Modal properties and especially damping of operational wind turbines can vary over short time periods as a consequence of environmental and operational variability. This study seeks to experimentally test and validate a recently proposed method for short-term damping and natural frequency estimation of structures under the influence of varying environmental and operational conditions from measured vibration responses. The method is based on Gaussian process time-dependent auto-regressive moving average (GP-TARMA) modelling and is tested via two applications: a laboratory three-storey shear frame structure with controllable, time-varying damping and a flutter test of a full-scale 7 MW wind turbine prototype, in which two edgewise modes become unstable. Damping estimates for the shear frame compare well with estimates obtained with stochastic subspace identification (SSI) and standard impact hammer tests. The efficacy of the GP-TARMA approach for short-term damping estimation is illustrated through comparison to short-term SSI estimates. For the full-scale flutter test, GP-TARMA model residuals imply that the model cannot be expected to be entirely accurate. However, the damping estimates are physically meaningful and compare well with a previous study. The study shows that the GP-TARMA approach is an effective method for short-term damping estimation from vibration response measurements, given that there are enough training data and that there is a representative model structure.

A novel operational modal analysis (OMA) method

The dynamic properties of wind turbines (i.e. natural frequencies, damping ratios, and mode shapes) can be sensitive to changing EOCs

Estimating aeroelastic (or operational) damping accurately is essential for improving the design of multi-megawatt wind turbines, as it is a crucial design parameter for modelling fatigue and aeroelastic instabilities, i.e. stall-induced vibrations (SIVs) and vortex-induced vibrations (VIVs)

OMA covers a broad class of output-only system identification methods for estimating modal parameters for structures in operating conditions where the input (i.e. forcing or excitation) is not measured. Standard OMA techniques, for instance, the covariance-driven stochastic subspace identification (COV-SSI)

When identifying modal damping from output-only measurements, the

Nonstationary auto-regressive moving average (ARMA) time series models offer an avenue for accounting for nonstationary input and time-varying system characteristics, including modal parameters. ARMA models closely resemble the mathematical structure of discrete-time equations of motion, where the auto-regressive (AR) part plays the role of the left-hand (homogeneous) side, and the AR

One approach to capture the effects of changing EOCs on vibrating structures is to embed measured environmental and operational variables (EOVs) into the model. Various approaches have been proposed for this. Multi-megawatt wind turbines pose a particular challenge due to the intricate aero–servo–elastic interactions.

The present work concerns short-term damping (and natural frequency) estimation based on output-only measurements for structures influenced by short-term varying EOCs, where short term is of the order of seconds. The methods mentioned above are based on models conditioned on (e.g. 10 min)

Verification and validation are integral parts of establishing any new method or model in structural dynamics, and it is essential for output-only damping estimation methods due to the latent and elusive nature of damping. This work contributes, in particular, to experimental validation of the GP-TARMA approach for short-term damping estimation, suitable for application to wind turbines. The method is validated using vibration measurements from two distinctly different experimental setups: a laboratory shear frame with abruptly changing damping realized with electromagnetic dampers and a full-scale 7 MW wind turbine prototype deliberately driven to flutter-like instabilities (measurements published by

The paper is structured as follows: Sect.

This section summarizes the procedures for estimating short-term damping ratios (and natural frequencies) from output-only measurements using a GP-TARMA model, which is introduced and presented in detail in

The GP-TARMA model introduced in

For example,

ARMA models are closely linked with discrete-time equations of motion (EOMs). The AR part resembles the left-hand side of discrete-time EOMs, which means the AR coefficients carry the physical characteristics of the system it models, i.e. natural frequencies and damping ratios. The MA part resembles the right-hand side of discrete-time EOMs as it can capture the effect of stochastic excitation on the measured response

Equations (

For a given data set

A procedure for maximum likelihood (ML) estimation of the hyper-parameters and innovation variance in

With the model parameter

The marginal likelihood of the response in Eq. (

The EM algorithm has been shown to converge to a

Once a GP-TARMA model is estimated, it is important to validate that it adequately represents the observations

An alternative whiteness test, fully applicable to the nonstationary case, is a simple

Cross-validation is performed by splitting a data set of a single recording in a training and a test set containing 75 % and 25 % of the data points, respectively. The model is solely estimated using the training set and is subsequently tested in terms of residual tests and whether the orders of magnitude of prediction errors are the same for the training and test set. If the prediction errors of the training set are much smaller than those of the test set, it suggests the model is over-fitted; i.e. the model excessively represents the measured realization of the stochastic response rather than the underlying system. However, it is only applicable to the nonstationary case if the two sets have comparable characteristics.

In this section a procedure for identifying a suitable model structure, i.e.

To compare the predictive performance (i.e. the prior one-step-ahead prediction errors) of the candidate models, the residual sum of squares normalized by the series sum of squares (RSS/SSS) and the Bayesian information criteria (BIC) are used. The RSS/SSS is given by

A simple backward regression scheme is employed to identify a suitable model structure, i.e. starting with high model orders,

Model structure identification procedure (adapted from

In this section the necessary results for estimating “frozen” modal parameters (excluding mode shapes) from an estimated GP-TARMA model and approximating the corresponding uncertainties are summarized. The frozen properties of a time-varying system represent the LTI properties at frozen time

The frozen modal parameters can be computed from the time-varying AR coefficients since an equivalent discrete-time state-space model with system matrix

The

The uncertainty in the modal parameter estimates can be approximated by propagating the estimated AR coefficient uncertainty (quantified by

The analytic uncertainty propagation method is only valid for

The procedure for computing natural frequencies and damping ratios and their uncertainties from AR coefficients is summarized in Table

Procedure for short-term modal parameter estimation using the GP-TARMA model (adapted from

In this section some considerations that may be useful for the practical application of the GP-TARMA approach are listed. These are based on using the GP-TARMA model for various applications, including the wind turbine blade response of an operational wind turbine during an extreme coherent wind gust (BHawC simulation; see

Reduce bandwidth of response (output). Low-pass filter and subsequently down sample the response prior to GP-TARMA modelling such that the reduced bandwidth only contains the modes of interest; i.e. unimportant response components at higher frequencies are filtered out. This reduces the computational cost of estimating the model and may help reduce the risk of over-fitting in cases with low amounts of training data.

Filter out high-frequency content of EOV signals by low-pass filtering. This may allow for a better fit, as modal properties usually do not vary at high frequencies (many times per second) and may prevent high-frequency noise propagating from AR coefficients to modal parameters.

Use zero-phase filtering for low-pass filtering of response and EOV signals. Consequently, the response and EOV signals are not phase shifted by filtering.

Add a small amount of artificial zero-mean NID noise to the (low-pass-filtered) response signal prior to GP-TARMA modelling. Adequate RMS noise levels relative to the signal are typically between 1 % and 5 %. This is an example of

This section presents an experimental setup of a shear frame (SF) structure with controlled time-varying damping and the experimental procedures used to generate response measurements for testing and validating the GP-TARMA approach. In

An experimental test rig with a three-storey SF structure was used to test modal parameter estimation methods for structures with time-varying damping. The structure has a pair of voltage-controlled electromagnetic dampers (EMDs) at each floor. Two types of tests were conducted: a base-excitation test, where the SF structure was excited with a shaker table, and a hammer test, where the structure was excited with a (measured) hammer impact. Figure

Experimental setup.

Laboratory shear frame test setup (cf. Fig.

As can be seen in Fig.

The input signal for the shaker table used for the base-excitation test is a stochastic normally distributed broadband signal (i.e. low-pass-filtered white noise) to achieve stochastic excitation of the first three modes of the SF structure. The test is run for 60 min, all signals are collected using a sampling frequency of 50 Hz, and all EMDs are turned on and off during the test with periods of constant voltage ranging from 240 to 540 s.

For the base-excitation test, the displacement response at the third floor is used and referred to as

Measured response comprising training data for

To validate modal estimates obtained from the output-only base-excitation tests, the modal parameters are also estimated by standard impact hammer tests (e.g.

A suitable model structure for modal analysis based on the third-floor response

The AR and MA model orders

Model order selection measures for the training and test set (circle and red crosses) for varying AR and MA model orders.

Inspecting the frequency stabilization diagram in Fig.

Frequency stabilization diagrams showing the time-averaged mean estimates of eigenfrequencies for increasing model orders, overlaid by the PSD (solid orange line) of response

In this section the model structure identified in Sect.

Residual analysis from the training and test sets (solid black line and solid red line).

The above whiteness tests, based on both standardized residuals and sign changes, suggest that the present GP-TARMA model is valid and well suited for downstream analysis.

This section presents modal parameter estimates computed from the validated GP-TARMA model of the third-floor displacement response

GP-TARMA estimates compared to SSI estimates.

The predicted response in Fig.

Good agreement for natural frequency estimates between GP-TARMA and SSI is observed. The modal parameter estimates are also summarized in Table

Natural frequency

The figure and table show good agreement between SSI and GP-TARMA damping estimates, where the best agreement is observed for the second mode. The recommended minimum measurement time

Convergence of damping estimates with respect to training data.

The widest damping estimate confidence intervals for the first to third mode are

Table

The SF test results presented in this section experimentally validate the efficacy of the GP-TARMA method in providing representative short-term damping estimates and illustrate its efficacy in the short-term case compared to a traditional OMA method.

In this section the GP-TARMA approach is applied to edgewise blade response measurements. Specifically, the method is used to estimate short-term, linear equivalent modal damping of edgewise rotor modes, which are deliberately driven to flutter-like instabilities corresponding to negative damping values.

The experiments were conducted with an SWT-7.0-154 prototype wind turbine located at the DTU Wind Test Centre in Østerild, Denmark. The measurements collected in December 2018 were published by

Figure

Measured response comprising training data for

The amount of available training data from this test is quite small relative to the model complexity (i.e. the number of parameters to be estimated) needed for the GP-TARMA model to represent the complex response and underlying system. The bandwidth of

The model structure is identified following the procedure in Table

Both EOVs are zero-phase low-pass filtered with a cut-off frequency of 0.3 Hz (see Sect.

Frequency stabilization diagrams showing the time-averaged mean estimates of eigenfrequencies for increasing model orders, overlaid by the PSD (solid orange line) of response

The model structure identified in Sect.

Residual analysis from the training and test sets (solid black line and solid red line).

The residual analysis implies that the NID assumption of the innovations is violated; i.e. the model does not completely represent the statistical structure of the response. However, the frequencies at which the residuals have the largest magnitudes do not coincide with the frequencies of the two modes of interest. Thus, the modal parameters computed from the GP-TARMA based on the present data set are not expected to be entirely accurate but may still offer some insight. More available measurement data (ideally including more instability measurements) for model training could improve the model accuracy by enabling more robust and accurate model parameter estimates. In addition, a single-output model (as the actual GP-TARMA model) cannot account for the whirling effect since the frequencies of the forward- and backward-whirling modes coincide in a response measured in the rotating frame. Such model-form error might cause correlated model residuals.

GP-TARMA model estimates as functions of time.

Figure

The uncertainty associated with damping estimates may seem considerable, as the confidence intervals cover both positive and negative damping values during the flutter-like instabilities. The uncertainty might be reduced by addressing the two limiting factors alluded to in Sect.

Figure

Modal damping of the first and second backward-whirling modes in terms of logarithmic decrement

The figure shows that the GP-TARMA estimates agree quite well with the exponential fit estimates, especially near the critical rotor speeds, but the GP-TARMA damping estimates tend to be higher than the exponential fit estimates. Table

Comparison of stability limit estimates of the first and second edgewise backward-whirling modes in terms of normalized rotor speed

A recently proposed approach based on a Gaussian process time-dependent auto-regressive moving average (GP-TARMA) model for short-term damping (and natural frequency) estimation from output-only vibration response measurements for vibrating structures influenced by environmental and operational variability has been experimentally tested and validated with two distinctly different experimental setups: a laboratory shear frame structure with time-varying damping properties achieved with electromagnetic dampers and a full-scale 7 MW wind turbine prototype which was deliberately driven to flutter-like instabilities. The primary idea of the GP-TARMA approach is to condition the model parameters on measured time series of environmental and operational variables, which may enable short-term tracking of system parameters like time-varying damping and natural frequencies.

An experimental setup consisting of a shear frame structure equipped with electromagnetic dampers was presented and shown to effectively realize a system with abruptly changing damping. Short-term natural frequencies and damping ratios were estimated using the GP-TARMA model and were shown to compare well to SSI and hammer test estimates in cases where the system was time invariant. Uncertainties were observed to be larger for the first mode compared to the second and third modes, but this could be explained by the first mode being trained on effectively less training data. GP-TARMA damping estimates were compared to short-term SSI estimates based on windows of 30 s measurements. The short-term SSI estimates were observed to be inconsistent and deviating from the remaining estimates, which illustrated the effectiveness of the GP-TARMA method for short-term damping estimation relative to traditional OMA methods. The laboratory test validated the efficacy of the GP-TARMA approach in short-term damping (and natural frequency) estimation, given a sufficient amount of training data and a representative model structure.

The GP-TARMA model was also tested using edgewise blade deflection measurements from a full-scale 7 MW wind turbine prototype during a flutter test. The first and second edgewise backward-whirling modes were found to exhibit flutter-like instabilities, in agreement with a previous study. The mean damping estimates were considered qualitatively meaningful, as the GP-TARMA model predicted negative damping for two modes coinciding in time and frequency with exponentially increasing vibration amplitude. The mean damping estimates also compared quite well with estimates from a previous study obtained from the same data. The estimated stability limits, i.e. the rotor speeds at which the damping becomes zero, showed quite good agreement with a previous study. However, the model validation implied that the model residuals did not resemble white noise, meaning that the GP-TARMA model trained on the available data cannot be expected to be entirely accurate. The correlated model residuals and uncertainties of the damping estimates could potentially be reduced by training the GP-TARMA model on more data and extending the GP-TARMA model to a multiple-output model to represent whirling modes better.

The GP-TARMA approach appears to be an effective way of estimating short-term damping based on output-only measurements, given enough training data and a representative model structure. The use of GP-TARMA models for analysing transient instabilities has been showcased. SSI and other standard OMA methods are easier to implement and apply than the GP-TARMA approach since it does not require much prior knowledge of the system; i.e. these should be used for applications where the LTI assumptions are valid but may be inadequate for applications with considerable short-term EOC variability.

Laboratory measurements are available upon request. Wind turbine test data are confidential. Code might be shared upon request.

Conceptualization and methodology: KLE, PC, and JJT; validation: KLE, PC, LMS, and JJT; investigation and visualization: KLE and LMS; writing (original draft): KLE and LMS; supervision and writing (review and editing): PC and JJT; software and formal analysis: KLE.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This work is partially funded by the Innovation Fund Denmark (grant 0153-00179B). The shaker table used in the laboratory setup was donated by COWIfonden (grant A-155.01).

This research has been supported by the Innovationsfonden (grant no. 0153-00179B) and the COWIfonden (grant no. A-155.01).

This paper was edited by Maurizio Collu and reviewed by two anonymous referees.