# Bayesian model updating of rotating wind turbines

Nina Delette, Jean-Lou Pfister, Enora Denimal Goy, Mohamed Reda El Amri, Laurent Mevel

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Nina Delette, Jean-Lou Pfister, Enora Denimal Goy, Mohamed Reda El Amri, Laurent Mevel. Bayesian model updating of rotating wind turbines. WESC 2025 - Wind Energy Science Conference, European Academy of Wind Energy, Jun 2025, Nantes, France. pp.1-3. hal-05380846

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Submitted on 25 Nov 2025

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# Bayesian model updating of rotating wind turbines

Nina Delette<sup>a,b</sup>, Jean-Lou Pfister<sup>a</sup>, Enora Denimal Goy<sup>c</sup>, Mohamed Reda El Amri<sup>a</sup>, and Laurent Mevel<sup>b</sup>

E-mail: nina.delette@ifpen.fr

**Keywords:** Model updating, Bayesian statistics, Vibration-based SHM, OMA, Floquet modes

## 1 Introduction

The rapid growth of wind energy has necessitated the development of larger-capacity turbines, featuring taller towers and longer blades. Yet, even as modeling capabilities have improved in recent years, the multi-physics nature of the problem combined with its structural complexity has limited prediction accuracy. Numerous unknown or uncertain properties within the system, particularly regarding material characteristics and geometric configurations, contribute to discrepancies between model simulations and actual turbine responses, making behavior prediction challenging. An effective approach to address these discrepancies is by tuning key model parameters using data from real-structure measurements.

This approach, known as model updating, has been extensively studied in the field of structural dynamics, with approaches ranging from deterministic to probabilistic. Deterministic methods focus on directly adjusting model parameters to minimize discrepancies, without accounting for uncertainties in the measurements or the model itself. Probabilistic methods, such as the Bayesian method, incorporate explicitly these inherent uncertainties. As detailed in [1], Bayesian approaches estimate a probability density function (PDF) of the parameters, offering a robust means of capturing their possible true values. These methods are particularly well-suited for applications involving uncertain data, making them highly relevant to wind turbine modeling. In this context, previous research has mostly focused on model updating for non-rotating turbines [2], or stationary components such as the tower and support structure [3], [4], avoiding the inherent complications caused by the periodic behavior of the system.

Although model updating has been extensively applied in civil engineering [1], wind turbines present unique challenges due to their time-periodic nature. In fact, the rotational motion of the blades induces time-periodic behavior, rendering it incompatible with classical vibration-based model updating techniques. However, recent research has demonstrated that such systems can be approximated by an equivalent Linear Time-Invariant (LTI) system by using the Floquet and Fourier decomposition [5]. The objective of this work is to propose a numerical framework to perform model updating on operating wind turbines. More precisely, a Bayesian framework will be employed on the Floquet-Fourier approximation of the operating wind turbines.

## 2 A new method for model updating of rotating wind turbine

In this study, synthetic vibration data (i.e. generated by a numerical model) will be considered as reference. To achieve model updating, relevant features must be defined to quantitatively assess the model's ability to replicate the data. In the context of vibration analysis, these features are typically modal frequencies and mode shapes. Operational Modal Analysis (OMA) can typically be used to extract these features from the measured vibration data, denoted as $\bar{d}$, assuming an underlying linear, time-invariant (LTI) system. On the other hand, modal frequencies and mode shapes can also be computed from the model, and then compared with the data-based estimation.

When the turbine is operating, the blades rotate with angular speed $\Omega$, making the system $\Omega$-periodic. This time-periodic behavior results in the emergence of Floquet modes, which describe the dynamic response of systems with periodic coefficients. An exact formalism thus requires to use Floquet theory [6], [7]. However, as demonstrated in [5], the system can still be approximated by a LTI system assembled from a truncated series of the Fourier components of the Floquet modes. These modes can then be identified using standard OMA algorithms like Stochastic Subspace Identification (SSI) algorithm [5]. This approximation will be used in our method.

<footnote><sup>a</sup>IFP Energies Nouvelles, 92852 Rueil-Malmaison, France</footnote>

<footnote><sup>b</sup>Univ. Gustave Eiffel, Inria, COSYS-SII, I4S, 35042 Rennes, France</footnote>

<footnote><sup>c</sup>Inria, CMAP, Ecole Polytechnique, IPP, Palaiseau, France</footnote>
<figure><img src="image_2.png" /><figcaption>Figure 1: Simplified wind turbine model with 5 DoF. [7]</figcaption></figure>

Using the Bayesian framework, the next step is to select the set of parameters, denoted with the vector $\theta_M$. The choice of a suitable set $\theta_M$ is not easy to do; it should be able to represent the unknown structural properties, but remains limited in order to avoid ill-conditioning problems [1]. A PDF is then defined for each parameter, representing the prior knowledge about the system. By comparing the measured modal properties with those calculated from the model, the likelihood function $L(\theta_M|\bar{d})$ can be calculated. This function reflects the likelihood of observing the data $\bar{d}$ given the model parameterized by the set $\theta_M$, accounting for uncertainties in the OMA process. This likelihood function can also be interpreted as the PDF of the experimental data, $p(\bar{d}|\theta_M)$, as it represents how well a model explains the observations $\bar{d}$. By applying the Bayes' theorem, the posterior distribution of the model parameters is obtained:

$$p(\theta_M|\bar{d}) = cp(\bar{d}|\theta_M)p(\theta_M) \quad (1)$$

With c the normalizing constant ensuring that the posterior PDF integrates to one. This process, which updates the distribution of parameters $\theta_M$ to align with the measured data $\bar{d}$, is the Bayesian model updating. Since computing the joint PDFs requires solving high-dimensional integrals to compute c, a sampling method is employed —specifically, Markov Chain Monte Carlo (MCMC).

## 3 Application to a rotating 5-DoF wind turbine

A simplified wind turbine model with 5 degrees of freedom (DoF) [7], shown in Figure 1, is employed to apply the presented framework. An arbitrary set of parameters is defined to simulate a hypothetical “real” turbine. Using this fixed parameter set, the model generates noisy acceleration time series for each DoF simulating the vibration measurements that could be obtained from an operational turbine. Bayesian model updating is then employed. The parameters to be updated are the rotational stiffness of each blade, denoted as $\theta_M = \{G_1, G_2, G_3\}$. The prior PDF of each of these parameters is chosen as a gamma distribution of mean value equal to 8 Nm. As previously mentioned, due to the blades' rotation, this system is periodic. The equivalent LTI system is approximated by retaining the 12 most significant Floquet harmonics, as listed in Table 1. Despite the reduced representation, this approximation effectively captures the majority of the system's dynamic behavior.

<table><caption>Table 1: The 12 Floquet harmonics with a participation factor > 1%, for the 5 DoF wind turbine with the prior parameters, $\theta_M = \{8Nm, 8Nm, 8Nm\}$ and with the parameters of the synthetic dataset, $\theta_d = \{8.6Nm, 7.3Nm, 9Nm\}$.</caption><thead><tr><th>Floquet mode</th><th>Participation factor prior [%]</th><th>Frequency prior [Hz]</th><th>Frequency updated [Hz]</th></tr></thead><tbody><tr><td rowspan="2">Backward Whirling</td><td>4.9</td><td>0.448</td><td>0.458</td></tr><tr><td>95</td><td>0.671</td><td>0.681</td></tr><tr><td rowspan="3">Forward Whirling</td><td>88</td><td>0.641</td><td>0.640</td></tr><tr><td>10</td><td>0.864</td><td>0.862</td></tr><tr><td>2.2</td><td>1.087</td><td>1.085</td></tr><tr><td rowspan="3">Tilt</td><td>50</td><td>1.369</td><td>1.376</td></tr><tr><td>28</td><td>1.592</td><td>1.599</td></tr><tr><td>22</td><td>1.815</td><td>1.822</td></tr><tr><td rowspan="3">Yaw</td><td>34</td><td>1.247</td><td>1.257</td></tr><tr><td>27</td><td>1.470</td><td>1.480</td></tr><tr><td>38</td><td>1.693</td><td>1.703</td></tr><tr><td>Symmetric</td><td>100</td><td>0.746</td><td>0.762</td></tr></tbody></table>

The Figure 2 illustrates the MCMC sampling of the updated PDF for each parameter. The sample size of
<figure><img src="image_3.png" /><figcaption>Figure 2: Prior and posterior distributions of the parameters in the rotating case.</figcaption></figure>

10,000 was chosen to ensure convergence of the updating process. The prior PDF, shown in red, represents the initial knowledge of the rotational stiffness for each blade, assumed to be identical. The histograms of the final distributions obtained through MCMC sampling are displayed in color for each parameter, with the associated dashed lines indicating the parameter values used to generate the synthetic dataset.

The Figure 2 shows that, despite the broad prior PDF reflecting limited prior knowledge, the method effectively updates the rotational stiffness of each blade. This results in a posterior PDF that is centered around the true parameter value and narrower than the prior, indicating a reduction in uncertainty about the parameter. The mean relative error for the parameters can be defined as:

$$ \bar{\epsilon} = \frac{1}{N} \sum_{i=0}^{N} \frac{\theta_d(i) - \theta_M(i)}{\theta_d(i)} \qquad (2) $$

Where $\theta_M$ are the parameters to update, and $\theta_d$ represents the parameters used to generate the synthetic dataset.

In this case, the initial relative error on the parameter vector was $\bar{\epsilon}_{ini} = 27.9\%$, while the relative error after updating is $\bar{\epsilon}_{up} = 0.8\%$, demonstrating the effectiveness of the model updating. The impact of this updating on the eigenfrequencies of the system can be seen in Table 1.

## 4 Conclusion

In this work, the effectiveness of applying Bayesian model updating on an operating wind turbine was shown. The method is based on an equivalent LTI formulation of the problem, that enables an easier data-based identification of the system. This method has been applied to a simplified wind turbine model and gives promising results for future application to realistic structures.

This approach relies on MCMC sampling, which demands numerous model evaluations and becomes computationally expensive as the model complexity increases. This bottleneck will be addressed in future work, using e.g. metamodels to accelerate the model evaluation.

## References

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