the selection of the best model structure is made in terms of the empirical risk [, Ch.], [5, Ch.]: R emp (S)= M M N val m= t= ( y (val) m [t] ŷ (val) ( y (val) m ) m [t t ] ) () [t] where N val is the length of the validation segment, ŷ m (val) [t t ] are the one-step-ahead model predictions evaluated on the validation segment employing the parameter estimates ˆϑ m obtained from the training segment []. The use of the empirical risk for model structure selection is important to achieve a representation that does not over-fit the training data and is capable to generalize to unseen data sets. Furthermore, the empirical risk tends to favor models with low complexity, thus enabling the selection of the most compact and efficient representation [, Ch.], [5, Ch.]. Nevertheless, criteria like the RSS (Residual Sum of Squares), RSS/SSS (Residual Sum of Squares over the Series Sum of Squares) and the BIC (Bayesian Information Criterion) may also be used to guide the decision of the best model structure [7]. A second issue of importance in the construction of a GMM RC representation is to decide the number of models to keep in the representation. To understand this, consider first that a set of M vibration response signals have been obtained from the structure from a single structural state with the purpose of constructing a respective GMM. Then, it must be assessed whether all the corresponding M models are actually necessary. For this purpose a measure determining the similarity between two models, like the Kullback Leibler (K-L) divergence can be used. The K-L divergence on the case of LPV AR models is actually of the form []: D KL (m a,m b )= N + d M(m a,m b )+ d s(m a,m b ) () ( dm (m a,m b )=(ϑ b ϑ a ) T Σ θ b (ϑ b ϑ a ) ds N σ (m w(a) (β a [t]) a,m b )= σw(b) (β a[t]) lnσ w(a) (β ) a[t]) lnσw(b) (β a[t]) The K-L divergence is a measure of the information lost when the model m b is used to represent the model m a. Thus, a large K-L divergence is an indicator that models m a and m b are very different between themselves and in consequence represent different dynamics in the vibration response. Likewise, if both models m a and m b are t= 8

9 similar, hence representing similar dynamic behaviors, then their divergence should be close to zero. In this sense, if two models appear to have a low divergence value, then one of them may be dropped from the representation. This plucking process may be repeated until a certain performance measure is met, such as the performance on damage diagnosis... Damage diagnosis based on GMM RC representations The GMM RC Marginal Likelihood (GMM-RC-ML) based damage diagnosis method attempts to determine the presence and type of damage in the structure in terms of the marginal likelihood of the test signal y u given the class v, namely p(y u v). The GMM-RC-ML method thus assigns the test signal to the class (structural state) with the highest marginal likelihood. The damage detection and identification tests derived from this concept, as presented in [], are summarized in Table. Two versions are considered according to the type of approximation used to compute the marginalizing integral, namely: (i) the GMM-RC-ML-sum method, which uses the finite sum approximation of the marginal likelihood; (ii) the GMM-RC-ML-max method, which uses the maximum approximation of the marginal likelihood. The weights π v,l and the threshold ρ lim are the variables determining the final performance of the method. As shall be shown in Section 5., an optimization method is used in order to find the best values for these variables. Table : The GMM RC Marginal Likelihood-based (GMM-RC-ML) method Damage detection: Given a test signal y u and the GMM of the healthy state M o = {m o,l },l =,...,L, with m o,l = { ˆθ o,l,s} and corresponding weights π o,l and a damage detection threshold ρ lim :. Compute the parameter vector estimate ˆθ u and its likelihood p(y u ˆθ u );. Compute the likelihoods of the individual models in the GMM p(y u ˆθ o,l ) for all l =,...,L;. Evaluate the marginal likelihood with one of the following approximations (i) (ii) p(y u o)= L l= π o,l p(y u ˆθ o,l ) Finite sum approximation (GMM-RC-ML-sum) p(y u o)=max l π o,l p(y u ˆθ o,l ) Maximum approximation (GMM-RC-ML-max). Damage detection test ρ d (y u )= p(y u ˆθ u ) p(y u o) ρ lim Otherwise The structure is damaged The structure is healthy where ρ d (y u ) is the statistical quantity for damage detection. Damage identification: Given a test signal y u and the GMMs of the damaged states M v ={m v,l } for l =,...,L and v={a,b,c,...}, withm v,l ={ ˆθ v,l,s} and corresponding weights π v,l :. Compute the likelihoods of the individual models of each one of the GMMs p(y u ˆθ v,l ) for all l =,...,L and v={a,b,c,...};. Evaluate the marginal likelihood with one of the following approximations (i) (ii) p(y u v)= L l= π v,l p(y u ˆθ v,l ) Finite sum approximation (GMM-RC-ML-sum) p(y u v)=max l π v,l p(y u ˆθ v,l ) Maximum approximation (GMM-RC-ML-max). Damage identification test ˆv= argmax p(y u v) v={a,b,c,...} The GMM RC Kullback-Leibler divergence (GMM-RC-KL) based damage diagnosis method attempts to determine the presence and type of damage/fault in the structure by comparing the likelihoods associated with the test 9

10 model and the models in the GMM of all the reference states. This comparison is made in terms of the Kullback- Leibler (K-L) divergence measure in Equation (). The damage/fault detection and identification tests derived from the K-L divergence are referred to as GMM-RC-KL methods and are summarized in Table. The fixing parameters of the GMM-RC-KL method are the weights λ v,k and the threshold d lim (only for the damage detection test). The optimization of these parameters for the maximization of the damage diagnosis performance is explained in Section 5.. Table : The GMM RC Kullback-Leibler divergence-based (GMM-RC-KL) method Damage detection: Given a test signal y u and the GMM of the healthy state M o = {m o,l }, with m o,l = { ˆθ o,l,s} and weights λ o,l for all l =,...,L, and a damage detection threshold d lim :. Compute the ML estimate ˆθ u and its corresponding covariance matrix Σ θ u ;. Evaluate the Kullback-Leibler divergence between the test model and all the models in M o as shown in Equation ().. Damage detection test min l=,...,l λ o,l D KL (m u,m o,l ) d lim Otherwise The structure is healthy The structure is damaged Damage identification: Given a test signal y u and the GMM of the damaged states M v ={m v,l }, with m v,l ={ ˆθ v,l,s} and weights λ v,l, for all l =,...,L and v={a,b,c,...}:. Compute the ML estimate ˆθ u and its corresponding covariance matrix Σ θ u ;. Evaluate the Kullback-Leibler divergence between the test model and all the models in M v, v = {a,b,c,...} as shown in Equation ().. Damage identification test ( ) ˆv = min min π v,l D KL (m u,m v,l ) v {a,b,c,...} l=,...,l. Modeling and Analysis of the Vibration Response Signals.. LPV-AR and FS-TAR based modeling Complete LPV AR structures are estimated via the Multi-Stage Maximum Likelihood (MS-ML) method [7, ], where the parameters and innovations variance are expanded on the functional subspace of the trigonometric type defined as follows [6, 7]: G (β[t])=, G k (β[t])= sin(kβ[t]), G k+ (β[t])= cos(kβ[t]), k=,,..., p a (5) only for p a odd (to keep the same number of sine and cosine components), where β[t] is the π modulus instantaneous angular position of the rotor at time t. The functional basis is defined in this form to account for the variability in the instantaneous angular speed of the rotor, which is constantly changing according to the upcoming wind. Simpler FS TAR models are also considered, this time utilizing a trigonometric basis function synchronized with the rated speed of the rotor for the expansion of the FS TAR parameters. Thus, the functional expansion basis is of the form [6, 5]: G [t]=, G k [t]=sin(kω o t), G k+ [t]=cos(kω o t), k=,,..., p a (6) also only for p a odd, where ω o = π f rot / f s and f rot =./6 =. Hz is the rated speed of the rotor in Hertz. The selection of the model structure is made in two stages: (i) Selection of n a, where FS TAR and LPV AR models are estimated with fixed p a = p s = and with increasing n a in the range 6 to 5; (ii) Selection

11 of p a and p s, where FS TAR and LPV AR models are estimated for the best n a in the previous step and with p a = p s ={,5,,9}. The selection of the model structure is guided by the empirical risk defined in Equation (). The RSS/SSS is also used as reference. Full details of parameter estimation and model order selection are provided in Table 5. Table 5: Details of the identification process for FS TAR and LPV AR models. Stage Details Vibration signal features: Total length N = 5 samples, initial samples used for training (N tr ), last samples used for validation (N val ) Estimation: Multi Stage Maximum Likelihood (MS-ML) method with instantaneous innovations variance estimate [7, ]. Stopping criteria: tolerance in the change of the parameters, tolerance in the change of the likelihood, maximum number of iterations. Model order selection: Estimate FS TAR/LPV AR(n) [pa,p s ] models with n=6,..., and p a = p s =. Basis order selection: Estimate FS TAR/LPV AR(n a ) [p,p] models with best n a of previous step and p a = p s =,,...,5. Figure displays the realization based empirical risk and sample average RSS/SSS curves obtained at the two stages of model order selection for vibration responses of the healthy structure measured at each one of the considered sensors. The empirical risk curves obtained during the model order selection procedure, shown in Figure (a) display clear minima according to each sensor. The minimum values are indicated with arrows on each plot. The curves obtained for the tower top in the lateral direction and at the blade in the edgewise direction also show other minimum values for higher orders, but the improvement is minimal compared with the selected order. The RSS/SSS curves have a similar behavior to the one displayed by the empirical risk, although these curves are continuously decreasing for increasing model order. Also, the empirical risk curves obtained for FS TAR models are slightly lower than those of the LPV AR models, while the RSS/SSS curves show the opposite, thus favouring LPV AR models. Figure (b) shows the results of basis order selection procedure, based on the best model orders obtained in the previous step. In this occasion, the empirical risk and RSS/SSS curves show opposite behaviors, where the RSS/SSS favors higher orders, while the empirical risk results suggest selecting lower orders. Thus, the decision in this step is not as simple as in the previous one. The basis dimensionality is ultimately selected as p a = p s = 7, so that the representation basis consists of the constant plus the first three sine/cosines. This basis dimensionality is selected for following reasons: (i) to reach a compromise between the empirical risk and RSS/SSS curves; (ii) to obtain a representation basis that may account for the first three harmonics of the blade rotation frequency; (iii) although some basis functions may not improve significantly the modeling performance at the healthy state of the structure, these may be helpful for the modeling unbalances found under certain damage types. A summary of some performance figures, including the empirical risk, RSS/SSS, log-likelihood, Bayesian Information Criterion (BIC), Condition Number (CN) of the inverted matrices in the computation of the parameter estimates, and Samples Per Parameter (SPP) criterion, of the selected model structures is shown in Table 6. Table 6: FS TAR/LPV AR model structure identification results for wind turbine vibration response simulations of the healthy state. Sensor Model R emp RSS/SSS ln p(y θ) BIC ( ) ( ) ( ) ( ) log CN SPP Tower top fore-aft FS TAR(5) [7,7] LPV AR(5) [7,7] Tower top lateral LPV AR(8) [7,7] FS TAR(8) [7,7] Blade flapwise LPV AR(7) [7,7] FS TAR(7) [7,7] Blade edgewise FS TAR(6) [7,7] LPV AR(6) [7,7]

12 (a) -. Empirical Risk: FS-TAR Empirical Risk: LPV-AR RSS/SSS: FS-TAR RSS/SSS: LPV-AR -. n a = 5 n a = 8 log Remp log Remp Tower top - fore aft, p a = p s = -. Tower top - lateral, p a = p s = log Remp (b) - n a = Blade - flapwise, p a = p s = Order n a -. log Remp -.6 n a = Blade - edgewise, p a = p s = Order n a -. log Remp Tower top - fore aft, n a = 5 log Remp Tower top - lateral, n a = log Remp Blade - flapwise, n a = Order p a log Remp Blade - edgewise, n a = Order p a Figure : Empirical risk and sample average RSS/SSS obtained at the two stages of model order selection for FS TAR and LPV AR structures for realizations of the simulated vibration responses of the healthy structure at each one of the sensors: (a) Selection of the model order with fixed p a = p s = ; (b) Selection of the basis order using best n a obtained in the previous step... Model based analysis of the dynamics The dynamic characteristics of the vibration response of the wind turbine are analyzed by means of the Time- Variant Power Spectral Density (TV-PSD) and the Spectral Correlation [, 7, 6]. A brief summary of these quantities and their calculation based on the parameters of a TARMA model are provided in Appendix A. The LPV ARMA model based Melard-Tjøsteim PSD is computed on 5 points over the frequency range [, 8] Hz (thus the frequency resolution is f o = 8/5=5.65 [Hz]), and evaluated for a single period of rotation of the wind turbine (thus β[t]=ω a t, with t =,,..., and ω a = π/). Likewise, the LPV ARMA model based spectral correlation is evaluated in the frequency range [, 8] [Hz] and for p up to p cyclic frequencies, where p =. [Hz] is the rotor frequency. Figure provides a comparison of the parametric Melard-Tjøsteim PSDs obtained from the sample average LPV AR models of the vibration response of the healthy state at each one of the considered sensors. Additionally, Figure 5 shows the parametric Melard-Tjøsteim PSD estimates derived from the sample average LPV AR models of the vibration response of the wind turbine at the tower top lateral direction at different structural states at the highest level of damage. The Melard-Tjøsteim PSD estimates from the vibration response at the healthy state show

13 the main frequency modes found in the stationary PSD estimates shown in Figure. Some of the unexplained frequency modes found in the stationary PSD are evidenced in the Melard-Tjøsteim PSD estimates as being part of a single mode modulated in amplitude or frequency. The presence of damage modifies the time-dependent behavior of some modes, according to the damage type. Figure 6 shows the spectral correlation derived from the sample average LPV AR model for the healthy and all the simulated damage scenarios at the highest damage level. The spectral correlations at p (corresponding to the constant terms in the PSD along time) and p (corresponding to modulations induced by the blade-toblade period) remain almost unchanged for most damage types at all the sensors. Only damages of types B and D introduce deviations from the normal state, which manifest differently at different sensor locations. Nonetheless, the most important changes in the dynamics are evident at p (related to modulations induced by the period of rotation of the rotor). This is due to the unbalance introduced by the presence of damage on the dynamics of the vibration response. It can also be seen that damage modifies the natural frequencies of some modes, as evidenced for example by the vibration response of the structure under damage A at the blade in the lateral direction (Figure 6(d)). (a) (b) (c) (d) Figure : Comparison of the parametric Melard-Tjøsteim TV-PSD estimates derived from the sample average LPV AR models of vibration responses of the wind turbine tower measured at different sensor locations: (a) tower top fore aft direction, (b) tower top lateral direction, (c) blade flapwise direction, (d) blade edgewise direction. 5. Damage and Fault Diagnosis Results Diagnosis results obtained by the GMM RC-based damage detection and identification methods, namely the approach based on the Marginal Likelihood (GMM-RC-ML sum and GMM-RC-ML max damage detection and identification methods) and the approach based on the Kullback-Leibler divergence (GMM-RC-KL method), are now presented. For the sake of comparison, the Single FS TAR/LPV AR Model Parameter-based (SMP Single Model Parameter based method) described in Appendix B is evaluated as well. The damage diagnosis methods are built upon the LPV AR and FS TAR models obtained in the previous section.

14 (a) (b) (c) (d) (e) (f) Figure 5: Comparison of the parametric Melard-Tjøsteim TV-PSD estimates derived from the sample average LPV AR model of vibration responses of the wind turbine tower at the tower top in the lateral direction for different structural states: (a) healthy, (b) damage A: increased mass blade tip, (c) damage B: decreased stiffness blade root, (d) damage C: decreased stiffness tower base, (e) damage D: yawing error, (f) damage E: decreased damping in the low speed shaft. 5.. Description of the optimization of the damage diagnosis methods via cross-validation In order to ensure the best performance of the methodologies, the fixing parameters of each one of the damage diagnosis methods are optimized within a cross-validation procedure. Cross-validation is a model assessment methodology in which the adjustment of the method (training) and its posterior performance evaluation (validation) are carried out on independent sets of data, with the purpose of assessing the behavior of the method in unseen sets of data. Then, the damage diagnosis methods utilized in this part of the work are evaluated and optimized within a -fold cross-validation approach, in which the whole data set is divided into different non-intersecting subsets.

15 (a) (b) (c) (d) Figure 6: Parametric estimates of the spectral correlation derived from the sample average LPV AR models of vibration response signals of the wind turbine from the sensors located at: (a) tower-top fore-aft direction, (b) tower-top lateral direction, (c) blade flapwise direction, (d) blade edgewise direction. Each column shows the spectral correlation S yy (α, f), with f = n f o, where f o = 5.65 [Hz] is the Fourier frequency, and α = k p and k=,...,, where p=. [Hz] is the frequency of rotation of the rotor. In the SMP method, since the training is based only on a single model, a different method to define the training and validation subsets must be followed. A summary of the training and validation subsets used for evaluation of the damage diagnosis methods is provided in Table 7. The performance of the damage diagnosis methods is evaluated in terms of the Receiver Operating Characteristic (ROC) curves and its corresponding Area Under the ROC Curve (AUC) [7]. ROC curves are constructed by plotting the True Positive Rate (TPR) vs. the False Positive Rate () of a detector or binary classifier as the decision threshold (ρ lim or d lim ) increases. The ideal ROC curve passes through the point (,), indicating zero and TPR equal to the unity, whereas a ROC curve moving through the diagonal TPR= indicates a 5/5% chance that the method decides for any of the classes. The ROC-AUC summarizes the information of the ROC curve in a single variable, and can take values between and, where the ideal performance is achieved when AUC =, while the worst is at AUC =.5. The AUC can be also associated to the probability that the detector or 5

16 Table 7: Definition of the number of records used for training and validation in the cross-validation for evaluation of damage diagnosis in single model and GMM methods. Single model method Damage detection Damage identification Structural state Training Validation Total Training Validation Total Healthy 99 Damage level 99 Damage level 99 Damage level 99 Damage level 99 Total per damage type 96 Total healthy and 5 damage types GMM methods Damage detection Damage identification Structural state Training Validation Total Training Validation Total Healthy 9 Damage level 9 Damage level 9 Damage level 9 Damage level 9 Total per damage type 6 Total healthy and 5 damage types 9 8 classifier will rank higher an sample from the target class (damaged state) than a sample from the reference class (healthy state), and in that sense, the AUC is also a related to the probability of detection of the method, regardless of the value of the decision threshold. Thus, the optimization of the weights of the damage detection methods is made in terms of the maximization of the ROC-AUC, defined as follows: π = argmax AUC(π), π s.t.: L l= π o,l =, and π o,l (7) where π = [ π o, π o, π o,l ], and AUC(π) is the ROC-AUC obtained with the GMM-RC-ML damage detection method (in its two versions) using weights π. Due to the complexity of the optimization problem and the possibility of several local maxima, a global optimization method is appraised for the search of optimal values of the weights. Therefore, the Generalized Pattern Search (GPS) method is selected for this task, which consists of the following steps [8]: (i) The GPS algorithm begins at a given starting point π. (ii) A mesh of points is created around the current point at the K points determined by the direction vectors [ ], [ ],..., [ ] and their negative complements. (iii) In a successful poll the GPS algorithm finds a point that improves the cost function, and then selects it as the next point. Afterwards, the algorithm duplicates the size of the mesh and polls again. (iv) In an unsuccessful poll the GPS algorithm does not find a point that improves the cost function. In that case, the current point is selected again as the next point, the mesh size is shrunk by a factor of two, and polls again. (v) The algorithm runs until a maximum number of iterations is reached, or until the tolerance values on the mesh size, the objective function or the change in the parameter size is met. The advantage of the GPS algorithm in the presently analyzed optimization problem is that it can search through several basins of attraction, thus improving the chances of finding a global minimum. The settings of the GPS algorithm used for the optimization problem in Equation (7) are summarized in Table 8. The computation of ROC curves comprises the computation of several binary tests between a reference and a target class, which may be cumbersome when the number of evaluated damage types is large, as may be the case of 6

17 Table 8: Settings used for the optimization of the damage diagnosis methods with the GPS algorithm. Method s implementation patternsearch (MATLAB s Global Optimization Toolbox) Stopping criteria Maximum number of iterations: L Tolerance value of the mesh size: 6 Tolerance value of the objective function: 6 Tolerance value of the change of the parameter size: 6 Initial values GMM-RC-ML method Damage detection: equal weights π o,l = /L, L=9; Damage identification: equal weights π v,l = / GMM-RC-KL method Damage detection: equal weights λ o,l = /L, L=9; Damage identification: equal weights λ v,l = / in a damage identification problem. Instead, the performance of the damage identification method can be measured in terms of other performance variables, such as the Correct Identification Rate, defined as the ratio between the number of correctly identified damages and the total number evaluated cases. The second option is preferred here for the evaluation of the damage identification methods, since it facilitates the optimization of the free parameters. Then, the optimization of the damage identification methods is based on the maximization of the correct identification rate. The fixing parameters in the damage identification methods are the weights of the GMM representations of each class, namely π v,l for all structural states v={a,a,...,a,b,b...,b,...,e,e,...,e } and l =,,L. Since the number of the weights and scale parameters is ( records of damages damage levels 5 damage types). Therefore, the use of any optimization method for this elevated number of fixing parameters is computationally prohibitive. For this reason, optimization is attempted only for the weights, and instead of optimizing individually the weights of each model of the GMMs, it is considered that all the weights of a class GMM are equal, and only differ among classes. Thus, only weights are optimized, each one for a single damage type and level. The optimization of the weights of the GMM representations is also performed with the GPS algorithm, using the same settings as in the optimization of the GMM-RC-ML damage detection method summarized in Table 8. The performance of the damage identification methods can be further analyzed by means of confusion matrices. Each entry (i, j) of a confusion matrix contains the number of instances belonging to class i identified as class j by the classifier (damage identification method). Therefore, the confusion matrix would tend to be diagonal if the method performs perfectly, while values out of the main diagonal of the matrix indicate that the method confuses different damage types. Hence, the confusion matrix is a tool that can be used to analyze the behavior of the damage identification method at individual damage types. 5.. GMM-RC-ML based methods The results obtained with the GMM-RC-ML damage diagnosis methods are, for both detection and identification, presented in the sequel Detection results The GMM-RC-ML sum and GMM-RC-ML max damage detection methods described in the first part of Table are used for the detection of damage in the wind turbine. The fixing parameters of this method in its both versions are the damage detection threshold ρ lim and the weights π o,l for l =,...,L, with L = 9 (according to the data distribution on the fold cross validation shown in Table 7). Initially, the values of the fixing parameters are set using equal weights π o,l = /L. Figure 7(a) shows the distribution of the statistical quantity ρ d (y u ) = p(y u ˆθ u )/p(y u o) used by the GMM-RC-ML-sum damage detection method on a single sensor of the wind turbine (tower top in the lateral direction) with the selections for the fixing parameters described before. Larger values of ρ d (y u ) are an indication of a larger deviation from the healthy state. Accordingly, the plot evidences the increasing values of ρ d (y u ) when the structure is in a damaged state, especially as the level of damage increases in the damages of types C, D and E. Figure 8(a) shows the ROC curves obtained with the GMM-RC-ML-sum damage detection method on the vibration signals from all the sensors and separated per individual damage types, using the selections described before for the fixing parameters of the method. From the obtained curves, it can be seen that the GMM-RC-MLsum method is insensitive for some combinations of sensor and damage types, since the ROC curves are running on the diagonal. However, very satisfactory detection performance can be found for all damage types at sensors. 7