To assess the structural health and remaining useful life of wind turbines within wind farms, the site-specific structural response and modal parameters of the primary structures are required. In this regard, a novel inverse-problem-based methodology is proposed here to identify the dynamic quantities of the drivetrain main shaft, i.e. torsional displacement and coupled stiffness. As a model-based approach, an inverse problem of a mathematical model concerning the coupled-shaft torsional dynamics with high-frequency SCADA (supervisory control and data acquisition) measurements as input is solved. It involves Tikhonov regularisation to minimise the measurement noise and irregularities on the shaft torsional displacement obtained from measured rotor and generator speed. Subsequently, the regularised torsional displacement along with necessary SCADA measurements is used as an input to the mathematical model, and a model-based system identification method called the collage method is employed to estimate the coupled torsional stiffness. It is also demonstrated that the estimated shaft torsional displacement and coupled stiffness can be used to identify the site-specific main-shaft torsional loads. It is shown that the torsional loads estimated by the proposed methodology is in good agreement with the aeroelastic simulations of the Vestas V52 wind turbine. Upon successful verification, the proposed methodology is applied to the V52 turbine to identify the site-specific main-shaft torsional loads and damage-equivalent load. Since the proposed methodology does not require a design basis or additional measurement sensors, it can be directly applied to wind turbines within a wind farm that possess high-frequency SCADA measurements.

Monitoring of wind turbines within wind farms is increasingly becoming very important due to the need to detect anomalous behaviour, plan inspections or preventive maintenance, and compute the remaining useful life of specific structures. The site-specific structural dynamic quantities such as structural response and modal parameters could assist in the condition monitoring of wind turbines.

The structural response of the wind turbines is measured using load instrumentation such as accelerometers

The addition of new instrumentation to existing turbines, such as the installation of strain gauges and accelerometers, can be costly and also require repetitive calibration and synchronisation of their measurement signals with the turbine computer. Most wind farm operators also do not possess the aeroelastic design parameters of their wind turbines and hence cannot simulate the mechanical loads acting on their wind turbines. Monitoring turbine primary structures through existing SCADA-based (supervisory control and data acquisition) measurements can yield a cost-effective solution and provide valuable information to the wind farm operator. Usually, such monitoring through SCADA only provides information on power performance without regarding turbine structural integrity. Since there are two SCADA signals (i.e. rotor speed and generator speed) related to torsional oscillations of the main shaft of wind turbine drivetrains, the same can be used to quantify the torsional dynamics of the main shaft.

For this purpose, an inverse-problem-based approach is developed here to determine the torsional stiffness and response of the main shaft of a wind turbine, using existing high-frequency SCADA measurements such as the rotor speed and generator speed. This is a cost-effective alternative approach that is being proposed for the main shaft without using any additional measurement sensors or an aeroelastic design basis of the wind turbine. The proposed inverse-problem approach is a model-based approach whereby a mathematical model concerning the shaft torsional dynamics will be utilised to obtain both the torsional displacement and the coupled torsional stiffness in a continuous time domain. It involves Tikhonov regularisation

The concerned mathematical model comprised of differential equations will be solved for the shaft torsional velocity with high-frequency rotor and generator speed measurements as inputs. Subsequently, the main-shaft torsional displacement is obtained by numerically integrating the shaft torsional velocity. However, numerical integration is based on time-marching algorithms, and the lack of initial conditions makes displacement reconstruction an ill-posed problem

Upon obtaining the regularised shaft torsional displacement, the same mathematical model is utilised to obtain the shaft stiffness. For this purpose, the collage method – a model-based system identification technique – is used

The estimated shaft torsional stiffness and displacement are further used to identify the site-specific shaft torsional load. This novel methodology can potentially benefit wind farm owners, since both the property of the structure in terms of its stiffness and the structural response and the site-specific load can be determined without requiring additional sensors or information from the wind turbine manufacturer. The main-shaft torsional load affects the fatigue performance of other drivetrain components such as the gearbox and planetary bearings

The rest of the paper is organised as follows: the problem formulation consisting of Tikhonov regularisation and the collage method is given in Sect. 2; Sect. 3 presents the verification of the proposed formulation; and application of the proposed formulation on measurements is presented in Sect. 4.

A two mass model of the wind turbine drivetrain.

As mentioned in the previous section, the main objective is to identify the shaft torsional displacement and coupled stiffness from SCADA measurements. This is achieved by solving the shaft torsional dynamical equations using a suitable inverse-problem algorithm and the estimated shaft torsional displacement

Given the modal parameters (

Flowchart depicting the inverse-problem algorithm.

In the following, implementation of Tikhonov regularisation and the collage method on the drivetrain torsional dynamics will be discussed in detail.

Given

Comparison of time integration displacement with actual displacement. The torsional displacement is in radians.

Implementation of Tikhonov regularisation on the velocity to obtain the displacement is not readily available in the literature, and hence the same is presented in Appendix A1 for the sake of completeness. By following the procedure outlined in Appendix A1, the regularised torsional displacement (

Comparison of the Tikhonov and time integration displacements with actual displacement.

Upon estimating

To test the applicability and efficiency of the collage method for the wind turbine drivetrain system, a verification study is undertaken by comparing the main-shaft torsional stiffness obtained using the collage method with its design value for two different wind turbines. This is done for the following two wind turbines, (i) DTU (Technical University of Denmark) 10 MW

Comparison of estimated

Aeroelastic simulations are performed for the Vestas V52 turbine corresponding to the design load case (DLC 1.2)

Estimated

Comparison of regularised

Since the displacement is reconstructed from the velocity, a dynamic quantity, the reconstructed displacement will have a mean of zero, and this displacement component is referred to as a dynamic component of the displacement (

Upon ensuring the correctness of

Even though the dynamic component of the displacement is sufficient enough for

Comparison of reconstructed time series and power spectral density (PSD) of the torsional load for two different mean wind speeds.

Upon estimating the torsional load, the torsional damage-equivalent load (DEL) at each mean wind speed is calculated using the following equation:

Comparison of the predicted DEL with the DEL computed from aeroelastic simulations over all mean wind speeds.

Regularised

Identified torsional loads (

Estimated torsional DEL from SCADA measurements of the Vestas V52 turbine.

Upon verifying the proposed method, the drivetrain main-shaft torsional loads are estimated from the SCADA measurements without any need for the aeroelastic model. SCADA measurements taken during January 2019 for the Vestas V52 850 kW research turbine installed at the DTU Risø site is utilised for this purpose. The measurement data consist of 4459 10 min recorded cases with 50 Hz sampling frequency. Generator torque is used as one of the SCADA signals instead of the generator speed for this part of the study, and the generator speed is obtained from the generator power and generator torque (on the low-speed side) SCADA signals as

Normal operation filter conditions.

Using the rotor speed and generator speed,

The regularisation parameter is obtained for each mean wind speed measurement using the L-curve criterion and the obtained values not presented here for the sake of brevity. Subsequently, by applying the collage method on Eq. (

After estimating the torsional loads for all 627 cases, the identified loads are grouped according to the mean wind speeds ranging from 6 to 22 m/s which are subdivided into nine wind speed bins of 2 m/s width each. Subsequently, the torsional DEL is calculated for each mean wind speed using Eq. (

A novel inverse-problem-based approach is developed for estimating the main-shaft torsional displacement and stiffness by using high-frequency SCADA measurements. A mathematical model describing the coupled-shaft torsional dynamics is used for this purpose. Numerical errors and the effect of measurement noise on the torsional displacement reconstruction are minimised through the Tikhonov regularisation technique. Subsequently, the collage method is used to estimate the main-shaft coupled torsional stiffness. The estimated main-shaft quantities are then used to identify the main-shaft site-specific torsional load. The proposed formulation is successfully verified for the main-shaft torsional loads with the aeroelastic simulation of the Vestas V52 turbine. Upon verification, the methodology is extended to identify the site-specific main-shaft torsional loads of the same turbine by using SCADA measurements. For this purpose, the measurement data from the DTU Risø site are utilised, and the measurement data are filtered and calibrated for the turbine normal operation. Using the identified torsional loads, the torsional DEL is obtained. Depending on the prior information about the stiffness value, one can either use the entire proposed methodology or follow the torsional displacement estimation part of the proposed methodology for the torsional-load identification. Since the site-specific SCADA measurements are used in the analysis, the obtained loads can give a better estimate of the remaining fatigue life of drivetrain components. Monitoring the estimated loads can help in inspection planning and scheduling maintenance activities. As the proposed methodology does not require any design basis or an aeroelastic design basis, it can be used for wind turbines that possess high-frequency SCADA measurements for the estimation of the main-shaft torsional load and DEL.

By definition, the velocity

The choice of regularisation parameter

methods based on knowledge of measurement errors

methods that do not require details about measurement errors.

For a given initial-value problem (IVP),

The exact solution

The code regarding the mathematical models developed in the article can be accessed at

The open-source DTU 10 MW wind turbine aeroelastic model can be accessed at

Vestas V52 wind turbine SCADA data and its other parameters are not publicly available. However, Vestas v52 SCADA data can be requested by signing a non-disclosure agreement.

WDR conceived the methodology; completed the formal analysis, investigation, and validation of the project; and wrote the original draft of the paper. AN conceived the original idea, developed the scientific methods, reviewed and edited the paper, and supervised the project.

The authors declare that they have no conflict of interest.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work has been funded by the Energy Technology Development and Demonstration Program (EUDP) of the Energistyrelsen, Denmark (LIFEWIND project; grant no. 64017-05114). Investigations were carried out at the Department of Wind Energy, Technical University of Denmark.

This paper was edited by Amir R. Nejad and reviewed by Edward Hart and one anonymous referee.