Modelling and Monitoring Erosion of the Leading Edge of Wind Turbine Blades

: Leading edge surface erosion is an emerging issue in wind turbine blade reliability, 1 causing reduction in power performance, aerodynamic loads imbalance, increased noise emission 2 and ultimately additional maintenance costs, and if left untreated, leads to the compromise 3 of the functionality of the blade. In this work, we ﬁrst propose an empirical spatio-temporal 4 stochastic model for simulating leading edge erosion, to be used in conjunction with aeroelastic 5 simulations, and subsequently propose a deep learning model trained on simulated data, which 6 aims to monitor leading edge erosion by detecting and classifying the degradation severity. The 7 main ingredients of the model include a damage process that progresses at random times, across 8 multiple discrete states characterized by a non-homogeneous compound Poisson process, which is 9 used to describe the random and time-dependent degradation of the blade surface, thus implicitly 10 affecting its aerodynamic properties. The model allows for one, or more, zones along the span of 11 the blades to be independently affected by erosion. The proposed model accounts for uncertainties 12 in the local airfoil aerodynamics via parameterization of the lift and drag coefﬁcients curves. 13 The proposed model is used to generate a stochastic ensemble of degrading airfoil aerodynamic 14 polars, for use in forward aero-servo-elastic simulations, where we compute the effect of leading 15 edge erosion degradation on the dynamic response of a wind turbine under varying turbulent 16 input inﬂow conditions. The dynamic response is chosen a deﬁning output as this relates to the 17 output variable that is most commonly monitored under a Structural Health Monitoring (SHM) 18 regime. In this context, we further propose an approach for spatio-temporal dependent diagnostics 19 of leading erosion, namely, a deep learning attention-based Transformer, which we modify for 20 classiﬁcation tasks on slow degradation processes with long sequence multivariate time-series as 21 inputs. We perform multiple sets of numerical experiments, aiming to evaluate the Transformer 22 for diagnostics and assess its limitations. The results reveal Transformers as a potent method for 23 diagnosis of such degradation processes. The attention-based mechanism allows the network to 24 focus on different features at different time intervals for better prediction accuracy, especially for 25 long time-series sequences representing a slow degradation process. 26


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A graphical abstract of this research is presented in Figure 2. The remainder of this article is organized as follows. In Section 2 we offer a review 108 of the state of the art in modelling and diagnostics of leading edge erosion. In Section 3 109 we present the details of the spatio-temporal stochastic model for leading edge erosion. 110 In section 4 we present the uncertainty modelling and the aeroelastic simulations setup.

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In section 5 we elaborate the theory of the deep learning multi-variate time-series based 112 Transformer for diagnostics and inference. In Section 6, we illustrate the novelty and The general thread in the first set of studies amounts to describing the constitu-123 tive laws, physical processes and instigators at the micro scale resulting in surface 124 erosion/degradation and the various stages and types of erosion severity e.g., [6][7][8][9][10]. 125 Such models incorporate the effect of wind speed, air density, particle size, incubation 126 time, and erosion intensity due to rainfall, snowfall, sea spray, and fog. It is unclear 127 however how the micro-scale models couple to the macroscale aspects of leading edge 128 erosion and consequently the effect on turbine performance. 129 The second set of studies quantifies the impact on wind turbine performance via 130 computational approaches (aeroelastic or CFD simulations) or extensive experimental 131 campaigns in wind tunnels, coupled to uncertainty propagation schemes e.g., [11][12][13][14][15][16]. 132 The limitations in all these works is that they do not allow for multiple zones along the 133 span of the blades to be independently affected by stochastic erosion processes.   Table 2. 315 We propose a two-step empirical spatio-temporal stochastic model of LEE.
Step 1 316 of the model includes a damage process that occurs at random times emulating the non-   The stochastic degradation process of a wind turbine blade LEE is given by a on seasonal variations such as rain intensity, and changes in environmental temperatures. 340 We thus infer that the rate of occurrence of shocks is periodic, as shown in Figure 5.
where, s is a a very short interval of time. Here o(s) is a function that is negligible compared to s, as s → 0. Assuming o(s) = g(s), then: One approach for generating Non-Homogeneous Poisson processes is the "process 360 analogue" of acceptance-rejection called thinning, which is the scheme we adopt in our 361 model. The procedure is as follows:

2.
Initialize t = 0 and I = 0 In the above procedure, the ratio λ u is the so-called thinning probability. In our 371 model λ u is chosen as the maximum of the rate function λ(t) (e.g. see Figure 5). For long interval. More specifically, we can write: The damage is always positive, i.e., that P(Y n ≥ 0) = 1, and the damage accumulates loss (e.g. rupture), but rather implying that the blade loses its ability to generate lift 398 efficiently. At this stage, we assume that once the blade section has reached level 9 of 399 erosion, the process stops under the assumption that any further degradation becomes 400 limited, since all the surface protective coatings and paint have already delaminated. 401 We adopt a classical approach whereby Y 1 , Y 2 , ..., Y n are each exponentially distributed 402 according to density: where µ is the mean jump (shock) magnitude. It follows that the sum Finally, sampled jumps of magnitude Y n > 0 for τ n < T are not allowed once the 412 compounded damage has reached its highest compounded severity class (Z max = 9) 413 before the end of time horizon T has been reached. This is reasonable as any additional 414 shocks (e.g. rain or hail) cannot not incur any additional erosion, as there are not any 415 more surface materials that can be easily eroded.  (C L ) and drag (C D ) coefficient to axial (a) and tangential (a ) induction factors, as follows: where σ is the rotor solidity, Φ is the angle of the incoming relative wind with the rotor 459 plane, λ = ωr V 0 , ω is the rotor speed, r is the radial distance from the rotor center, and V 0 is The conditional dependence between the turbulence σ U and the mean wind speed U is 514 defined in the Normal Turbulence Model described in the wind turbine design standard 515 [78]. Here, we elect to use a reference ambient turbulence intensity I re f = .16 (the 516 expected value of the turbulence intensity at 15 m/s is called I re f ). This dependency is 517 given by the local statistical moments of σ u ∼ LN µ σ U , σ 2 σ u as:

D R A F T
As a result, the turbulence intensity is expressed as: T i = σ u u .

519
The wind profile above ground level is expressed using the power law relationship, 520 which defines the mean wind speed u at height Z above ground as a function of the 521 mean wind speed u h , measured at hub height Z h as reference: where α is a constant called the shear exponent. The conditional dependence between 523 the wind shear exponent α ∼ N µ α , σ 2 α and the mean wind speed U is given by [79]: We define a custom conditional dependence between the inflow horizontal skew-525 ness Ψ and the mean wind speed U, truncated to [−11, 11] deg., The conditional dependence between the inflow vertical skewness Σ and the mean 527 wind speed U and turbulence intensity T i , truncated to [−6, 6] deg., is designed such as: The blade leading edge degradation scenario occurs over a 240-months period; 529 consequently, we introduce an additional element of uncertainty by allowing a time-530 varying aspect of the inflow conditions, namely, the mean wind speed E(U), the shape 531 parameter K U of the Weibull distribution, and the reference ambient turbulence intensity 532 I re f as shown in the example in Figure 12. 533 We choose to sample the wind inflow and the aerodynamic RVs using the Sobol   The layout of the airfoils along the span of the blade are shown in Figure 15. We also list 551 some of the more important properties of the simulated wind turbine in Table 3.

552
OpenFAST is a wind-turbine-specific nonlinear time-domain simulator that em-

579
As mentioned in the beginning of the paper, the motivation for this research is the  Table 4: Retained sensors output of aero-servo-elastic simulations.

Sensor name Description Time
Time steps of the simulations Wind1VelX X-direction wind velocity at hub-height B1N9Cl Lift force coefficient at Blade 1, Aerosense Node at 0.96R B1N9Cd Drag force coefficient at Blade 1, Aerosense Node at 0.96R

B1N9Al pha
Angle of attack at Blade 1, Aerosense Node at 0.96R

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The simulation pipeline described above allows us to generate data which replicate 589 the sensory output of Aerosense nodes. Thus, we are able to produce datasets to train  can be transferred to a label-scarce task, thereby improving the generalization 626 performance of the model as well as reducing the risk of overfitting.

627
• Physics-constraints shall be built-in the loss or likelihood function.

628
• The output predictions should be probabilistic in nature.

630
We utilize multivariate time-series data generated via the coupled simulation

2.
In a continuous monitoring context, are we able to diagnose jumps in LEE severity and 645 therefore identify the degradation path that the system takes?

660
The second set of experiments aims to assess whether using full degradation paths 661 in a continuous monitoring setup to train the diagnostics method is a suitable strategy.

662
Here, the datasets are comprised of full NHCPP degradation paths, albeit with reduced 663 uncertainty. Due to the stochastic degradation, the datasets do not have balanced classes.

664
Assuming that a continuous monitoring system is in place, we therefore have access to 665 the previous degradation states. This allows for degradation monotonicity to be enforced 666 in the prediction, thus constraining the output to physically possible solutions (i.e., it 667 is physically impossible to have a state which is less degraded than previous states -668 except for the case of direct service intervention for repair and maintenance, which is out 669 of scope in this work). Another objective is to evaluate whether severity grouping is a 670 viable strategy. Indeed, grouping the LEE stages by type (see Table 1) could be beneficial  Table 5 summarizes the differences between the datasets of the three sets of experi-686 ments, including the partition of data between the different subsets.

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where the projections are parameter matrices If we aim to predict the degradation at multiple zone along the blade, we can 739 simply stack multiple MLPs, one for each zone.   742 We first train and evaluate the Transformer model on the datasets with equal 743 amounts of degradation classes, in the sense of a traditional classification problem. Thus, 744 the objective function for this first task is the standard cross-entropy loss:

Loss functions
where y is a one-hot label vector indicating the correct class,ŷ is a vector containing the 746 predicted softmax probabilities for each class and n is the number of classes.

747
In the second and third set of experiments, it is assumed that the state of the system 748 is known for the previous sampling period. We use this information to enforce physical-749 ity; we add a second term to the loss function that penalizes a predicted degradation 750 class that is lower than the known previous class. This second term is the margin ranking 751 loss: whereĉ is the predicted class, c prev is the degradation class of the previous known state, 753 and m is the margin hyperparameter. Overall, the objective function for the second and 754 third experiments is therefore: where α is a hyperparameter used to balance the two components.     Here, we assess the use of stochastic degradation paths as diagnostic training points.
810 Table 8 shows accuracy scores gathered on an un-seen degradation path for experiments

Experiment set 3 843
In experiments 3.1 and 3.2, we assess how the diagnostics method performs on more 844 uncertain degradation paths, with multiple sources of variability. Not only are there 845 large class imbalances due to the stochastic degradation, but aerodynamic uncertainty 846 (see Figure 7) and long-term weather fluctuations makes it challenging to distinguish 847 between different LEE severities. Weather variability is not commonly used for long-term 848 aero-elastic modelling in the wind energy industry, and is somewhat unrealistic, but 849 the goal is to understand the limitations of the Transformer model by making inference 850 extremely challenging. We report in Table 9 the accuracy scores gathered on an un-seen