LEE is primarily driven by material stresses due to the collision of particles (e.g., rain drops) with the blade surface . This has the effect of damaging the blade coating, ultimately leading to surface roughening and the growth of pits and gouges . As the blade is roughened, the transition between laminar and turbulent airflow shifts, , reducing lift and increasing drag . Rather than using computationally expensive CFD simulations to model airflow over rough surfaces, for the results in this paper we adapt a phenomenological LEE model developed by that is straightforward to implement within the AeroDyn module of OpenFAST. With this simplified model, the effect that erosion has on the aerodynamic properties of the blade edge is represented via a spatial perturbation of the lift and drag polars as a function of the angle of attack. based these perturbations on wind tunnel experiments in which erosion defects were simulated using roughened tape applied to the blade .

As in , we quantify the severity of erosion using a finite ordered set of damage classes, which encode the overall erosion level of the blade. These classes provide a practical link between observed surface degradation, aerodynamic performance loss, and potential remedial actions .

Although we do not model the time evolution of erosion, the idea is that erosion severity will be greater for older blades that have experienced more impact events. In addition, it is reasonable to assume that, to first order, the amount of erosion depends linearly on position along the blade. This erosion is expected to initiate and progress more rapidly in blade regions that are further from the hub . Finally, to take into account the inherent variability in erosion, we model the spatial dependence of erosion stochastically .

We now describe the details of this erosion model. For this study, we use five erosion severity classes parameterized by α∈{0.00,0.25,0.50,0.75,1.00}. We divide the blade into six erosion regions via a grouping of the AeroDyn nodes defined in the NREL 5MW RWT (see Table ; ) summarized in Table  and shown in Fig. . We let ei∈[0,1] denote the erosion level of the ith blade region, where ei=0 corresponds to a clean surface, and ei=1 corresponds to the maximum modeled erosion state. We model the erosion vector as a correlated random field, e=(e1,⋯,e6)∼MVN(μ(α),Σ(α)), where both the mean and covariance depend on the erosion severity class, α. To incorporate the linear relationship between blade velocity and radial distance into the model, we define the mean erosion level by μ(α)=(αx1,…,αx6), where xi is the normalized radial distance from the hub to the midpoint of the ith blade region. The covariance matrix is defined by Σ(α)i,j=min⁡(xi,xj)α102exp⁡-(xi-xj)2xi+xj. This heuristic covariance model imposes stronger correlation between the erosion level in nearby blade regions while allowing erosion variability to increase both toward the blade tip and with erosion severity. The increase in erosion variability with erosion severity accounts for the fact that severe erosion can be due to a range of physical states, including coating loss, roughness patches, pits, gouges, and localized defects, whereas mild erosion is comparatively more uniform . This model takes values sampled from the normal distribution that lie in the interval [0,1], replacing values that are <0 by 0 and values >1 by 1.

Once we have sampled the erosion levels, ei, we model the aerodynamic effect of erosion through a simplified spatial perturbation of the lift and drag polars. This approach is consistent with prior erosion models that interpolate or perturb aerodynamic polars according to local damage severity . Based on reported severe-erosion effects, including lift losses up to 53 % and drag increases up to 500 % , we scale the lift and drag coefficients in the ith blade region according to cL,i=1-0.53ei and cD,i=1+4ei.

This simplified erosion model has several limitations. First, our model lacks realistic temporal resolution of erosion. More detailed stochastic models represent erosion as a random accumulation process, where damage arrival and severity depend on quantities such as blade radius, wind speed, precipitation, and exposure history . In addition, experimental descriptions of liquid droplet erosion often identify a multi-stage progression, including an incubation period with little apparent mass loss and a nonlinear acceleration phase, followed by an approximately linear damage–growth regime . By contrast, the present model assigns blades directly to ordered erosion classes and does not attempt to reproduce these intermediate growth dynamics.

Second, we use a simplified aerodynamic perturbation model which interpolates between clean and maximally eroded behavior using scalar modifications to the lift and drag coefficients. In reality, surface roughness can alter the shape of lift and drag polars in an angle-of-attack-dependent and nonlinear manner. For example, model eroded lift polars using multiple aerodynamic parameters, while show that erosion effects may be negligible over some angle-of-attack ranges. A more detailed modeling approach would explicitly modify the blade surface geometry, compute the resulting aerodynamic polars using CFD, and then pass the modified polars to a turbine-level simulator .

Finally, our use of five erosion classes necessarily coarsens a gradual physical process. This representation does not resolve subtle differences between early-stage erosion states or the full diversity of possible damage patterns along the blade. However, our simplified stochastic parameterization is not intended to model the complete time evolution of LEE. Rather, it is designed to generate physically motivated families of aerodynamic perturbations for evaluating whether emulator-generated data can support erosion-severity classification. For this purpose, the model preserves several important qualitative features. Namely, (1) erosion damage increases monotonically with class severity, (2) erosion accumulation is weighted toward the tip, and (3) the induced aerodynamic changes reduce lift and increase drag within reported ranges. Since practical erosion assessment often relies on coarse-level observed blade deterioration, which is then connected to performance loss and maintenance decisions , this class-based proxy provides a useful proof-of-concept setting for testing the proposed emulator-based workflow.

We use five sensors to monitor erosion damage: generator power, aerodynamic lift and drag, blade tip acceleration, and blade root force moment . Although lift and drag measurements are not commonly available, new sensors under development measure aerodynamic pressure over an aero-foil section to calculate the dynamic lift coefficient, which can be used to detect changes in airflow patterns due to surface roughness . Similar studies have used these data to track the progression of LEE using OpenFAST software . Note that we only track lift and drag at the tip of the blade, AeroDyn Node 18, since it is the location where erosion will accumulate most rapidly .

Since aerodynamic changes alter blade kinematics, we track flap-wise acceleration of the tip of the blade, used by and , and the edge-wise moment of the root of the blade, used by and .

We use feature extraction to reduce the dimension of the time series outputs from our OpenFAST simulation, a vector with 160 entries per second of simulation time per sensor. Following a similar approach to , we use statistical moments of each of the five output time series, specifically the mean, standard deviation, skew, and kurtosis , rather than the time series itself.

also extracted richer features from time series data such as higher-order crossings, mobility, and complexity features, but in Sect.  in Figs.  and we show high classification accuracy without additional feature extraction. A plausible reason for this may be that (1) we look at a relatively coarse set of erosion levels that naturally have distinct features because they do not naturally overlap, and (2) we are using bulk output statistical features which contain a large amount of information for steady-state wind flow. In Table  we summarize the simulator outputs, along with their variable names in OpenFAST and their units. These quantities are the inputs to the random forest classifier which we use to determine erosion accumulation damage.

We begin with a discussion of the basic GP formalism and then describe the extensions we employ for emulation of wind turbines. For scalar-valued output, the wind turbine simulator is treated as a function, f, from the space of input parameters, u∈U⊂Rp, to the output quantity of interest, v∈V⊂R. For wind turbine erosion modeling, inputs include (for example) the wind velocity and the blade damage level, with generator power as an output. The unknown deterministic function f:U→V is then modeled as a draw from a random Gaussian process (GP), f̃, with the property that for any finite subset, {u1,…,un}∈ U, the random variables, {f̃(u1),…,f̃(un)}∈V, follow a multivariate Gaussian distribution . The modeling begins with the assumption of a GP prior with a mean trend μ:U→V and covariance C:U×U→R. The mean trend is of the form μ(u)=hT(u)θ for some choice of q basis functions (often taken to be a constant or linear function of the input parameters), h(u)=(h1(u),…,hq(u))T, and regression coefficients, θ. The covariance is of the form C(u1,u2)=σ2c(u1,u2), where σ2 is the variance (output scaling) of the GP, and c:U×U→R is the correlation function, which is specified by the choice of a kernel . In this work we use the Matérn 5/2 kernel (, , ) given by c(u1,u2)=∏k=1p(1+5dk+53dk2)exp⁡(-5dk). Here dk=1γk|u1,k-u2,k| is a normalized distance between the kth entries of the vectors u1 and u2. The correlation coefficients, γ=(γ1,γ2,…,γp)T, determine how rapidly the emulated output changes with respect to each input dimension. Letting UiD (i=1,…,n) represent the design of training inputs (e.g., different settings of wind speed and blade damage), the corresponding response vector of outputs (power generated for each setting) is given by vD=(v1D,…,vnD)T. We denote the training input/response data together as DGP={(U1D,v1D),…,(UnD,vnD)}.

The GP emulator's predictive posterior distribution evaluated at a new input, v*∼f̃(u*), is conditioned on the training design/response, DGP. It follows Student's t-distribution with n-q degrees of freedom , 2v*∣vD,UD,γ,θ,σ2∼St(m(u*),σ2c′′(u*),n-q). The predictive mean, m(u*), and predictive variance, σ2c′′(u*), are given by 3m(u*)=hT(u*)θ+rT(u*)R-1(vD-Hθ),4c′′(u*)=1-rT(u*)R-1r(u*)+w(HTR-1H)-1wT, where R is an n×n matrix of correlations between input design points, Ri,j=c(UiD,UjD). Likewise r is a vector of correlations between the new input and each of the inputs in the design, ri(u*)=c(u*,UiD) and w=(rT(u*)R-1H-hT(u*)). The GP predictive mean given in Eq. () is a best linear unbiased predictor . Also note that the rule of thumb for the minimum size of a GP design is 10× the number of input dimensions (10×p) .

To specify f̃, we need estimates of σ2,θ, and γ. Finding good estimates of the correlation lengths is key to fitting a GP emulator. We use the R package RobustGASP , which considers the marginal posterior density for γ to obtain the maximum a posteriori (MAP) estimate of γ . With γ in hand, we can estimate the trend parameters and the scalar variance, respectively, as 5θ=(HTR-1H)-1HTR-1vD,and6σ2=(n-q)-1(vD-Hθ)TR-1(vD-Hθ).

In the next two sections we introduce parallel partial emulation to handle vector-valued QoIs and zero-censored emulation to handle QoIs with range constraints.

For simulators with vector-valued output, practitioners following the standard scalar GP methodology typically either train a separate GP for each component of the vector or use dimension reduction on the output vector before fitting GPs . The former approach can be computationally prohibitive, while the latter emulator is no longer an interpolator of the data. Instead, we use a parallel partial Gaussian process emulator (PPE) which approximates the entire output vector f:U→Rs via a single emulator . The PPE starts with the assumption that output components can be treated independently, each with their own scalar variance, σj2, and trend parameters, θj, j=1,…,s. Yet all output components share a common correlation structure with respect to dependence on input scenarios.

For a design of n input scenarios, the vector-valued responses are now collected in an n×s matrix VD, where the jth column, VjD, corresponds to all n responses of the jth output component in the training data. Much like the scalar GP, the PPE approximates the jth component of v*=f̃(u*) at an untested input u* as a sample from the Student t-distribution with n-q degrees of freedom given by 7vj*∣VjD,UD,γ,θj,σj2∼St(mj(u*),σj2c′′(u*),n-q). Here θj is calculated by replacing vD by VjD in Eq. () and where mj(u*) and σj2 are calculated by replacing vD by VjD and θ by θj in Eqs. () and (), respectively.

As each component of the output vector shares a common correlational structure, R is the only matrix which must be inverted. Thus, the cost of training and predicting with a PPE is comparable to that of a scalar GP.

Computer model QoIs are often required to be positive or to take values within a given interval, as is the case for wind turbine data. For example, for the NREL reference turbine, the generator power cannot exceed 5 MW, and the standard deviation of any random variable (sensor data) must remain positive. Restrictions on the range of f pose a challenge for GP emulation because Gaussian processes have full support. That is, a GP's predictive outputs take values in the interval from (-∞,∞). Further, this severe form of non-stationarity (outputs varying smoothly as inputs change versus outputs not varying at all as inputs change) violates standard GP assumptions. To mitigate the difficulty of fitting a GP to a range-limited QoI, proposed the zero-censored Gaussian process emulator (zGP).

Regardless of whether the scalar output v is in reality bounded above, v∈(-∞,vmax], or bounded below, v∈[vmin,∞), by applying a linear transformation, we can assume that the transformed output is bounded below by zero. The main objective of the zGP is to replace all of the zero outputs with negative values that are consistent with a GP fit only to the positive outputs. To start the process, zero-output training data are initialized with negative values (this can be done with a deterministic rule or by sampling GPs fit to positive data; see Algorithm 2 in , for more detail). Then for each input in the design that led to a zero output, a GP is fit to all other (imputed negative and positive) output responses. At the left-out design point, this GP's truncated normal distribution (on (-∞,0)) is sampled. This sample replaces the previous negative sample for that design point in a Gibbs-sampled Markov chain Monte Carlo (MCMC) scheme. The process is then repeated for each of the other design points that led to zero outputs to complete one step in the MCMC chain.

We observe that this imputation process (which can be expensive but is only done once as preprocessing) converges after a few thousand iterations. One hundred samples are kept from this chain (accounting for burn-in and thinning) for each input that originally led to a zero output. For each such input, we take the average of the hundred negative MCMC samples as the final imputed negative response. We then use the imputed negative- and original positive-output-response QoIs to fit a GP. For prediction at an untested input, we take the GP's prediction if positive or zero if the GP's prediction is negative.

For vector-valued outputs, the zGP is a preprocessing step which is independently applied to each component of the output vector that is range-constrained. This process can be sped up via parallelization. The resulting imputed values are used for the training of a PPE (see ).

The parallel partial and zero-censored Gaussian process emulator (PPzGP) has several advantages that make it an effective emulator for wind turbine simulation. First, it predicts multiple outputs simultaneously, eliminating the need for multiple, independent surrogate models, and second, it satisfies known range constraints on sensor outputs. While scalar GP emulators have been used to analyze WT power production , wake steering , blade loads , and mono-pile reliability , the results in this paper are the first where zero-censored and parallel partial emulation have been combined for WT modeling.

We trained and evaluated the GP emulators on the third dataset in Table . These simulations were generated using OpenFAST with 50 samples for each of the four eroded blade classes and 10 for the clean blade class. The training dataset varies the three environmental inputs identified by the GSA using Latin hypercube sampling over the ranges in Table . The erosion level in blade sections 1–6 are varied according to values given in Table , which we selected to ensure coverage of the stochastic erosion classes from Sect. .

First, we compare the training efficiency and prediction accuracy of scalar-GP emulation to that of standard parallel partial Gaussian process (PPGP) emulation, with and without the additional preprocessing step of zero-censored emulation. We test the emulators and compute 95 % confidence levels using five repeated 5-fold cross-validation sets of the OpenFAST dataset. Each split allocates 168 training points proportionally across erosion severity classes, with the remaining 42 data points reserved for testing.

In Table  we show that the time taken to train the parallel partial emulator is 8 times less than with 8-fold scalar emulation, which is to be expected because the output vector has eight components. In Table  we show the normalized root mean square error (NRMSE), which, to allow for comparisons between outputs with different scales, is defined to be the root mean square error divided by the output range. The NRMSE is slightly less with the PPzGP emulator than with the standard GP emulator, which is trained to predict each output separately, except for the standard deviation of the lift coefficient, σ-CL, and the mean of the generator power, μ-Pgen.

In Table , we show the average percentage of predictions falling within the 95 % credible intervals of the PPzGP emulator for each of the eight sensor quantities. The 95 % GP credible interval contains the true function value with a rate of 0.95. If the GP model is calibrated well, under repeated sampling, the empirical coverage of the credible interval will also converge to 95 %. We observe coverage between 84 % and 95 % for the PPzGP. We attribute this to model misspecification (choice of kernel, violations of the stationarity assumption, etc.) and effects of sampling noise in the out QoI moment calculations, which we do not model directly.

In Table  we compare the performance of the standard GP and PPzGP emulators across the selected sensor outputs using paired 95 % confidence intervals for differences in NRMSE and empirical coverage. A positive difference in NRMSE indicates the PPzGP emulator has a lower error, while a negative difference in coverage implies the PPzGP has better coverage. These results show that PPzGP emulation provides improved uncertainty calibration for all sensor outputs while maintaining prediction accuracy comparable to the standard GP. For most outputs, the paired NRMSE confidence intervals include zero, indicating that the mean prediction errors of the two emulators are not significantly different. In contrast, the coverage intervals are mostly negative, indicating that the PPzGP intervals are generally closer to the nominal 95 % level than those of the standard GP. This suggests that incorporating physical output constraints can improve emulator uncertainty quantification without degrading mean prediction accuracy. These benefits are obtained alongside the reduced training time and storage requirements of the PPzGP formulation.

In Fig. , we compare the PPzGP emulator predictions (blue) with the simulated outputs (red), along with the associated 95 % credible intervals for four sensor outputs for one test split. For each output, the majority of true values fall within the credible intervals, indicating good emulator calibration. The generator power predictions exhibit narrow credible intervals, and the zGP ensures that the mean power does not exceed the maximum rated power of 5 MW. The statistical constraint that the blade tip acceleration standard deviation is non-negative is also enforced.

In Table , we distinguish between the cost of a single simulation, training, and generating samples with the GP emulator. All timings are wall-clock times measured on a machine with an Intel(R) Core™ Ultra 7 155H processor (3.80 GHz) and 32 GB RAM using OpenFAST-v3.5.0. Simulator timings were estimated from 25 sequential OpenFAST runs at randomly selected inputs. Emulator timings were estimated from five repeated 5-fold splits on 210 data points; prediction timings correspond to generating 10 000 emulator samples and were repeated 25 times.

Our OpenFAST simulation requires 28.4±0.3 s, whereas generating 10 000 PPzGP samples requires 0.93±0.01 s, corresponding to approximately 9.3×10-5 s per emulator sample. Thus, evaluating the fitted PPzGP is roughly 3×105 times faster than running OpenFAST once. This prediction-only speedup does not include the cost of constructing the emulator. The full PPzGP workflow includes the 210 OpenFAST simulations used to train the emulator, zero-censored preprocessing for range-limited outputs, PPzGP fitting, and generation of the emulator-based classifier-training data. Using the measured OpenFAST timing, the simulator portion of this workflow costs approximately 5964 s. Zero-censored preprocessing is the most expensive part of fitting the PPzGP, but it can be sped up by preparing each output in parallel, approximately 280 s on our laptop. It takes 3.7 s for PPzGP fitting and 0.93 s to generate 10 000 samples; the full surrogate workflow requires approximately 6.2×103 s. In contrast, directly generating 10 000 OpenFAST samples would require approximately 2.84×105 s, giving a full-workflow speedup of about 46×. The prediction-only speedup and full-workflow speedup indicate that querying the fitted emulator is extremely fast, and the full workflow remains dominated by the initial OpenFAST simulations needed to train the emulator.

In Fig.  we show that the PPzGP emulator captures the essential wind turbine dynamics represented by the full simulator. For this analysis, erosion is set deterministically for the values of α given in the legend with e=(αx1,…,αx6). In the left panel, we show generator power as a function of wind speed across five erosion severity classes. The PPzGP accurately reproduces the power curve, with higher erosion levels requiring greater wind speeds to achieve an equivalent power relative to the clean blades. The non-eroded power-curve aligns with findings from , who used a GP to estimate the first and second statistical moments of generator power for a different turbine model. The PPzGP provides vector-valued predictions across multiple sensor outputs. In the right panel we demonstrate that the model captures two key physical trends: the power increases with air density and decreases when the rotor is misaligned with the wind. These relationships are consistent with prior studies and are also evident in Fig. . Power loss due to erosion is proportionally highest at lower power levels, remains significant up to the rated wind speed, and disappears beyond that threshold . However, as we see in Fig. , model limitations exist. The prediction performance deteriorates in extreme loading conditions, particularly for high values of tip acceleration and root moment mean. These deficiencies likely stem from a lack of training data in more extreme turbine operational states.

To assess whether the high classification performance in the baseline study is due to an overly clean and easily separable dataset, we performed an additional noisy-sensor classification experiment. Sensor noise was modeled by adding both a sample-level Gaussian bias and time-varying Gaussian noise to each classifier input channel. The bias standard deviation was set to 10 % of the average absolute mean sensor value, and the pointwise noise standard deviation was set to 10 % of the corresponding time series standard deviation. We kept the same sensor outputs described in Table . We then repeat the classifier input ranking using a noisy version of dataset 2 from Table  and the selection procedure from Sect. , also shown in Fig. , retaining μ-CD, σ-CD, γ-atip, σ-CL, μ-CL, μ-Mroot, μ-atip, κ-CD, and κ-atip. The emulator is then trained on a noisy version of its original training dataset. We compare the simulator-trained classifier to the emulator-trained classifier. We fit the emulator using a noisy version of dataset 3 from Table  and generated 10 000 data points for training the classifier. We use a noisy version of dataset 5 from Table  for 10 repeated 5-fold cross-validation studies.

In Table  we show that noisy sensing changes the class-wise error distribution. Compared to Table , all recall values are lower as expected. In comparison to the simulator-trained model on noisy data, the emulator-trained classifier has lower recall for the lowest erosion classes. However, recall improves for the intermediate classes. Recall for the most severe class remains statistically comparable between the two classifiers.

To further assess the dependence of the classifier on direct aerodynamic sensing, we repeat the classification study after removing the lift and drag coefficient channels. In addition to Mroot,atip, and Pgen, we consider an expanded set of structural outputs in Table . These channels are not directly tied to the imposed aerodynamic erosion perturbation. We calculated six additional features shown in Table  from the time series data and their first difference . Two features were based on frequency response using the spectrum of the time series data . For a signal x(t) with power spectral density Sxx(f), the band power over a frequency interval [f1,f2] is P[f1,f2]=∫f1f2Sxx(f)df. In this study, we selected frequency bands using the range of rotor speeds observed in the data to capture energy near rotor-related frequencies. For the three-bladed turbine, both the once-per-revolution band (1P) and blade-passing band (3P) were considered.

We repeated the predictor selection process from Sect.  four times, ranking and selecting the top 14 quantities from an initial list of 108: γ-ax,tip, μ-Mx,root, μ-Fy,root, ρ1-az,tip, BP3P-Mz,root, NFD-az,tip, ρ1-My,tower, ρ1-Mz,shaft, κ-Mz,shaft, ρ1-My,shaft, BP3P-My,tower, ρ1-Mx,tower, κ-Mz,root, and κ-ax,tip. The two highest-ranked predictors, γ-ax,tip and μ-Mx,root, were also ranked highly in the primary study, suggesting their importance even without the aerodynamic sensor channels.

We train the emulator on a separate dataset with 210 samples and then use 10 repeated 5-fold cross-validations and inner-loop hyperparameter tuning to compare the simulator-trained and emulator-trained classifiers with dataset 5 from Table . In this experiment, we used the emulator to generate a dataset of 50 000 samples.

In Fig.  we show that, after removing the aerodynamic lift and drag channels, classifier performance decreases substantially relative to the full-sensing case shown in Fig. , confirming that with less access to aerodynamics-based sensors the classification problem is more difficult. However, the simulator-trained and emulator-trained classifiers have comparable class-wise behavior. Nonetheless, the confusion matrices are dominated by their diagonal entries, indicating that misclassifications typically occur between adjacent erosion classes rather than between widely separated damage states. The severe erosion class remains identifiable, with both classifiers correctly classifying 73 % of α=1.00 samples. Consistent with the confusion matrices, in Table  we observe nearly identical balanced accuracy and macro-F1 values for the two classifiers, indicating that the emulator-trained classifier remains statistically comparable to the simulator-trained classifier in the case of no direct aerodynamic sensors.