3.1.1. Automated OMA

Modal parameters are essential for vibration-based SHM and are determined through the automated OMA, a process that is executed on discrete segments of ACC data, specifically 10-minute intervals for this study. Multiple algorithms exist for MPE, including frequency domain decomposition (FDD), least squares complex frequency domain (LSCF), and covariance-driven stochastic subspace identification (SSI-COV; Magalhães and Álvaro Cunha, Reference Magalhães and Cunha2011; Reynders, Reference Reynders2012). For this study, the methodology developed by Devriendt et al. (Reference Devriendt, Magalhães, Weijtjens, De Sitter, Cunha and Guillaume2014), utilizing the LSCF algorithm for MPE, is implemented to identify all modes from the vibrations in the initialization dataset (cf. Figure 4). As previously mentioned, all clusters identified in the MPE are kept for delayed DBSCAN tracking, as detailed in the subsequent section. This results in the natural frequency, damping ratio, and cluster size for every cluster of the stabilization diagram of 10-minute vibration data. Mode shapes are not considered, as the method applies to a single-sensor SHM setup. The calculated modal parameters form the basis for the next section, consisting of clustering the structural modes and removing spurious modes and non-structural effects.

3.1.2. DBSCAN-based unsupervised clustering

The initialization process builds an initial structural modes dataset through unsupervised clustering. As previously mentioned, this is achieved through the DBSCAN algorithm (Ester et al., Reference Ester, Kriegel, Sander and Xu1996) on an initialization dataset. The relevant clusters are then used to train ML models for the smart tracking methodology detailed in Section 3.3.1. The novel insight in this method is to add the time dimension to allow for slow variations of the natural frequency over time. This is done by adding the time difference between the different identified modal parameters as an additional feature to the DBSCAN algorithm. The clustering can thus be done on the frequency and time difference only, as done in a previous SHM research on a high-voltage transition tower (Bel-Hadj et al., Reference Bel-Hadj, Weil, Weijtjens and Devriendt2024). However, due to EOV of natural frequencies in OWTs, this process required corrections to the cluster through physical knowledge of the structure and the supervision of an interpreter, as detailed in Section 3.1.3.

Before clustering, the search space must be transformed to allow for the correct clustering of the structural modes. This is done by scaling every dimension to transform the hyperspace on which the DBSCAN algorithm clusters the data. In this case, the input dimensions are the frequency ( $ f $ ), cluster size from the automated OMA ( $ \mathrm{size} $ ), and time ( $ t $ ). Although new scaling parameters ( $ {\gamma}_{\bullet } $ to scale the parameter $ \bullet $ ) are introduced, this enables the DBSCAN algorithm to accurately cluster the modes. Together with the inherent DBSCAN parameters: $ \unicode{x025B} $ (maximum distance between two points for them to be considered neighbors) and $ \mathrm{minPts} $ (minimum number of neighbors a point needs to have to be considered a core point), these new parameters should be optimized through the corrections process detailed below. However, this optimization only needs to be run once for a population of similar structures, such as OWTs in an offshore wind farm, as the same parameters can be used. This methodology is implemented in a separate Python package (Weil, Reference Weil2024c). The previous work on transmission towers contains an in-depth description of this approach (Bel-Hadj et al., Reference Bel-Hadj, Weil, Weijtjens and Devriendt2024).

3.1.3. Cluster corrections

Following the initial mode clustering, corrections are made to ensure accurate mode labeling under the supervision of an interpreter, in a human-in-the-loop fashion, to ensure engineering judgment is added to the selection process. This involves both an optimization of the DBSCAN parameters and manual correction for remaining confounded modes. By training ML models on this curated data and using OOD detection through UQ (cf. Section 3.2.4), the engineering judgment is automatically transferred to new data during the tracking process, as detailed in Section 3.3.1. This approach facilitates the optimization of the clustering algorithm’s parameters and ensures that the resulting clusters exclude obfuscated modes.

Interpreter and physical knowledge: Iterative adjustments are made to both the added parameters for the hyperspace scaling ( $ {\gamma}_{\bullet } $ ) and the inherent DBSCAN parameters ( $ \unicode{x025B} $ and $ minPts $ ). Alternatively, the optimization of the latter can be automated through the use of hierarchical DBSCAN (McInnes and Healy, Reference McInnes and Healy2017), but this is outside the scope of the current research. The obtained optimal parameters are given for the FA and SS direction in Table 1. Using these parameters for the clustering yields a set of initial structural mode clusters. However, these can still have non-structural components. Therefore a manual data selection based on engineering judgment is performed on these clusters.

Table 1. DBSCAN parameters for unsupervised clustering in FA and SS directions

Manual data selection: There are instances where modes resulting from the automated OMA cannot be trusted, for example, due to nonstructural mode interference. In these scenarios, only the data representing the mode’s expected behavior, as informed by prior knowledge of the engineer, can be manually isolated from the clusters. Such a process can readily be implanted with a graphical user interface, reducing the process to the engineer encircling the desirable results on the initialization dataset only. An implementation in Python is made available at Weil (Reference Weil2024a). When sufficient data are gathered to represent all relevant EOV, the data are used to train the ML models for smart tracking, leveraging OOD and UQ of the ML predictions to automatically remove modes with nonstructural interference on new arriving data (cf. Section 3.3.1).

Reproducibility: While the human-in-the-loop approach ensures that expert engineering judgment is incorporated into the mode selection process, it introduces a challenge to reproducibility. Although the DBSCAN parameters can easily be transferred, different interpreters may make slightly different decisions when selecting data, potentially leading to variations in the final outcome. Therefore, different solutions are proposed to enhance reproducibility:

• • Detailed guidelines and criteria for decision-making based on the monitored structure should be established and documented, ensuring consistency across different users. In future research, this could be incorporated within the graphical user interface for manual data selection. In the case of the current research, different rules are outlined below:

  • – The clusters resulting from the DBSCAN implementation are used as an initial set from which to select the structural modes.

  • – Only modes identified with a damping ratio lower than 5% are considered (Van Der Tempel, Reference Van Der Tempel2006). This threshold can be set higher for FA modes in the rated operational state (Devriendt and Weijtjens, Reference Devriendt and Weijtjens2017).

  • – Only modes identified with a cluster size higher than 5 are considered (Devriendt et al., Reference Devriendt, Magalhães, Weijtjens, De Sitter, Cunha and Guillaume2014).

  • – Modes from the rotor dynamics are known to have a one-on-one relationship with the RPM (cf. Equation (1)) and are removed through the Campbell diagram (Jahani et al., Reference Jahani, Langlois and Afagh2022).

  • – It is known that the natural frequencies of structural modes do not jump abruptly over large frequency ranges. Therefore, these jumps are probably caused by perturbations in the OMA result and are not considered (Weijtjens et al., Reference Weijtjens, Verbelen, De Sitter and Devriendt2016).

  • – The selection performed on the SS2 mode is made only when the mode exhibits the expected dependency to the tidal levels for monopile-supported OWTs (Weijtjens et al., Reference Weijtjens, Verbelen, De Sitter and Devriendt2016).

• • Maintaining a detailed log of the human-in-the-loop adjustments, including the specific parameters and data selections made, can help in tracking decisions and understanding their impact on the final results.

  • – In the current research, only corrections based on damping ratio and cluster size are made on the clusters identified for the FA1 and SS1 mode, while only five time periods exhibiting the expected dependency to the tidal levels and have no perturbation from extraneous modes are used as the final SS2 cluster. The data selected for SS2 and the operational states represented in the dataset are later described in the charts of Figure 8.

3.2.1. Data preprocessing and synchronization

In the ML workflow, the data preparation step always precedes the ML model development (Géron, Reference Géron2019). In this case, the data preparation step involves preprocessing and synchronizing the data collected from the different sources.

The preprocessing step is crucial in the ML workflow to ensure the model interprets the data accurately (Al-jabery et al., Reference Al-jabery, Obafemi-Ajayi, Olbricht and Wunsch2020). The preprocessing involves encoding circular data, including angles with a 360° range, such as yaw and wind direction, as well as temporal data (e.g., time of day, month of the year). The challenge with these data is that, for example, an angle of 350° is closer to 10° than to an angle of 300°. To enable ML models to understand circular relationships, the sine and cosine transformations are applied to encode the circular data (Van Wyk, Reference Van Wyk2022). In the studied case, this includes the wind direction and the yaw angle or orientation of the OWT nacelle.

Additionally, preprocessing involves data scaling or normalization, such as standardization or min–max normalization, to prevent the magnitude of the data from skewing the ML predictions. However, this step is model-dependent; tree-based learners, for instance, do not require data scaling due to their inherent handling of feature scales (Hehn et al., Reference Hehn, Kooij and Hamprecht2020). The data normalization used in this research is the min–max normalization, performed on the selected data during the model comparison on the input of certain models. The used minimum and maximum values are given in Appendix A.

Data synchronization is then undertaken to align input and target data across corresponding timestamps and interpolate data with different sampling frequencies, resulting in a consolidated dataset ready for ML training and testing.

3.2.2. Feature selection

The next step in the ML workflow is feature selection, determining the most relevant features to predict the natural frequencies. As stated in Guyon and Elisseeff (Reference Guyon and Elisseeff2003), the objective of the feature selection process is three-fold: to improve the prediction performance of the model, to make the model faster and more cost-effective, and to better understand the parameters influencing the natural frequencies.

Feature selection is performed using recursive feature elimination with cross-validation (RFECV) on the training data for automatic tuning of the number of features selected. RFECV consists of fitting a model and recursively removing the weakest feature, according to the model’s feature importance, until a specified amount of features are obtained (Guyon et al., Reference Guyon, Weston, Barnhill and Vapnik2002; Sanz et al., Reference Sanz, Valim, Vegas, Oller and Reverter2018). The addition of cross-validation (CV) identifies the best number of features within a k-fold CV loop.

In this case, the XGBoost regression model (Chen and Guestrin, Reference Chen and Guestrin2016) is selected for its execution speed, good predictive performance, and its built-in feature importance metric. The RFECV is implemented using the scikit-learn Python package (Pedregosa et al., Reference Pedregosa, Varoquaux, Gramfort, Michel, Thirion, Grisel, Blondel, Prettenhofer, Weiss, Dubourg, Vanderplas, Passos, Cournapeau, Brucher, Perrot and Duchesnay2011), with XGBRegressor as the estimator from the XGBoost library with standard hyperparameters. The RFECV is configured with the following parameters: $ \mathrm{step}=1, $ $ cv=5, $ $ \mathrm{stepscoring}="\mathrm{neg}\_\mathrm{mean}\_\mathrm{squared}\_\mathrm{error}" $ , $ \min \_\mathrm{features}\_\mathrm{to}\_\mathrm{select}=1 $ . The results of applying this RFECV to the selected SS2 data are summarized in Table 2, where the last column indicates whether each feature is selected.

Table 2. Overview of the available data with the source of the data, the measurement units, and availability

Note: The last column states if the features are selected by the RFECV.

^a Pulled from a second source to increase availability.

^b Even though 4% seems enough to detect slow evolving damages in SHM, this is concentrated in two regions of data for the entire monitoring period, sometimes leading to months without accurate data to rely on.

Table 2 provides an overview of the feature selection from METEO and SCADA data, used as ML model inputs for predicting natural frequencies of OWTs (i.e., the targets). Only input variables with over 90% availability are considered to ensure adequate coverage of the ML model. However, the tidal-level data were initially available only 50% of the time, this is problematic as tidal level is known to influence higher order structural modes of OWTs (Weijtjens et al., Reference Weijtjens, Verbelen, De Sitter and Devriendt2016). By supplementing this with data from another nearby weather station, availability was increased to 99%.

The data selected by RFECV include parameters that are known to affect the natural frequencies of OWTs. First, all SCADA features are selected. Parameters such as pitch, power, RPM, and wind speed describe the turbine’s operational condition, which directly influences its dynamics (Jahani et al., Reference Jahani, Langlois and Afagh2022). Yaw and wind direction can impact dynamics, particularly in cases of misalignment (Song et al., Reference Song, Mehr, Moaveni, Hines, Ebrahimian and Bajric2023). Not all features from the weather station are selected; however, the tidal level, wave height, and 10% highest waves are included due to their significant influence on modal frequencies. Sea water temperature affects modal parameters, contributing to seasonal variability, while air pressure influences the interaction between wind and structure, affecting dynamics (Weijtjens et al., Reference Weijtjens, Verbelen, De Sitter and Devriendt2016). Among the parameters not selected, many are correlated with the selected features. However, average wave period, identified as relevant by Xiang et al. (Reference Xiang, Wang, Wang, Pan, Liu, Le and Cai2024), is not included in the selected features in this case.

The SS1 and FA1 modes show high (87%) and moderate (41%) availability, but the SS2 mode has high interference from nonstructural modes and could only be considered when the OWT is parked. But this translates into low availability (4%) concentrated in two regions of the year, leading to long periods without reliable data. Therefore, a manual data selection, as detailed in Section 3.1.3, is implemented to train a model on structural modal data only for predicting the SS2 frequency. The ML model can then be used for smart tracking, normalization, and anomaly detection.

3.2.3. Model optimization, comparison, and selection

Once the features are selected, a model comparison is performed to guide the ML model selection. Model performance is evaluated on the test data using two key metrics: the mean squared error (MSE) and the coefficient of determination (R ²), as defined in Equations (3) and (4), respectively. These metrics provide insights into the accuracy and goodness of fit for each model.

(3) $$ {\mathrm{MSE}}_m=\frac{1}{n}\sum \limits_{t=1}^n\Big({f}_m(t)-{\hat{f}}_m{\left(\boldsymbol{x}(t)\right)}^2\hskip1em \forall m, $$

(4) $$ {R}_m^2=1-\frac{\sum \limits_{t=1}^n\Big({f}_m(t)-{\hat{f}}_m{\left(\boldsymbol{x}(t)\right)}^2}{\sum \limits_{t=1}^n{\left({f}_m(t)-{\overline{f}}_m\right)}^2}\hskip1em \forall m. $$

In these equations, $ n $ denotes the number of samples considered, $ {f}_m(t) $ is the target value, the natural frequency of mode $ m $ in this case, for timestamp $ t $ , while $ {\hat{f}}_m\left(\boldsymbol{x}(t)\right) $ represents the ML model prediction based on input $ \boldsymbol{x}(t) $ , and $ {\overline{f}}_m $ the average value of $ {f}_m(t) $ over the $ n $ considered samples. For this study, eight ML models, categorized into three distinct types, are considered to provide a comprehensive comparison of their predictive capabilities.

LR models are first considered, as these were previously used by Magalhães and Álvaro Cunha (Reference Magalhães and Cunha2011) and Peeters and De Roeck (Reference Peeters and De Roeck2001a). Both a traditional and multiple linear regression (MLR) are implemented. In the case of MLR, one LR is trained for each operational condition of the OWT (e.g., parked, idling, etc.) based on SCADA data (Weijtjens et al., Reference Weijtjens, Verbelen, De Sitter and Devriendt2016). The MLR extension reduces MSE and increases R ² for all three modes considered, when compared to LR, as shown in Table 3. However, it requires manually splitting the data according to operational conditions, which other ML models do not require. Despite their limitations, LRs are useful as benchmarks for more advanced ML models due to their simplicity and interpretability.

Table 3. Results obtained for the model comparison for OWT natural frequency predictions on the test data after BHPO. The metric of the best performing model is highlighted in the table as bold-faced.

Next, tree-based learners are implemented, including random forest (RF), which uses the Bagging principle (Altman and Krzywinski, Reference Altman and Krzywinski2017), and gradient boosting methods (Friedman, Reference Friedman2001) such as eXtreme gradient boosting (XGB; Chen and Guestrin, Reference Chen and Guestrin2016) and categorical boosting (CB; Prokhorenkova et al., Reference Prokhorenkova, Gusev, Vorobev, Dorogush and Gulin2018). These models capture complex, nonlinear relationships in the data, with XGB focusing on regularization, efficient tree-building, and pruning to enhance performance, while CB emphasizes efficiency and accuracy through ordered boosting and symmetric trees.

Finally, artificial neural networks (ANNs) with one, two, and three hidden layers are explored for predicting natural frequencies. Data and target normalization are applied to improve training efficiency and model performance (Arnekvist et al., Reference Arnekvist, Carvalho, Kragic and Stork2020).

Hyperparameter optimization is crucial for enhancing the performance of tree-based learners and ANNs. Bayesian hyperParameter optimization (BHPO) as implemented in the Hyperopt Python package (Bergstra et al., Reference Bergstra, Komer, Eliasmith, Yamins and Cox2015) is used to systematically fine-tune models, aiming to achieve the lowest possible MSE on the selected data. A full overview of optimized parameters is provided in Appendix B.

The optimized models’ performances, based on test data predictions and targets, are detailed in Table 3.

Table 3 presents a comparative overview of the different ML models applied to predict monopile-supported OWT natural frequencies SS1, SS2, and FA1. First a reference value of the MSE and R ² are given, by calculating them for a hypothetical model that would give the mean natural frequency of the training data as prediction, replacing $ {\hat{f}}_m\left(\boldsymbol{x}(t)\right) $ by $ {\overline{f}}_{m_{tr}} $ in Equations (3) and (4). Subsequently, MSE and R ² values are calculated for each model. In general, the R ² values for the SS2 mode are much higher than for both FA1 and SS1 regardless of the model. This is because of the high EOV of SS2, mainly driven by the tidal level and wave height, which is well caught by the models.

The relatively low R ² values observed for FA1 and SS1 indicate that the input variables used in the ML models cannot explain much of the variability in these modes. Thus, the low R ² values for FA1 and SS1 suggest inherently low variability in these modes rather than poor model performance. It is important to note that R ² serves as an effective metric to compare models predicting the same variable but is less suitable to compare models predicting different variables due to inherent differences in the variability of the data to predict.

This higher variability captured by the model when compared to simply predicting the mean frequency is also observed by comparing the MSE of the models to the reference value. SS2 shows a greater reduction in MSE compared to FA1 and SS1. SS1 shows the lowest MSE value, but by comparing to the reference value, this can be interpreted as an inherent low variability in the natural frequency of this mode.

When comparing the models, the LR model shows in general a comparatively higher MSE and lower R ². Applying MLR greatly improves model performance, especially for SS1, where R ² almost doubles, and MLR even outperforms the other models in terms of MSE for the FA1 frequency predictions. However, MLR requires the definition of operational cases. Tree-based models and ANNs generally outperform the linear models without prior knowledge of the operational cases, with tree-based models slightly outperforming ANNs. This aligns with expectations set by Grinsztajn et al. (Reference Grinsztajn, Oyallon and Varoquaux2022) regarding the efficacy of tree-based algorithms for tabular data. In general, RF and CB perform best on all three natural frequencies. In the end, CB is selected because the UQ methodology, as described in Section 3.2.4, is already implemented in the Catboost Python package (Dorogush et al., Reference Dorogush, Ershov and Gulin2018).

3.2.4. UQ

Once the models are trained, a prediction of the natural frequencies can be made and used to remove the identified EOV, isolating the residual $ {\unicode{x025B}}_m(t) $ as given in Equation (2). Assuming an additive error model the residual $ {\unicode{x025B}}_m(t) $ can be further decomposed to isolate noise and modeling error components, as shown in Equation (5).

(5) $$ {\unicode{x025B}}_m(t)={\nu}_m(t)+{\xi}_m(t)+{\delta}_m(t)\hskip1em \forall m $$

The first term $ {\nu}_m(t) $ denotes the noise present in the natural frequency data. The second term $ {\xi}_m(t) $ represents the ML modeling error. This consists of EOV in the natural frequency that the model could not adequately predict. Finally, the third term $ {\delta}_m(t) $ is the damage-sensitive part of the residual, independent of EOV, which remains zero as long as the structure’s state is unchanged and gradually increases with anomaly. However, it is not straightforward to dissociate the term $ {\delta}_m(t) $ from $ {\nu}_m(t) $ and $ {\xi}_m(t) $ .

The noise component $ {\nu}_m(t) $ is assumed to follow a zero-mean Gaussian distribution, allowing for its reduction through signal averaging (Hassan and Anwar, Reference Hassan and Anwar2010). In prior work (Weil et al., Reference Weil, Sastre Jurado, Chaar, Winkler, Weijtjens and Devriendt2023), the frequentist confidence interval (CI; Dekking et al., Reference Dekking, Kraaikamp, Lopuhaä and Meester2006) was used to determine the number of residuals to average to reduce the influence of the process noise below a certain threshold. However, as stated in Hespanhol et al. (Reference Hespanhol, Vallio, Costa and Saragiotto2019), the Bayesian credible intervals (BCI) are a more appropriate measure for this purpose. Through BCIs, it is found that by averaging residuals $ {\unicode{x025B}}_m(t) $ over 1-week timestamps, the noise in $ {\unicode{x025B}}_m(t) $ is reduced to below 1%, increasing the probability of detecting genuine shifts, despite the inherent noise in the natural frequency. However, $ {\xi}_m(t) $ and $ {\delta}_m(t) $ could still vary within this time window. To address this, UQ methods for OOD detection are applied to ensure that the inputs to the ML model are drawn from the training distribution. By focusing only on reliable predictions, the contribution of $ {\xi}_m(t) $ to the residual is minimized, as later shown in Section 3.3.2. Consequently, if a shift in the averaged residual is detected during periods of low uncertainty, it can be attributed to $ {\delta}_m(t) $ , indicating the detection of an anomaly.

This strategy requires incorporating UQ into the predictions. However, deterministic point predictors, like those discussed in Section 3.2.3, while straightforward to train and implement with strong predictive capabilities, do not provide inherent UQ. Fortunately, recent advancements in UQ methodologies now enable uncertainty estimation for both tree-based models (Malinin et al., Reference Malinin, Prokhorenkova and Ustimenko2020; Mondal, Reference Mondal2021) and ANNs (Gawlikowski et al., Reference Gawlikowski, Tassi, Ali, Lee, Humt, Feng, Kruspe, Triebel, Jung and Roscher2021; Pearce et al., Reference Pearce, Leibfried and Brintrup2020). These techniques allow for more robust predictions by quantifying uncertainty, making them suitable for improving the reliability of the proposed SHM framework.

Uncertainty can be divided into either aleatoric, relating to variability which is due to inherently random effects (Hüllermeier and Waegeman, Reference Hüllermeier and Waegeman2021), or epistemic, stemming from incomplete model knowledge (Bastani and Alur, Reference Bastani and Alur2024). Aleatoric or data uncertainty arises from physical phenomena that are random by nature. Therefore, the aleatoric uncertainty cannot be easily quantified and eliminated during modeling calibration (Wang et al., Reference Wang, Nord, Ziemer and Li2023). On the other hand, epistemic, model, or knowledge uncertainty arises from inadequate knowledge of the model to explain the data from inputs from regions either far from the training data or sparsely covered by it. This uncertainty is reducible by providing more knowledge about the problem to the ML model (e.g., providing more and new training data or optimizing the model’s hyperparameters). Various empirical UQ techniques exist, as detailed in Poggi et al. (Reference Poggi, Aleotti, Tosi and Mattoccia2020) and Mondal (Reference Mondal2021), offering methods to approximate the epistemic uncertainty such as Bayesian neural networks (BNN) and Bayesian ensembles. Because BNNs entail significantly higher computational costs compared to point-wise predictors (Bai and Chandra, Reference Bai and Chandra2023), it is decided to explore Bayesian ensembles.

Bayesian ensembles consist of an ensemble or collection of $ {n}_{\mathrm{BE}} $ models with different configurations that all make predictions on the same data. The final prediction is an average of the predictions of all models, and epistemic uncertainty is quantified by measuring the variance or diversity of these predictions (Chipman et al., Reference Chipman, George and McCulloch2006). As it has been shown in Section 3.2.3 that tree-based learners outperform ANNs for the analyzed tabular data, this research focuses on Bayesian ensembles constructed from tree-based models for UQ (Chipman et al., Reference Chipman, George and McCulloch2006; Duan et al., Reference Duan, Anand, Ding, Thai, Basu, Ng and Schuler2020). The construction of $ {n}_{\mathrm{BE}} $ different models of the Bayesian ensemble is detailed by Malinin et al. (Reference Malinin, Prokhorenkova and Ustimenko2020) for gradient boosted decision trees (GBDTs), using stochastic gradient Langevin dynamics. However, the creation, storage, and prediction of these Bayesian ensembles can be computation- and memory-intensive, as it requires training and storing of the $ {n}_{\mathrm{BE}} $ models.

Therefore, Malinin et al. (Reference Malinin, Prokhorenkova and Ustimenko2020) offer a practical solution to the computational demands of these Bayesian ensembles through virtual ensembles. By recognizing that GBDTs are inherently an ensemble of individual decision trees, a virtual ensemble can be constructed from a single GBDT model. This virtual ensemble comprises $ {n}_{\mathrm{BE}} $ truncated submodels, each representing a stage in the GBDT’s creation process. The final proposed strategy uses the virtual ensemble as implemented in the Catboost Python package (Prokhorenkova, Reference Prokhorenkova2020) to quantify the knowledge uncertainty on the frequency predictions from a CB regression model.

3.3.1. Smart tracking

Traditional mode tracking from OMA often relies on reference thresholds. However, as demonstrated in Xiang et al. (Reference Xiang, Wang, Wang, Pan, Liu, Le and Cai2024), model-based strategies employing RF and XGB models offer an alternative. Instead of setting static upper and lower thresholds on the natural frequency for tracking, only results $ {f}_i(t) $ that are sufficiently close to the ML model predictions $ \hat{f}{\left(\boldsymbol{x}(t)\right)}_m $ are tracked as mode $ m $ . The proposed methodology integrates prediction uncertainties obtained through UQ to assess the validity of the model itself given the prevailing EOV. This translates the tracking strategy into the rules given in Equation (6).

(6) $$ {U}_m\left(\boldsymbol{x}(t)\right)\le {\tau}_{u_m}\hskip1em \wedge \hskip1em \mid {\hat{f}}_m\left(\boldsymbol{x}(t)\right)-{f}_i(t)\mid \le {r}_{f_m}\hskip1em \Rightarrow \hskip1.12em {f}_i(t)\in m. $$

In this equation, $ {U}_m\left(\boldsymbol{x}(t)\right) $ denotes the knowledge uncertainty obtained from the virtual ensemble for input variables $ \boldsymbol{x}(t) $ . The uncertainty threshold $ {\tau}_{u_m} $ is chosen as the 90th percentile of uncertainty from the training data $ {U}_{m_{tr}} $ . The tracking range is set as a $ 3\sigma $ band around the error predictions from the training data, or $ {r}_{f_m}=3\sigma \left({\hat{f}}_m-{f}_m\right) $ .

The concept is illustrated in Figure 6 in comparison to a classic reference-based tracking of a monopile-supported OWT’s SS2 mode. The traditional reference-based tracking bounds (blue) make no distinction in data quality and require a wide tracking range to accommodate the EOV. Meanwhile, in smart tracking, only results close to the ML model predictions are used, and only when uncertainty is lower than the uncertainty threshold $ {\tau}_{u_m} $ (orange). The resulting smart tracked modes (green) are then stored as values for the structural mode for further analysis.

Figure 6. Conceptual illustration contrasting smart tracking (based on ML and UQ) with traditional reference-based tracking for an OWT’s SS2 mode.

Figure 6 also illustrates the advantageous use of UQ in the smart tracking principle. The SS2 mode is expected to exhibit cyclic variations corresponding to tidal levels. However, discrepancies in this cyclic pattern, indicative of rotor harmonic interference, are flagged by a high uncertainty of the virtual ensemble. Meanwhile, (a simple) reference-based tracking would have still considered these results, potentially obfuscating changes in the natural frequency.

3.3.2. Decision-making

The final step in the vibration-based SHM framework is the decision-making process. In the case of natural frequency monitoring, frequency shifts after normalization are used to determine the structural health of the structure. In the proposed methodology, this is obtained by calculating the residual $ {\unicode{x025B}}_m(t) $ and separating parameter $ {\delta}_m(t) $ as defined in Equation (5). A substantial deviation in $ {\delta}_m(t) $ for any monitored frequency $ {f}_m(t) $ should trigger a structural health alert.

However, as stated in Section 3.2.4, the parameter $ {\delta}_m(t) $ cannot be directly measured. Therefore, an average of the residual $ {\unicode{x025B}}_m(t) $ is considered to remove $ {\nu}_m(t) $ , and only low uncertainty periods are considered, to ensure a low $ {\xi}_m(t) $ . This latter relationship is highlighted by the sparsification error curves (SECs) as proposed by Poggi et al. (Reference Poggi, Aleotti, Tosi and Mattoccia2020). SECs are drawn by sequentially removing the prediction for timestamp $ t $ with the highest knowledge uncertainty $ \max\;\left({U}_m(t)\right) $ from the predictions and then recalculating the MSE (cf. Equation (3)) on the new dataset with one uncertain sample less. The SECs depicted in Figure 7 for modes SS1, SS2, and FA1, illustrate a general MSE reduction with the exclusion of high-uncertainty samples, thereby refining the predictive accuracy and reducing modeling error $ {\xi}_m(t) $ .

Figure 7. Sparsification error curves for the physical modes of an OWT, obtained by sequentially removing the most uncertain predictions and showing the effect on the model performance through MSE.

Finally, either a distribution of the residuals over a time period or a control chart of the averaged residuals is drawn to monitor the changes in the residual for low-uncertainty predictions. Both the distribution and control charts are shown in the results of Section 4. The frequency shift that should trigger an alarm depends on the use case. For instance, Prendergast et al. (Reference Prendergast, Gavin and Doherty2015) associated a 5% shift in the SS1 mode with significant scour. Alternatively, it is possible to draw the limits on the frequency changes through physics-based models as demonstrated in the study by Weil et al. (Reference Weil, Sastre Jurado, Chaar, Winkler, Weijtjens and Devriendt2023) for the SS1 mode of a monopile-supported OWT. However, this is currently not incorporated into this research but will be explored in a subsequent study.