POD-Based Sparse Stochastic Estimation of Wind Turbine Blade Vibrations

TL;DR

The study addresses real-time reconstruction of full blade deformations from sparse measurements by combining a data-driven POD reduced-order representation with an azimuthally periodic stochastic ROM and Kalman fusion. The method projects blade displacements onto POD modes, uses Linear Stochastic Estimation for sparse sensing, and incorporates an azimuthal Fourier model to regularize dynamics across rotor azimuth; the fused estimator updates via a Kalman gain $\mathbf{K}$ to produce $\hat{\mathbf{a}}(t)$ and $\hat{\boldsymbol{\Sigma}}(t)$. Key contributions include a sensor-placement strategy based on QR factorization, a Fourier-based azimuthal prior for ROMs, and demonstration on OpenFAST/GEBT simulations showing accurate reconstruction of 3D blade motions under turbulence with only four sensors. The approach enables real-time blade monitoring, sensor optimization, and potential active load control, with implications for digital twins and broader aeroelastic applications.

Abstract

This study presents a framework for estimating the full vibrational state of wind turbine blades from sparse deflection measurements. The identification is performed in a reduced-order space obtained from a Proper Orthogonal Decomposition (POD) of high-fidelity aeroelastic simulations based on Geometrically Exact Beam Theory (GEBT). In this space, a Reduced Order Model (ROM) is constructed using a linear stochastic estimator, and further enhanced through Kalman fusion with a quasi-steady model of azimuthal dynamics driven by measured wind speed. The performance of the proposed estimator is assessed in a synthetic environment replicating turbulent inflow and measurement noise over a wide range of operating conditions. Results demonstrate the method's ability to accurately reconstruct three-dimensional deformations and accelerations using noisy displacement and acceleration measurements at only four spatial locations. These findings highlight the potential of the proposed framework for real-time blade monitoring, optimal sensor placement, and active load control in wind turbine systems.

Figures (16)

• Figure 1: Blade system of reference. CAD made available by gaertner2020definition.
• Figure 2: Example of free stream velocity $\mathbf{U}_\infty = (U_\infty, V_\infty, W_\infty)$ generated with Turbsim, for a mean horizontal wind speed of $\bar{U}_\infty = 10.6$ m/s and TI=5%.
• Figure 3: Analysis of the influence of the operative condition (wind speed, control) over the amplitude and frequency characteristics of $\bm{u}(z=L_b)$. The left column shows violin plots of blade tip displacement as a function of mean hub-height wind speed ($\bar{U}_\infty$), showed for TI=5% (blue distributions) and TI=15% (orange distributions). The right panels show their respective PSD grouped for all turbulence intensities. The raw spectra are smoothed with a Savgol filter using a window size of 33 and a third-order polynomial. The frequency axis is normalized with the 1P rotor frequency, i.e. $\hat{f}=f/f_{1P}$, and dashed vertical lines highlight the first $nP$ harmonics with $n=6$. Each row corresponds to a direction of displacement: the first row analyses the flapwise ($u_x$), the second row addresses the edgewise ($u_y$) and the last illustrates the axial response ($u_z$).
• Figure 4: Modal energetic decay of POMs and LNMs. The LNM have been sorted according to their amplitude and not according to the associated eigenfrequency $\hat{\omega}_n$, to ease comparison with the POD. The canonical ordering according to $\hat{\omega}_n$ is kept as a reference and indicated on top of each marker. The left $y$ axis shows the energy associated with each mode in the logarithmic axis. The right $y$ axis expresses it in percentage terms computed over the whole modal basis, illustrated by bars.
• Figure 5: Comparison of POD modes, indicated as solid blue lines, and LNMs expansions computed for the stand-alone blade, depicted with dashed black lines. The modal expansions are truncated at $n=4$ (see Figure \ref{['fig:sigmas_tot']}).