This study aims to investigate the performance of a data-driven methodology for quantifying damage based on the use of a metamodel obtained from the Polynomial Chaos-Kriging method (PC-Kriging). The investigation seeks to quantify the severity of the damage, described by a speciﬁc type of debonding in a wind turbine blade as a function of a damage index. The damage indexes used are computed using a data-driven vibration Structural Health Monitoring (VSHM) methodology. The blade’s debonding damage is introduced artiﬁcially, and the blade is excited with an electromechanical actuator that introduces a mechanical impulse causing the impact on the blade. The acceleration responses’ vibrations are measured by accelerometers distributed along the trailing and the wind turbine blade. A metamodel is formerly obtained through the PC-Kriging method based on the damage indexes, trained with the blade’s healthy condition and four damage conditions, and tested with the other two damage conditions. The PC-Kriging manifests promising results for capturing the proper trend for the severity of the damage as a function of the damage index. This research complements the damage detection analyzes previously performed on the same blade. Keywords Structural Health Monitoring, wind turbine blades, damage quantiﬁcation, damage features, data-driven metamodel, Polynomial Chaos-Kriging Introduction Currently, the society aims for a future where the generation of energy is clearer. Thus, there is an expansion in offshore wind turbines, making studies in this area necessary. Additionally, the maintenance of wind turbine blades involves, in most part, methods based on visual inspection, which can be dangerous and expensive (Garc ´ıa and Tcherniak 2019). Therefore, a system for monitoring wind turbines’ conditions is of great industrial interest, demanding further development of methodologies for detecting and quantifying damage to these structures (Ciang et al. 2008). Structural Health Monitoring (SHM) methods are used to diagnose and extract meaningful information about the health from a structure of interest, based on the measured data from sensors distributed and permanently installed along the structure (Larrosa et al. 2014). SHM techniques have four functional levels of the classiﬁcation proposed by Rytter (1993): Level 1 – damage detection, Level 2 – damage location, Level 3 – damage quantiﬁcation, and Level 4 – remaining useful life estimate. In the literature, the ﬁrst three levels are also categorized as diagnosis and the last as prognosis. This work focuses on damage quantiﬁcation to complement a methodology for damage detection by Garc ´ıa and Tcherniak (2019) and thus provide a full damage diagnosis of debonding wind turbine blades. The contribution concerns the quantiﬁcation of the damage, obtaining the severity by the damage indices, assuming the inherent uncertainties to obtain a robust damage quantiﬁcation. The damage detection methodology used in Garc ´ıa and Tcherniak (2019) followed the idea of Vibration-basedStructural Health Monitoring (VSHM), which measures vibration responses from a artiﬁcial excitation. Thus, the structure’s health/current state can be analyzed by monitoring its vibration response changes. The methodology was applied to an SSP 34 m wind turbine blade. Conventional classiﬁcation of VSHM methodologies is the division between model-based (Maes et al. 2016) and non-model- based (or data-driven) methods (Love 2002; Avenda ˜no- Valencia et al. 2015). In the methodology of this work, a data-driven approach is used. Non-model-based methods depend exclusively on the data measured from the structure under study. These methods also involve the construction of a model, but that model is based on data rather than numerical or analytical models (Avenda ˜no-Valencia et al. 2015). The measurements in the structure’s initial state are considered reference states, where these observations can be compared with the structure measurements in operation. Any deviation of the new observations from 1UNESP - Universidade Estadual Paulista, Departamento de Engen- haria Mec ˆanica, Ilha Solteira, SP , Brasil 2IFMS - Instituto Federal de Mato Grosso do Sul, Tr ˆes Lagoas, MS, Brasil 3Universidade do Estado do Rio de Janeiro, Rio de Janeiro, RJ, Brasil 4University of Edinburgh, School of Engineering, Institute for Infrastruc- ture and Environment, Scotland, UK Corresponding author: Bruna Pavlack, UNESP - Universidade Estadual Paulista, Departamento de Engenharia Mec ˆanica, Av. Brasil, 56, Ilha Solteira, 15385-000, SP , Brasil. IFMS - Instituto Federal de Mato Grosso do Sul, Tr ˆes Lagoas, MS, Brasil Email: bruna.pavlack@ifms.edu.br Prepared using sagej.cls [Version: 2017/01/17 v1.20] 2 Journal Title XX(X) the reference state might be considered an indication of damage. In Tcherniak and Mølgaard (2015), an analysis of an unsupervised methodology was carried out that combined the correlation between the signals measured from different accelerometers. This study demonstrated that methodologies based on data-driven techniques have a big potential for detecting damage on large structures. Currently, many data- driven methodologies are being developed; for example, in Soize and Orcesi (2020) a machine learning approach was presented using only an experimental database, consisting of a small number of records, to detect changes in the rigidity of engineering structures. The damage quantiﬁcation is not yet extensively addressed by the scientiﬁc community, but it is crucial to improve the safety and useful life of structures, and thus motivating the scientiﬁc community to develop some damage quantiﬁcation methods. Paix ˜ao et al. (2020) used AutoRegressive (AR) models on Lamb wave signals measured on a composite material plate to calculate damage-sensitive features for calculating indexes using the Mahalanobis square distance. Quantiﬁcation was achieved by training a deﬁned curve, using cubic spline functions to predict the delamination area. Da Silva et al. (2019) executed a similar idea when manipulating the possibility of extrapolating these trend curves to future prognostic states when the delamination grows in the same place and with a similar effect. These studies demonstrated that the resources that use AR models are precisely correlated with the structural state and with a smooth tendency that allows the use of cubic spline functions. Many studies have addressed experiments in wind turbines in the literature, but there is a lack of methodologies to quantify the damage to them. One of the difﬁculties in quantifying wind turbine blades’ damage is that these structures work under variability (temperature, climate, etc.). Therefore, it is of great importance to consider uncertainty quantiﬁcation. A method with valid results in obtaining uncertainty quantiﬁcation models is the Polynomial Chaos Expansion (PCE). The PCE method with application in engineering was introduced in Ghanem and Spanos (1990). They proposed a new method using convergent orthogonal expansion for solving problems involving material variability. The material property was modeled as a random ﬁeld. The results found had a good agreement with the results obtained through a Monte Carlo simulation. In Ghanem and Spanos (1991), the PCE was used to quantify the uncertainty applied to some problems involving mechanical systems. The use of different types of orthogonal polynomials to represent non-Gaussian processes was introduced in Xiu and Karniadakis (2002), which presents a method for solving stochastic differential equations based on Galerkin projections on a polynomial chaos basis. They represented stochastic processes based on an Askey family of orthogonal polynomials, that reduced the dimensionality of the system and led to the exponential convergence of the error. This new methodology, which is reviewed in Xiu (2010), presents satisfactory results in terms of computational cost and precision. One of the signiﬁcant advantages seen in using the PCE is its rapid convergence and expressing the ﬁnal solution as a random process and not just as a set of statistics. Bogoevskaet al. (2017) mentions that operational structures such as wind turbines have complex dynamic behavior that challenges the applicability of existing SHM strategies for condition evaluation. Thus, Bogoevska et al. (2017) proposes a structure based on the symbiotic treatment of environmental/operational variables acting on the structure’s vibration response. A probabilistic model of the PCE was used for uncertainty quantiﬁcation in the identiﬁed structural performance indicators. Avenda ˜no-Valencia et al. (2017) emphasized that effective fatigue monitoring and prediction algorithms for structures such as wind turbines require an accurate representation of their dynamic response in the short and long term scale. The long term can be achieved in a computationally efﬁcient way through the use of metamodels. That article discussed a two-step methodology, in which the ﬁrst consisted of projecting the Power Spectral Density (PSD) of the measured dynamic response of the wind turbine linearly in an alternative representation space through Principal Component Analysis (PCA). In the second stage, the coefﬁcients of the PCA-based projection were used as a vector of characteristics, represented by a probability density model in the characteristic space, which is associated with environmental/operational variables measured by the Supervisory Control and Data Acquisition (SCADA) of the wind turbine through the PCE. The proposed methodology facilitated the detection of different wind turbine modes, although it can still be used for fatigue simulation and prediction, only by sampling from the resource space. This methodology was demonstrated with real data measured on a wind turbine located in L ¨ubbenau, Germany, measured over three months. In this study, the methodology discussed for quantifying the size of damage like trailing edge debonding of a wind turbine blade is based on obtaining a metamodel using the Polynomial Chaos-Kriging (PC-Kriging) method. Polynomial Chaos Expansions (PCE) and Kriging are two popular non-intrusive metamodeling techniques (they do not modify or adapt the original model equations). The PCE replaces the computational model with a series of orthonormal polynomials in the input variables, where the polynomials are chosen in coherence with the probability distributions of these input variables (Ghanem and Spanos 1991; Ghanem et al. 2017). The Kriging method assumes that the computational model behaves as a realization of a Gaussian random process whose parameters are estimated from the available computer executions, that is, input vectors and response values (Lataniotis et al. 2015). The PC-Kriging presents itself as a new non-intrusive metamodel approach combining PCE and Kriging. The PCE is close to the computational model’s global behavior, while Kriging is responsible for its local variability. Combining these two methods leads to better accuracy, or at least as good, as either method alone (Sch ¨obi et al. 2014; Sch ¨obi et al. 2015). That is why the choice of applying both methods as a combined approach in this work. Some works were carried out using the combination of the PCE and Kriging methods, and thus it was possible to identify some advantages and disadvantages of this combination. Some of the PC-Kriging method’s main advantages are the ease model construction, the low computational cost, the analytical calculation of classical Prepared using sagej.cls Pavlack et al. 3 statistical measures of the quantity of interest, and its simplicity compared to other machine learning techniques. However, the PC-Kriging is very sensitive to data quality, just like any other machine learning technique, and impossible to apply to large problems, these being some of its disadvantages (Sch ¨obi and Sudret 2014; Du and Leifsson 2020). In Kersaudy et al. (2015), the Speciﬁc Absorption Rate (SAR) was evaluated using a surrogate model to reduce the computational cost. Thus, it was considered a sparse representation of the PCE using minimal angle regression as a selection algorithm to retain the most inﬂuential polynomials, and the selected polynomials are used as regression functions for the universal Kriging model. This combination proposal was applied to three benchmark examples, and the performances were compared with a standard Kriging model and a sparse PCE classic. The combination of the methods showed an adequate performance. In the literature, some studies used PC-Kriging for quantiﬁcation problems considering uncertainties. In Sch¨obi et al. (2016), a new structural reliability method was developed based on the PC-kriging approach, which was coupled to an active learning algorithm known as adaptive Kriging-Monte Carlo Simulation (AK-MCS). The problem was formulated so that the calculation of small probabilities of failure and extreme quantiles were uniﬁed. Dubreuil et al. (2018) carried out a parametric study of engineering models under uncertainty, using the PC-Kriging approach. The advantage of the approach developed in this article was the reduction in computational cost, which was demonstrated in several numerical examples and also illustrated in the parametric study of an aircraft wing under uncertainty. In this study PC-Kriging is used to deﬁne a trend curve that associates a damage index with an estimate of the damaged area, considering the uncertainties. Each damage index is calculated based on the data-driven algorithm using the Mahalanobis distance (MD), considering a baseline condition as a reference condition. The work is organized as follows: ﬁrst, a statement of the problem in question is performed, then the methodologies that will be adopted for the detection and quantiﬁcation of damage in wind turbine blades are presented. The experiment setup and the introduction of artiﬁcial damage, and the data collection procedure are presented. Finally, the results are investigated, and the conclusions have discussed the performances of the proposed VSHM methodology for detecting damage in different locations of the accelerometer and the methodology used to quantify the damage. Problem statement The study presented in this work is carried out on an SSP 34 m wind turbine blade. The blade is instrumented with 20 triaxial accelerometers, ten along the trailing edge (TE) and ten along the leading edge (LE). The results of this study consist of two parts: 1) Damage detection: the methodology in Garc ´ıa and Tcherniak (2019) is used. Such methodology consists of 4 steps: data collection, the reference state, feature extraction, inspection phase, and decision making. Each of these steps is detailed throughout the text. An actuator is used to excitethe blade, which is an impact test. An investigation is carried out on the sensitivity of detection and damage progression. 2) Damage quantiﬁcation: to quantify the damage’s size, a metamodel obtained from the PC-Kriging method is used. The metamodel generates a trend curve related to the damage indexes obtained in the detection part with the severity of the damage. Figure 1 shows a schematic of the entire methodology of this work, with the damage quantiﬁcation part being the main contribution of this work. Damage assesssment Methodology for damage assessment in wind turbine blades The methodology presented in this study is available in Garc ´ıa and Tcherniak (2019). This methodology is considered a simple nonparametric method for data compression and information extraction. The procedure is divided into four steps: data collection, the reference state, feature extraction, and inspection phase for decision-making (Garc ´ıa and Trendaﬁlova 2014). Data collection The ﬁrst step is to collect the data from the struc- ture/system. Acceleration signals are measured and dis- cretized into a vector. Each measured signal is ﬁrst stan- dardized to have zero mean and unit variance and secondly transformed into the frequency domain. Each signal vector realization is arranged in the columns of the matrix Z, i.e., Z= (Z1;Z2;:::;ZM): (1) The matrix Zis constructed from signal vectors obtained on the pristine/healthy state of the wind turbine blade, and it is used for creating the reference state. Creation of the reference state A reference state is created based on the matrix Z, where the observation signal vectors can be compared. First, each vector signal Zmis embedded into a matrix Zmby W-lagged copies of itself. All matrices Zmare used to create the full embedded matrix Z, i.e., Z=Z1;Z2;:::; ZM : (2) The covariance matrix of Z, which deﬁnes the covariance between the different signal vector realizations, is estimated by CZ=ZtZ N: (3) The eigendecomposition of CZyieldskandEk eigenvectors in the same order as their corresponding eigenvalues. Each eigenvalue deﬁnes the partial variance in the direction of its corresponding eigenvector. A matrix EZ contains all eigenvectors Ek. Each Principal Component (PC)Akassociated with each eigenvector Ekis calculated by projecting the matrix ZontoEZ. The Reconstructed Components (RCs) are calculated by convolving the PCs with the associated Ek. The RCs are then arranged in Prepared using sagej.cls 4 Journal Title XX(X) I) T raining II) T est Data collection Cr eation of the r efer ence state Featur es extractionDamage index Data collection Metamodels for damage quantification Lerned modelData collection Featur es extraction Damage index D i > D T Damage quantificationY esNo Pr ogr essive damaged conditions Refer ence condition Unknown condition Figure 1. Schematic representation of the proposed methodology for quantifying damage to wind turbine blades. columns into the matrix R. Therefore, Rcan be used as the reference state of the structure/system to which the observation signal vectors are compared. Feature extraction A Feature Vector (FV) is obtained for each new observation signal vector, which will be subjected to damage evaluation by comparing its similarity to the reference state deﬁned by R. An FV is calculated by multiplying an observation signal vector zwith each RC in the reference stateR, as shown in (4) where j= 1;:::W . Tj=NX n=1znRnj (4) EachTjvalue represents the inner product between an observation signal vector and each RC. All Tjare arranged into a vector Twith dimension W. The FV Tcharacterizes the observation signal vector onto the feature space. Inspection phase and decision making The baseline feature matrix TBis created. Once the baseline is deﬁned, an observation FV is then compared with the baseline TB. Using the Mahalanobis distance damage index is obtained, Di=q (TiB)t1(TiB); (5) where Bis the mean row of the baseline feature matrix TB;is its corresponding covariance matrix, and Diis the damage index. It is necessary to set a threshold against which damage rates can be assessed to label an observation as an outlier or inlier. A probabilistic threshold DTbased on the probability density function (pdf) of the distances measured by the baseline FVs for the baseline matrix is calculated TB. As the damage indexes are always positive (Di>0), a lognormal probability density function was used to approximately adjust the data considered as a training set,to deﬁne a limit to distinguish between observations of the healthy and damaged structure. The DTlimit is calculated by the inverse of the lognormal cumulative density. Thus, the deﬁnition of whether or not there is damage is given by the decision below: H0:DiDT) Undamage wind turbine blade H1:Di>DT) Damaged wind turbine blade Damage quantiﬁcation methodology After detecting the initial trailing edge debonding in the wind turbine blades using a data-driven approach, the user needs to decide if there is an imminent structural failure or if the system can be kept in operation under monitoring to track the damage progression and its impact on structural safety conditions. Therefore, it is imperative to obtain the quantiﬁcation of the debonding length. Computer simulation of many problems in modern engineering and applied sciences has a high computational cost. In this context, the metamodelling tries to reduce computational costs and perform sophisticated analyzes, such as reliability analysis and design optimizations (Sch ¨obi et al. 2017). One method for obtaining metamodels is investigated: Polynomial Chaos-Kriging (PC-Kriging). In the next subsection, the method algorithm is presented. Polynomial Chaos-Kriging The PCE method works as a type of “response surface”, which locally interpolates the model hypersurface. This method obtains the computational model by sum the orthonormal polynomials to the input variables. This orthogonal expansion decouples stochastic and deterministic objects, that is, the polynomial basis is random and the numerical coefﬁcients are deterministic, obtained from the data. This property made the metamodel construction easier. In this context, consider a ﬁnite-variance computational modelM:DXRM7!R, which receives and input a Mdimensional vector X= (X1;;XM)2RMwith a given probability density function (PDF) fXdeﬁned on the Prepared using sagej.cls Pavlack et al. 5 supportDx, and returns as output the scalar quantity of interestY=M(X)2R, such as illustrated in Figure 2. computational model input output Figure 2. Computational model under uncertainty. According to Marelli and Sudret (2015), the Polynomial Chaos Expansion of M(X)is given by Y=M(X)MPC(X) =X 2Ay (X); (6) whereANMis a subset of the polynomial indexes, (X)is a family of orthonormal polynomials with respect tofX, andy are real-valued deterministic coefﬁcients to be determined. The appropriate family of polynomials is chosen according to the probability distributions of the input variables (Xiu 2010; Ghanem et al. 2017). One of the signiﬁcant advantages seen in using the PCE is its rapid convergence and expressing the ﬁnal solution as a random process and not just as a set of statistics. The Kriging method, also known as Gaussian Process Regression (GPR), is a non-parametric Bayesian approach that has the advantages of working on small data sets and providing measurements that consider the predictions’ uncertainties. A Kriging model is called ordinary Kriging when the trend is a single parameter with an unknown value. When the trend is a sum of functions, it is called universal Kriging. The combination of the Polynomial Chaos Expansion (PCE) and Kriging methods results in the method called Polynomial Chaos-Kriging (PC-Kriging). This combination results in technique metamodeling more accurately than the PCE and Kriging separately. The PC-Kriging uses the PCE type regression to capture the computational model’s global behavior and the interpolation type Kriging to capture the variations. The PC-Kriging is considered a universal Kriging technique that obtains the trend from a set of orthonormal polynomials (Sch ¨obi et al. 2014). One of PC-Kriging’s main advantages is the ease in building the model and the low computational cost; however, PC-Kriging may not perform well on high-dimensional problems. The construction of the metamodel by PC-Kriging consists of two stages: (i) the determination of a set of polynomials that deﬁnes the trend and (ii) the determination of the ideal correlation parameters and the trend parameters. The polynomials that deﬁne the trend are calculated using the PCE by employing the Least Angle Regression (LARS) algorithm. The trend parameters and correlation parts are calculated as in the universal Kriging algorithm by solving the log-likelihood function’s maximization using a gradient- based optimization algorithm. These two steps are processed in series, as the set of polynomials can be determined independently of Kriging’s settings (Sch ¨obi et al. 2015, 2016; Kersaudy et al. 2015; Sch ¨obi et al. 2017). The idea of applying PC-Kriging in the damage quantiﬁcation level is to employ a metamodel to capturethe trend between the local damage index and damage size, which has been observed by Garc ´ıa and Tcherniak (2019). The local damage index is used as input, and the damage size is used as output for the construction of a PC-Kriging model. The PC-Kriging can be described as S=X 2Ay (Di) +2Z(Di;!); (7) whereDi2Ris the local damage index with a given Probability Density Function (PDF) fX, S is the damage size, withS2R. TheP 2Ay (Di)is a weighted sum of orthonormal polynomials that describes the PC-Kriging model trend, where (Di)are orthonormal polynomials in relation to fX, 2A are the indices and y are the corresponding coefﬁcients; 2is the variance of the process, andZ((Di);!)is a Gaussian random process with zero mean.!describes outcomes of the underlying probability space with a correlation family Rand its hyperparameters . That is, the correlation function R=R(x;x0;)describes the correlation between two samples of the input space. For example, xandx0depends on the hyperparameters  (Lataniotis et al. 2015). The PC-Kriging can be interpreted as a universal model of Kriging with a speciﬁc trend. The PC-Kriging algorithm can be done in two ways: sequential and optimal. In sequential PC-Kriging (SPCK), the set of polynomials and the Kriging metamodel are determined sequentially. First, the ideal set of polynomials is determined by the PCE based on LARS. Every set of polynomials is incorporated into the PC-Kriging equation, and then the PC-Kriging metamodel is calibrated as a usual Kriging model, including the calculation of the coefﬁcients y . In the optimal PC-Kriging (OPCK), the model is obtained iteratively. As in SPCK, the ideal set of polynomials is determined by LARS. The LARS algorithm results in a dispersion of the set of polynomials classiﬁed according to their correlation with the current residual in each LARS iteration (in decreasing order). Each polynomial is then added individually to the trend of a PC-Kriging model. In each iteration, a new PC-Kriging model is calibrated. At the end of this process, the PC-Kriging models are compared using their leave-one-out (LOO) error estimators. The PC- Kriging metamodel optimal is chosen according to the one that minimizes the LOO error (Sch ¨obi et al. 2017). The LOO error can be deﬁned as (Sch ¨obi et al. 2015) LOO =1 NNX i=1 Y(i)^y;(i)(X(i))2 ; (8) where^y;(i)(X(i))is the prediction mean ^yof sample X(i)by a Kriging metamodel based on the experimental designX(i)=XnX(i)andY=fY(i);i= 1;;Ngis the exact model response. In Sch ¨obi et al. (2015), the performances of the Kriging, PCE, SPCK, and OPCK methods, in terms of generalization of relative error in the analytical reference functions, are compared. The results showed that PC-Kriging is better than, or at least as good as, Kriging and PCE methods separately for small experimental projects. Moreover, it was concluded that OPCK is preferable to SPCK because it reduces the number of polynomials in the regression part and, therefore, reduces the metamodel’s complexity. Based Prepared using sagej.cls 6 Journal Title XX(X) on this conclusion, the OPCK algorithm was employed in this work. The implementation of the PC-Kriging algorithms was performed using the UQLab *, which is a MATLAB- based software framework designed to bring uncertainty quantiﬁcation (UQ) techniques and algorithms to a broad audience. The UQLab offers an extensive list of algorithms for UQ, including the PC-Kriging method (Marelli and Sudret 2014). Experimental Application The data set used in the experimental application proposed in this work belongs to Bruel &Kjaer. This data set has been explored recently to validate different methodologies (Garc ´ıa and Tcherniak 2019; Ulriksen et al. 2016; Hernandez Crespo 2016). A brief description of the experimental setup will be provided below. A detailed description of the experimental setup can be found in Nielsen et al. (2010). The experimental application of the methodology pro- posed was performed in the SSP 34 m wind turbine blade. The blade was manufactured by SSP-Technology A/S, and the experiments were performed on a test rig at the Wind Energy department, the Technical University of Denmark. The blade of the wind turbine and the experiment’s facilities can be seen in Figure 3(b). The blade was instrumented with 20 triaxial accelerom- eters, model Bruel & Kjaer Type 4524-B, positioned as represented in the scheme in Figure 3(a). Ten accelerom- eters were placed in the trailing edge (TE) and ten in the leading edge (LE). In the experiment, acceleration signals were collected for the blade’s impact response under healthy and progressive damaged conditions. Figure 3(c) show the electromechanical actuator used to generate the impact in the structure, and it was placed on the surface outside of the blade at the position indicated in Figure 3(a). In total, 386 signals were collected for seven structural health conditions simulated, being a healthy condition, and six with damage (see Table 1). Condition Damage Size Number of Signals H - 53 D20 20 cm 70 D40 40 cm 61 D60 60 cm 60 D80 80 cm 49 D100 100 cm 54 D120 120 cm 39 Total 386 Table 1. Number of signals measured on each experimental test (Garc ´ıa and Tcherniak 2019). One of the main types of damage in wind turbine blades is the adhesive joint debonding (Montesano et al. 2016). This type of damage occurs when an adhesive bond between the laminates of the pressure and suction sides of the blade breaks can happen on both leading and trailing edges. Small debonding size can grow up to a level at which repair is impossible, and the entire blade should be replaced (Garc ´ıa and Tcherniak 2019). First, a series of holes through theadhesive between the blade’s pressure and suction sides were drilled. Then, using a saw and a chisel, the holes were merged, forming an opening that was gradually extended from 20 cm up to 120 cm by increments of 20 cm. The debonded parts were connected by bolts, placed at 10 cm intervals. The healthy condition was then simulated by tightening all the bolts, and the progressive damaged conditions were reproduced by loosening some bolts. The number of loosened bolts deﬁned the damage size. The location of the damage can be seen in Figure 3(a). More details about the experiment can be found in Garc ´ıa and Tcherniak (2019). Results and Discussions This section presents the results of the damage detection and the quantiﬁcation of the area of the trailing edge debonding. To obtain the results is used the software MATLAB. For damage quantiﬁcation results is used the toolbox UQLab. It is used to estimate the trailing edge debonding size, using the damage index and optimizing the PC-Kriging model. Damage detection The methodology for damage detection described previously is applied in the dataset of SSP 34 m wind turbine blade. The data set provides acceleration signals collected from sensors placed in the trailing and leading edges. In Garc ´ıa and Tcherniak (2019), the authors deﬁned the parameters chosen for the best methodology performance, which are employed in this work. In the creation of the reference state, it is considered a M= 10 signal vector realizations with a sliding window size of W= 10 . The feature vector dimension is set to p= 5. The baseline matrix construction is performed using s= 26 feature vectors of dimension p= 5 extracted from healthy condition signals. In the inspection phase, the risk of false alarm probability is set to = 0:01in the lognormal density function. A detailed discussion of each one of these parameters on the methodology performance can be found in Garc ´ıa and Tcherniak (2019). Figure 4 shows the damage index by damage size obtained, considering the accelerometers in the TE to the actuator’s position. Figure 5 shows the damage index by damage size obtained, considering the accelerometers in the LE. It can be seen that the accelerometers along the TE detected the damage better than the accelerometers along with the LE. The damage detection results present three scenarios: i) Damage indexes can detect and track the severity of the damage. This case can be seen in Figure 4, in sensors 2 and 4; ii) The damage is not well detected, as it has many false negatives. However, as the damage increases, there is a tendency for the damage indexes, so the damage’s progression is somehow detected. In this case, there is a time- dependent feature for damage detection. This can be seen in Figure 4, in sensors 1, 3 and 8, and Figure 5 in sensors 4 and 5; iii) A ﬁnal scenario is when the damage indexes cannot detect the damage or track its progress. In Figure 4, this can be seen in sensors 6 and 9 and Figure 5, in almost https://www.uqlab.com/ Prepared using sagej.cls Pavlack et al. 7 Sensor Number: 10 9 8 76 5 4321 Location: 3m 8m12m 15m18m 20m22.5m 25m27m 29m(5m) Sensor Actuator outside blade DamageTrailling Edge Leading Edge (a) Accelerometers, damage and actuator locations scheme. (b) test rig set-up. (c) Electromechanical actuator. Figure 3. Experimental setup of the SSP 34 meters wind turbine blade manufactured by SSP-Technology A/S. The blade was instrumented with 10 accelerometers along the leading edge (LE) and 10 along the trailing edge (TE) and was excited by an electromechanical actuator. The debonding damage was introduced into the blade (Garc ´ıa and Tcherniak 2019). Figure 4. Damage index (Di)by damage size (S)for accelerometers in the trailing edge (TE). The damage index in the healthy () and damaged ( x) conditions The dashed line ( ) corresponds to the threshold deﬁned by a risk of false alarm probability equal set to = 0:01. Prepared using sagej.cls 8 Journal Title XX(X) Figure 5. Damage index (Di)by damage size (S)for accelerometers in the leading edge (LE). The damage index in the healthy () and damaged ( x) conditions The dashed line ( ) corresponds to the threshold deﬁned by a risk of false alarm probability equal set to = 0:01. all sensors. These results of damage detection inﬂuence the results of damage quantiﬁcation because damage indexes are used. Damage quantiﬁcation The damage quantiﬁcation methodology used is based on the construction of a PC-Kriging metamodel, which obtains the trailing edge debonding size as a function of the local damage index. The local damage index is used as input for building a PC-Kriging metamodel. In the model training stage, the following conditions’ local damage index is considered: H, D40, D80, D100, and D120. The damaged conditions D20 and D60 are used in the test steps to validate the prediction of damage quantiﬁcation using metamodel. In the ﬁrst stage of the PC-Kriging metamodel construc- tion, the polynomials set that deﬁne the trend through the PCE are deﬁned using the LARS algorithm. In the second stage, the ideal correlation parameters and the tendency parameters are calculated the same way they are calculated in the universal Kriging algorithm. For this work results, an optimal PC-Kriging approach is used; that is, the metamodel is obtained iteratively, and the one chosen is the one that minimized the leave-one- out (LOO) error. The optimal PC-Kriging is chosen because it already has better performance in the literature than the sequential PC-Kriging (Sch ¨obi et al. 2015). Figure 6 shows a schematic representation of the Optimal Polynomial Chaos- Kriging algorithm. The damage index is always positive, so in the PCE settings, the distribution used is lognormal, and the moments of the PCE settings are the mean and standard deviation of the training data. In the Kriging conﬁguration, a correlation function based on an exponential and ellipsoidal family is used to optimize the maximum likelihood estimateperformed by a gradient method to deﬁne the GPR model (Lataniotis et al. 2015). The trained models represent a mean and 95% conﬁdence interval of the predicted distribution. Figure 7 attests to the severity, that is, the size of the trailing edge debonding as a function of the damage index obtained by the PC-Kriging method for the sensors in the TE, and Figure 8 present the results for each sensor in the LE. Figures 7 and 8 show that most of the damage indexes used in the training and test stages are within the region of the conﬁdence interval, inferring an adequate selection of the training parameters. It is also noted that the PC-Kriging model captured the damage index trend adequately well. The Root Mean Squared Error (RMSE) metric is chosen to validate the model prediction for test conditions D20 and D60. This metric is frequently used measure of the differences between values estimated by a model or an estimator and the values measured. The RMSE in this work can be deﬁned as RMSE =vuutnX i=1(SestSmea)2 n; (9) whereSestare estimated values for the quantiﬁcation of the debonding area,Smeaare measured values of the debonding area andnis number of observations. The validations of the PC-Kriging model prediction for the test conditions in the TE and LE are presented in Figures 9 and 10. Ideally, the points would concentrate on the diagonal line, as this is where the estimated damage size coincides with the measured damage size. Table 2 presents the RMSE values and the mean of the estimated value for the test conditions (D20 and D60) for each sensor in the TE and LE. Prepared using sagej.cls Pavlack et al. 9 Damage indexPCE Universal Kriging Input distribution Autocorrelation function Optimal PC-Kriging LARS (iteration 1) LARS (iteration 2) LARS (iteration p) ... PC-Kriging metamodel Quantification of the debonding area in large wind turbine blades Figure 6. Schematic representation of the Optimal Polynomial Chaos - Kriging algorithm. Table 2. Value of the root-mean-square error (RMSE) and relative error of the test conditions (D20 and D60) for each sensor along TE and LE by the PC-Kriging metamodel. The sensors that have detected the damage better have the values shown in blue. Sensor 1 2 3 4 5 6 7 8 9 10 TERMSE 16.8 10.0 11.4 5.83 31.1 32.0 30.6 16.4 33.6 38.5 D20: Mean est. damage[cm] 25.81 15.58 3.31 13.99 23.31 49.04 22.89 25.14 50.40 57.52 D60: Mean est. damage[cm] 36.77 47.95 52.25 63.54 15.78 56.23 17.86 72.04 71.38 83.89 LERMSE 34.3 32.3 32.3 28.5 26.5 38.5 35.5 32.9 36.1 32.4 D20: Mean est. damage[cm] 56.40 52.14 44.43 50.69 28.47 58.38 52.94 55.69 55.33 62.37 D60: Mean est. damage[cm] 46.99 72.16 59.80 52.44 56.62 78.79 74.91 60.41 82.85 64.50 The sensors located in the TE also had a better performance for the quantiﬁcation of the damage. The results show that the location of the sensor in the TE and LE inﬂuences the results. The sensors that fall under scenarios i) and ii) referred to in damage detection are the ones that present the best results. In TE, sensors 1, 2, 3, 4, and 8 had the best results. Sensor 10 had the worst result in the TE, which did not detect the damage well. The results of the LE sensors were not good, with sensors 4 and 5 that best quantiﬁed the damage, and these sensors were the ones that detected a trend in the progression of damage. The metamodel obtained by PC-Kriging presents promising results for the quantiﬁcation of the damage to the DIs that present good detection of the damage or capture the damage progression trend. Conclusions This work approached the debonding area’s quantiﬁcation due to the debonding in wind turbine blades, using a new methodology considering the uncertainties and interpolation through a metamodel obtained by the Polynomial Chaos- Kriging (PC-Kriging) method. The metamodel obtained relates the damage indexes to trailing edge debonding. To obtain the damage indexes, the methodology presented inGarc ´ıa and Tcherniak (2019) was used. This methodology was applied to an SSP 34 m wind turbine blade, instrumented with one actuator, ten accelerometers in the trailing edge (TE), and ten in the leading edge (LE). It is observed that the accelerometers located in the TE detected the damage better than those located in the LE. The excellent performance of accelerometer 4 in the TE stands out, it is located close to the damage and far from the actuator. It is also noted that the damage introduced is located in the TE, where the accelerometers obtained more favorable results. The PC-Kriging method was used to obtain a metamodel that relates the damage index with severity to quantify the trailing edge debonding. A trend curve was obtained for this relationship, considering a 95% conﬁdence interval. The PCE method captures the computational model’s global behavior, while the Kriging method, of the interpolation type, captures local variations. For this reason, PC-Kriging, which is the combination of the two methods, presents itself as a more robust method for obtaining metamodels. In this study, the quantiﬁcation of the damage using the PC-Kriging showed better TE’s accelerometers’ per- formances, highlighting the TE’s accelerometer 4. With that, it can be concluded that the accelerometers that bet- ter detected the damage obtained better damage indexes, Prepared using sagej.cls 10 Journal Title XX(X) Figure 7. Damage severity (S)by the damage index (Di)for accelerometers 1 to 10 on the trailing edge (TE). The metamodel was trained using ﬁve conditions ()and tested with two conditions (x). The bold line ()corresponds to the trend mean and the gray-colored region ()to the 95 %of conﬁdence interval. Prepared using sagej.cls Pavlack et al. 11 Figure 8. Damage severity (S)by the damage index (Di)for accelerometers 1 to 10 on the leading edge (LE). The metamodel was trained using ﬁve conditions ()and tested with two conditions (x). The bold line ()corresponds to the trend mean and the gray-colored region ()to the 95 %of conﬁdence interval. Prepared using sagej.cls 12 Journal Title XX(X) 20 6020406080100120Estimated Size [cm]Sensor 1 - TE 20 6020406080100120Sensor 2 - TE 20 60 Measured Size [cm]20406080100120Sensor 3 - TE 20 6020406080100120Sensor 4 - TE 20 6020406080100120Sensor 5 - TE 20 6020406080100120Estimated Size [cm]Sensor 6 - TE 20 6020406080100120Sensor 7 - TE 20 60 Measured Size [cm]20406080100120Sensor 8 - TE 20 6020406080100120Sensor 9 - TE 20 6020406080100120Sensor 10 - TE Figure 9. Validation of the estimated damage size using the PC-Kriging metamodel by the actual damage size for each sensor in the TE and test condition. The estimated damage size for all damage indexes ()and the mean of estimated damage size ()for each test condition. 20 6020406080100120Estimated Size [cm]Sensor 1 - LE 20 6020406080100120Sensor 2 - LE 20 60 Measured Size [cm]20406080100120Sensor 3 - LE 20 6020406080100120Sensor 4 - LE 20 6020406080100120Sensor 5 - LE 20 6020406080100120Estimated Size [cm]Sensor 6 - LE 20 6020406080100120Sensor 7 - LE 20 60 Measured Size [cm]20406080100120Sensor 8 - LE 20 6020406080100120Sensor 9 - LE 20 6020406080100120Sensor 10 - LE Figure 10. Validation of the estimated damage size using the PC-Kriging metamodel by the actual damage size for each sensor in the LE and test condition. The estimated damage size for all damage indexes ()and the mean of estimated damage size ()for each test condition. and consequently resulted in a good performance for the quantiﬁcation. It is also observed that the location of the accelerometers inﬂuences the results. The PC-Kriging trend curve, in general, managed to capture a monotonic increase in damage indexes, showing promising results for quantiﬁ- cation. The advantages observed in the use of PC-Kriging were its simplicity and ease in constructing the metamodel since the toolbox UQLab is available, which allows an easy implementation of the method. It is also observed the low computational cost to obtain the results.This study presents a contribution to the data-driven SHM methodology regarding the quantiﬁcation of damages in mechanical structures, as it addresses the use of a method that is not yet widely explored in this area. This study collaborates with the development of research in damage quantiﬁcation so that better results are always obtained, and thus, it is possible to apply in the industrial context and contribute to society. Prepared using sagej.cls Pavlack et al. 13 Acknowledgments The authors of this work would like to acknowledge the generous input of Dr. Dmitri Tcherniak, from Br ¨uel & Kjær Sound and Vibration Measurements, who kindly provided the data from the experimental regime performed on the SSP 34 m wind turbine blade. Declaration of conﬂicting interests The authors declared no potential conﬂicts of interest concerning the research, authorship, and publication of this article. Funding The authors are thankful for the ﬁnancial support provided by the S ˜ao Paulo Research Foundation (FAPESP) grants 2017/15512- 8, 2018/15671-1, 2019/11755-9 and 2019/19684-3; the Brazilian National Council for Scientiﬁc and Technological Development (CNPq/Brazil), grant number 306526/2019-0. We would also like to acknowledge the Carlos Chagas Filho Research Foundation of Rio de Janeiro State (FAPERJ) under grants 210.021/2018 and 211.037/2019, and Coordenac ¸ ˜ao de Aperfeic ¸oamento de Pessoal de N´ıvel Superior - Brasil (CAPES) - Finance Code 001.