The high-fidelity simulations that were run for this paper were performed on the Adaptive Mesh Refinement for Wind (AMR-Wind) simulator . AMR-Wind is part of the ExaWind modeling and simulation environment , built on top of the AmRex library . The simulator solves the three-dimensional incompressible Navier–Stokes equations in a spatially filtered resolved-scale formulation and employs the subgrid-scale one-equation turbulence model  for smaller eddy dynamics.

Within the simulations, the turbines were controlled using the reference open-source controller (ROSCO) toolbox for wind turbine applications . In the baseline control case, the turbines were operated to maximize their own power production. The WFFC methods that are considered for the load surrogate model include wake steering and the helix method . Although we only consider these methods, the load surrogate model can easily be extended to include other strategies, such as static induction control. Implementing wake steering in the simulations is straightforward and consists of adding a constant offset to the turbine yaw angle.

With the helix control strategy, individual blade pitch control is used to exert time-varying blade moments on the incoming flow. These moments are designed in such a way that, when transformed from a rotating to a non-rotating reference frame , slowly varying yaw and tilt moments are applied to the flow. These moments create the characteristic helical shape of the wake. An example of the blade pitch signals is given in Fig. . Each of the sinusoidal reference signals is offset by a phase of 120°. Increasing the amplitude (Aβ) of the pitch signals results in stronger yaw and tilt moments, and hence, in a more pronounced helix shape. The excitation frequency (fe) at which the yaw and tilt moments move across the rotor plane is generally expressed using the dimensionless Strouhal number (St): 1fe=St⋅U∞D, with U∞ indicating the freestream inflow velocity. The pitch actuation frequency (fβ) is then defined as 2fβ=fr±fe, with fr denoting the rotational frequency of the rotor. Adding the excitation frequency (fe) to the rotational frequency (fr) results in a counterclockwise-rotating helix wake, while subtracting the excitation frequency yields a clockwise-rotating helix wake. For the simulations, we used the built-in functionality of the Active Wake Control module in ROSCO to implement the helix approach, which requires settings for the amplitude, excitation frequency, and direction. The resulting pitch reference signal for each of the blades (βi) at time t is then defined as 3βi(t)=β*+Aβsin⁡2π(fr+fe)t+ψ0,i, where β* is the optimal collective pitch angle and ψ0,i represents the azimuth angle of blade i at t=0. Higher pitch amplitudes result in larger wake deflections, but they are accompanied by higher power losses and increased structural loading .

The LES described in the previous sections are very computationally expensive. Each wind turbine simulation requires several days of simulation time on a high-performance computing (HPC) cluster. Therefore, simulating a full sweep of all control parameters for the helix method becomes very time-consuming, especially when multiple ambient conditions need to be considered. When one also wants to evaluate the effects of WFFC methods on fatigue loads under different alignment and spacing conditions, simulating this becomes unfeasible. To make this problem tractable, this paper uses surrogate models trained on a limited number of LES.

Before introducing the mathematical formulation, we briefly summarize the role of GP regression in this framework. A GP allows us to construct a computationally cheap (compared to LES) surrogate model of a relationship that would otherwise require many expensive LES or aeroelastic simulations to characterize, such as turbine power as a function of wake-mixing control settings, or fatigue loads as a function of inflow and control conditions. This is achieved in three steps. First, a limited set of simulations is used to generate training data, consisting of pairs of known inputs (e.g., control settings or inflow conditions) and their corresponding outputs (e.g., power or fatigue load). Second, the GP model is trained on this dataset by tuning a set of hyperparameters that determine how strongly the modeled function is correlated between different input locations. Third, the trained model is used to predict the output along with its associated uncertainty for new input conditions, without requiring additional simulations. In this paper, this framework is applied twice: first, to model the power output of a two-turbine array as a function of the helix frequency and amplitude (Sect. ) and, second, to predict fatigue loads as a function of inflow and control conditions (Sect. ). These two applications place different demands on the GP model. For the load surrogate model, the training data are generated in advance from a fixed database of simulations, and the aim is to obtain an accurate representation of the fatigue loads across the full range of inflow and control conditions, since predictions may later be required for arbitrary combinations of these conditions. In the controller tuning framework, the GP serves a dual aim, which is to obtain a model that shows where, and to what extent, the helix method is effective across the frequency–amplitude domain, and to use this model to make informed decisions on which settings to simulate next.

When considering a GP, we assume that the output data, collected in a one-dimensional vector y, belong to a multivariate Gaussian distribution 4:yy*∼N0,Kx,x+σn2IKx,x*Kx*,xKx*,x*. The distribution of the output depends on one or multiple inputs collected in a matrix x. The vector y* refers to the inferred function value for possible test inputs x*. The covariance matrix K can be computed from a wide range of kernel functions and depends on the type of function one would like to model. Two popular options are the squared exponential and Matérn functions , which will both be used in this work. Each Kernel function is equipped with a set of hyperparameters that control the output scaling, correlation, and level of smoothness. An additional hyperparameter is given in the form of σn, which represents the noise or uncertainty in the output data. In this work, the hyperparameters are obtained by maximizing the log marginal likelihood. After determining the hyperparameters, we compute the posterior distribution of the model, resulting in the mean (μ*) and variance (Σ*) for the test locations (x*) using the following set of equations: 5μ*=Kx*,xKx,x+σn2I-1y,6Σ*=Kx*,x*-Kx*,xKx,x+σn2I-1Kx,x*. The content of the test input matrix (x*) depends on the application. When we want to determine the optimal control settings for the helix method, this matrix consists of different combinations of actuation frequency and pitch amplitude. In the case of evaluating fatigue loads, this matrix consists of the inflow conditions, as well as some control inputs. Both cases will be considered in more detail in the subsequent sections.

This section describes how the load surrogate model is constructed. The overall workflow to obtain the surrogate model is illustrated in Fig. . In short, cross-stream velocity flow slices are extracted from the LES at different downstream distances and wake displacements, representing a range of wake conditions. These flow slices are then used as time-varying inflow for standalone aeroelastic simulations in OpenFAST, from which structural load time series are obtained for the turbine components listed in Table , accompanied by a brief description. The resulting load time series are converted into damage equivalent loads (DELs), while the corresponding flow slices are reduced to four sector-averaged velocity and turbulence intensity values. Finally, these sector-averaged inflow quantities, together with the relevant control settings (i.e., helix pitch amplitude Aβ and yaw misalignment γ), are used as inputs to train a separate GP model for each load channel, allowing fatigue loads to be predicted directly from inflow and control conditions without additional aeroelastic simulations. The remainder of this section details each of these steps.

Cross-stream flow slices were collected at a 1 s interval from the wake of a single turbine at distances between 3D and 8D, as well as a slice of the undisturbed precursor inflow. Subsequently, these time series were cut into blocks of 660 s to mimic the effect of different turbulence seeds of the inflow conditions. Next, the flow slices were projected on a rectangular grid of 340 m (≈1.20 D) width by 330 m (≈ 1.16 D) height, with grid spacing of 10 m (≈ 0.04 D), and exported to a .wnd file compatible with OpenFAST. To simulate different levels of wake overlap, a displacement parameter Δy was introduced to move the grid laterally over the cross-stream flow slices. In this way, the aeroelastic simulations covered all wake conditions from full overlap (Δy=0 m) to no wake overlap (at close turbine spacing with baseline control and for Δy=±390 m ≈1.37 D). It should be noted that the 1 Hz sampling frequency of the extracted flow slices could limit the high-frequency inflow-driven load content, and some structural excitation present in a full LES-driven simulation may not be captured by standalone OpenFAST simulations at higher frequencies. However, sampling the inflow at a frequency similar to that of the LES was not feasible from a data storage perspective.

In the LES cases used to obtain the flow slices, we employed different control strategies, including greedy control, wake steering, and the helix method. Furthermore, several simulations with three turbines, each separated by a distance of 4.5 D, were conducted to include the effect of accumulated turbulence in the wake. From these simulations, flow slices from the wake of the second and third turbines were collected. For now, we assume that these inflow conditions are representative of larger wind farm configurations, but this requires additional validation at a later stage. An overview of all the control strategies that were simulated is provided in Table . These control cases were simulated with the precursor wind speeds specified in Table , resulting in a total of approximately 30 LES runs for the load surrogate database. The standalone OpenFAST simulations for the loads database also include the turbine using different control strategies to assess the impact on loads. For wake steering, the turbine was yawed with γ=-30:10:30°, and for the helix method, the pitch amplitude was varied between Aβ=1.0:1.0:4.0°.

In terms of computational cost, each LES run requires between 1 and 2 d of runtime on an HPC cluster. Extracting and post-processing the flow slices, and subsequently running the standalone aeroelastic simulations for the range of wake displacements and control settings described above, is comparably expensive and adds a further several days of computation time. Once the resulting loads database is available, however, training the GP surrogate model is considerably cheaper. Optimizing the hyperparameters for a single load channel takes around 10–20 min. In the current framework, the computational bottleneck lies almost entirely in generating the training data rather than in training the surrogate model itself.

The entries for the load surrogate database were obtained after post-processing the time series of the inflow planes and the load channels. The first set of inputs sent to the load surrogate model are the sector-averaged inflow quantities, as proposed by . In this case, the inflow consisting of streamwise velocity and turbulence intensity is averaged over four sectors spanning the rotor plane. For the sector-averaged wind speed (SAWS) USA(s), the wind speed value for each sector s is determined in the following way: 7USA(s)=∫ψ=(s-1)⋅π2-π4ψ=s⋅π2-π4∫r=0r=Rr⋅u¯(ψ,r)drdψ. In the previous equation, the local time-averaged streamwise wind speed u¯ from the LES flow planes is integrated over the radial distance r and the azimuth angle ψ, where the latter has been discretized into four sectors. The same equation is used to determine the sector-averaged turbulence intensity (SATI), where turbulence intensity is defined as 8TI=13u′u′‾+v′v′‾+w′w′‾U∞. Here, u′, v′, and w′ indicate the instantaneous velocity deviations from the average for the streamwise, transverse, and vertical velocity components, respectively. Furthermore, u′u′‾ denotes the time average of the velocity fluctuations squared. Note that the freestream wind speed velocity U∞ is selected to normalize the root mean square of the average velocity fluctuations instead of the local average wind speed. In this case, the TI sectors provided as input to the load surrogate model are higher for turbines operating in a helix wake. Hence, the surrogate model can predict the additional fatigue loading resulting from the helix wake. This was done to distinguish between the wake of a turbine operated with greedy control and the helix method, as the latter generally causes higher velocity fluctuations.

To assess the effect of control strategies and wake conditions on the fatigue loads, time series from the load channels are converted into 1 Hz DELs using the following equation: 9Leq1Hz=∑i=1Nbinsni⋅LimNcycles1m. The number of load cycles ni and load amplitude Li for a bin i are obtained from a standard rainflow counting algorithm for Nbins=200. The Wöhler slope coefficient is set to m=10 for the composite blades and m=4 for the steel tower components. Finally, Ncycles refers to the number of 1 Hz cycles in the 10 min time series, as per convention, equaling a value of 600. The DEL can be interpreted as the load amplitude that does a similar amount of damage at a 1 Hz interval as the load time series.

The use of wake-mixing techniques can also have a significant impact on the blade pitch bearings due to the combination of continuous movement and alternating load cycles. One method to express the loads that the pitch bearings experience is using the pitch bearing damage equivalent load (PBDEL) . The PBDEL (LPB) is determined using the following equation: 10LPB(χ)=∑k=1Nstepsδβ(k)⋅max⁡cos⁡(χ)⋅Mflp(k)+sin⁡(χ)⋅Medg(k),0mNref1m. The PBDELs are determined for the full range of radial positions of the bearings with 10° increments, i.e.,  χ={0:10:360°}. The degree of pitch travel is denoted by δβ(k) for a time step k. The blade root flapwise and edgewise bending moments are given by Mflp and Medg, respectively. Nref indicates the reference degree of pitch travel during a 10 min interval. In this study, we set Nref=600°, which matches the distance a blade travels when applying the helix with an amplitude of Aβ=2.0° and with St=0.25 at a wind speed of U∞=10 m s⁻¹. For the pitch bearings, a Wöhler coefficient of m=3 is used . For the load surrogate model, we always consider the radial position that experiences the highest load.

The simulation data required for the data-driven modeling framework are extracted as time series from the OpenFAST output files. An example of such a simulation output is given in Fig. , showing the generator power of the upstream (T1) and downstream (T2) turbines for the baseline case and the helix method. The time series show a slight decrease in power for T1, and a significant increase in power for T2, when the helix method is used. The 1 h measurement domain is then divided into multiple 10 min periods, and the power is averaged over time, as shown on the right-hand side of Fig. . Note that the relative power of the helix method over the baseline can vary depending on the 10 min time interval, due to the turbulent eddies that hit the turbines. This indicates the importance of sufficient simulation time and the use of multiple data points for a single helix case.

The time-averaged power data are subsequently used to compute the power ratio (PWFFCPBL) of the helix method over baseline control. All the power ratio measurements are then collected and scaled to a zero mean and unity standard deviation dataset y. Similarly, we collect the training locations, consisting of the helix amplitude and frequency, in a matrix x and scale it before feeding it to the GP model as training data. For the GP to effectively identify the contribution of both turbines to the total power of the array, the model is split into two parts with their own set of hyperparameters. Hence, we obtain a GP model that estimates the power loss of T1 (μT1, σT1) due to the helix actuation, and a GP model that estimates the power gain or loss for turbine T2 (μT2, σT2) due to the affected wake recovery. A useful property of Gaussian processes is that the sum of two Gaussian processes is also a Gaussian process. Therefore, we can model the mean and variance of the total power gain as follows: 11μarray=μT1+μT2,12σarray2=σT12+σT22, where μarray is the mean power gain of the two turbines, and σarray2 is the variance of the estimated mean power gain.

After extracting power data from the simulations and post-processing them, we can iteratively determine the optimal settings for the helix method for the current ambient conditions. We used a Matérn covariance function with a smoothness parameter of ν=3/2 to get a smooth but flexible model. The GP modeling framework presented in the previous section will now be used to model the power increase in the counterclockwise (CCW) helix implementation. For the sake of demonstration, we first focus on a single input GP model where the Strouhal number is the free variable, and the pitch amplitude is fixed at a value of Aβ=2.5°. Using only one input also allows for easier initialization of the model, when the quantity of data is still small. Figure  shows the total power gain as a function of the Strouhal number. For each of the subplots, the hyperparameters are first optimized, and the mean power ratio and variance are inferred for a range of other frequencies. Based on the inferred power ratio, a new frequency is selected and subsequently applied in a new LES run, after which the process is repeated.

The model is initialized (see Fig. a) using 10 min average data samples from three separate simulations that employed the helix method at different frequencies. For a Strouhal number just above St=0, we see a power ratio below 100 %, indicating a loss of power due to the actuation of the helix method. However, this ratio increases as the Strouhal number rises to values of St=0.2. For the next two simulations (Fig. b, c), we focus on exploring the power ratio function by selecting Strouhal numbers of 0.5 and 1.0, respectively. Based on previous studies , we do not expect the optimal frequency to lie within this range, but we are interested in determining the shape of the function and reducing the uncertainty. After updating the GP with these simulation results, the model predicts the highest power ratios to be obtained in the range of St=0.2–0.4. After the latest iteration (Fig. f), we notice that the GP estimates show little variation in this frequency range. The power increase for the considered pitch amplitude seems to settle at approximately 4.3 %. The measurement samples also indicate that where the helix method is effective, large variations in the power ratio occur.

The next step in the modeling framework is to expand the GP model to include pitch amplitude. This requires only a small alteration to the GP model, consisting of adding an additional input (Aβ) and a corresponding input lengthscale (hyperparameter). The model is then trained using all prior data. The same approach for selecting the control settings for the LES runs is adopted as in the previous example, focusing first on exploration, followed by maximizing the power gain. The final versions of the GP models are presented in Fig.  as contour maps of the power ratio as a function of frequency and pitch amplitude. From the GP model of turbine T1 shown in Fig. a, we see a clear decrease in power for lower frequencies and increasing pitch amplitudes. For the second turbine (Fig. b), the power ratio keeps rising for increasing pitch amplitudes. When combining the models of both turbines, we obtain the contour map in Fig. c for the total power ratio, indicating a clear optimum. The optimum indicates that for pitch amplitudes above 4°, the additional power loss (with respect to the optimal settings) of the actuated turbine (T1) cannot be compensated for by the increased power of the downstream turbine (T2). The optimal power ratio of 7.5 % was achieved with an amplitude of Aβ=4° and a frequency of St=0.25. While the optimal identified frequency agrees well with previous studies , showed an increasing power trend for even higher pitch amplitudes. However, their simulations were performed with a different turbine model and for a different wind speed.

Figure  also shows the two-dimensional estimates of the GP models with uncertainty bounds for the optimal frequency and amplitude. The model for T1 (Fig. a) can estimate the power loss with high certainty, with the only uncertainty arising from the lack of samples at higher frequencies. Power losses of up to 5 % are visible for increasing amplitudes at the optimal frequency. The models for T2 (Fig. b) and the overall power (Fig. c) look very similar to each other, with higher uncertainty in the estimates. Interestingly, the range of optimal frequencies narrows as the pitch amplitude increases. When operating the helix with pitch amplitudes up to 2.5°, this could offer some additional flexibility on the selected frequency. In this case, the optimal frequency is determined not only by the maximum power gain but also by the minimum fatigue loading.

Modeling the performance of wake-mixing techniques has thus far been limited to the CCW helix method. In Fig. , the model is modified to include clockwise (CW) simulations of the helix method. For this model, we assume the shape of the GP model for the CW helix is similar to the CCW version. The model is therefore extended by mirroring the frequency axis around the rotational frequency fr. For the CW excitation of the helix, the blade pitching frequency will be below the rotational frequency of the turbine, as seen in Eq. (). Figure  shows that the CW helix method achieves similar power gains to CCW for pitch amplitudes up to 2°. For higher pitch amplitudes, the gain remains fairly constant, which means that the power loss of the actuated turbine is not overcompensated for by the power gain of the downstream turbine.

The tuning framework has so far been demonstrated for a single free-stream wind speed and turbulence level. However, the controller performance for additional simulation conditions could potentially be modeled by adding inputs covering wind speed, turbulence, and atmospheric stability, among others. This extension remains for future work. It should be noted, however, that GP regression scales cubically with the number of training points, since it requires the inversion of the covariance matrix (Eq. ). Adding input dimensions does not directly change this scaling but typically requires more training data to adequately characterize the response surface, so extending the model to more inputs still drives up the training cost in practice. The load surrogate model presented in Sect. , which already uses 10 inputs and is trained on several thousand samples, illustrates that this remains computationally manageable at the scale considered in this study. Nevertheless, as additional inputs and training data are added, the cubic scaling of GP training may become a practical constraint, motivating the use of sparse GP methods in future work.

The load database consists of 10 410 distinct simulation cases (i.e., cases with the same turbine spacing, wake overlap, and control strategy) with six turbulence seeds, resulting in 62 460 simulation results. Prior to feeding the simulation results to the GP, the results of each simulation case are averaged over the six turbulence seeds to obtain converged fatigue load quantities, helping towards predicting them from a steady-state representation of the inflow. Next, the dataset is divided into a training and a test set with a 70:30 ratio. The training set is then used to optimize the hyperparameters of the GP as discussed in Sect. . The squared-exponential function is employed to ensure smoothness in the prediction results.

The load prediction capabilities of the load surrogate model are validated against the load database test set. From the test set, we take the sector-averaged inflow quantities and control inputs and feed these as inputs to the surrogate model. The predicted load values are then compared to the test set loads. Given the large number of load channels, the validation procedure will mainly focus on four channels: the blade root flapwise bending moment (RootMflp), tower base fore-aft bending moment (TwrBsMyt), low-speed shaft yaw bending moment (LSSGagMys), and the pitch bearing (PitchBr). The validation results of the GP load surrogate model for these components are presented in Fig. . To quantify the performance of each surrogate model, the root mean square percentage error (RMSPE) is computed for both the training and test set as 13RMSPE=1n∑i=1nL^eq1Hz,i-Leq1Hz,iLeq1Hz,i2⋅100%, with n denoting the number of cases in the test set and L^eq1Hz the DEL prediction from the surrogate model.

For the blade root, low-speed shaft, and pitch bearing components, we observe similar prediction accuracies. In the case of the tower base, the surrogate model performs slightly worse, with some outliers visible in the higher load range. For each component, the surrogate model shows a slight decline in prediction capabilities for the test set compared to the training set, although the differences are minimal. To assess whether prediction accuracy varies across control strategies, Fig.  distinguishes between baseline, helix, and wake steering cases. No systematic bias is observed for any particular strategy, confirming that the surrogate model generalizes well across all operating conditions included in the training data. For the remaining components, the accuracy of the surrogate models is provided in Table . For the majority of the components, the fit with the test data is in the same range. The only outliers are the tower components in the side–side direction. The mismatch between the prediction and test data is seen primarily in the higher load range, corresponding to scenarios with full wake overlap. Adding more turbulence seeds to the loads database is expected to improve the prediction performance of the load surrogate model.

In this section, we will perform a case study featuring a two-turbine array. The upstream turbine (T1) is operating in freestream conditions, and the second turbine (T2) is located 5 D downstream. Different control strategies will be tested on the upstream turbine, consisting of baseline (greedy) control and the helix method with different pitch amplitudes.

For the comparison, we used cross-stream flow slices of the streamwise velocity and turbulence intensity from the LES. An example of the flow slices is shown in Fig. . For T1, a flow slice of the undisturbed velocity field is selected with a hub-height velocity of U=10 m s⁻¹. This velocity field is subsequently decomposed into the four sector-averaged quantities that are used for the load surrogate model. The same approach is used for the turbulence intensity profiles. For T2, we use the flow slices extracted at 5D in the case when the upstream turbine is operating with baseline control, as well as the helix method with amplitudes of Aβ=2.0° and Aβ=4.0°. To replicate the effect of different alignment angles between the two turbines, we vary the lateral displacement d of the extracted rotor plane for T2.

The DELs of several components for the different cases as predicted by the load surrogate model are presented in Fig. . In all plots, the predictions are compared to the simulated loads from the test set and are seen to match well. The figure also contains the confidence interval of the GP's prediction. For each component, except the tower base (Fig. b), the surrogate model is very confident of the predicted mean. Note that the confidence interval here refers to the uncertainty in the expected mean function, and does not capture the variation in loads from the different turbulence seeds.

The first column of the figure shows the effect of a rising helix pitch amplitude on the fatigue loads. All components in the figure show an increase in loads, but the most significant rise is seen for the pitch bearings in Fig. d. This is evident from its definition in Eq. (), which considers the degree of travel of the bearing δβ. For the blade root (Fig. a) and low-speed shaft (Fig. c), the increase in load with the helix is on the same order of magnitude as a turbine experiences when operating in a wake. Moving on to the loads of the downstream turbine (T2), we see a double Gaussian shape for the blade root bending moment when T1 uses baseline control. This load profile originates from operation with partial wake overlap, where the blades experience both high and low wind speeds over a single rotation. When the helix method is active on T1, the shape remains the same, but the peaks and the valley in the middle see a small rise for increasing pitch amplitudes. This rise can be seen as the combined result of a higher wind speed and turbulence downstream with the helix. In the case of the tower base (Fig. d), the loads initially decrease for a helix amplitude of 2.0°. This results from the turbine's blade passing frequency (3P) coinciding with the first tower mode at lower wind speeds . As the wind speed increases with the helix, the 3P frequency moves away from the tower mode. Then, as we apply higher pitch amplitudes, the load increases once more due to the additional turbulence in the wake. Both the low-speed shaft and pitch bearings see a small increase in loading with the helix method, although for the latter, the increase in loading is insignificant when compared to the additional loading seen for T1.

Previously, we only considered the effect of the helix method on the fatigue loading of the two turbines. As with the various load channels, we also trained a surrogate model to predict generator power (extracted from the OpenFAST simulation database) based on inflow sectors. In this way, we quantify the cost in terms of loads associated with a potential energy gain. For this purpose, we consider the same two-turbine array as before and evaluate the performance within a small wind direction range of ϕ={-4:0.5:4°} where the helix will typically be active. The helix pitch amplitude is varied between 0 and 4°. The wind direction misalignment is achieved by feeding the same angle as yaw misalignment to the surrogate models and by displacing the incoming flow field for T2 as indicated in Fig. . The resulting load and power estimates are presented in Fig. . The average total power of the two turbines is computed for each pitch amplitude by weighting the individual estimates with a Gaussian distribution of the wind direction as proposed by : 14P¯array=∑ϕ=-4°ϕ=4°pϕ(ϕ)⋅P^arrayϕ,γ,Aβ. The Gaussian distribution pϕ uses a standard deviation of σϕ=2.5° and is discretized over the selected wind direction range with 0.5° increments. Likewise, the average DELs of each component are determined by weighting the predictions with the Gaussian distribution. Analyzing the turbine array in this way replicates the effect of a realistic time-varying wind direction signal.

The first column in Fig.  shows the tradeoff in loads and power gain for the upstream turbine (T1). For the helix cases, the relative power increase is compared to the baseline case with the same wind direction misalignment. The load differences in each of the control cases are small, as it is primarily influenced by wind direction and thus small yaw misalignments. However, the power gain varies to a greater extent depending on the wind direction angle. For increasing pitch amplitudes, the average power gain increases but so do the loads of each component due to the periodic pitch actuation. The results from the downstream turbine show greater load variations across changing inflow conditions. Similar to T1, we also observe higher loading for increasing pitch amplitudes, due to the additional turbulence introduced in the wake. Only in the tower base load channel do we see a decrease in loading as pitch amplitude increases, indicating that not all loads are driven by turbulence but can also be affected by the average inflow wind speed. For the blade root, tower base, and low-speed shaft, the increased loads of T1 are in the same range as those experienced by T2. This means that the loading of these components is dominated by wake conditions. The same does not apply to the pitch bearings, which see a large increase in loading on the actuated turbine (T1), whereas the bearings of T2 are barely affected.

The maximum power gain achieved in this scenario is approximately 4 %, which is below the maximum gain achieved in Sect. . The difference in gain is explained by the different wind speed for this case study (i.e., U∞=10 m s⁻¹ instead of U∞=8 m s⁻¹) and the simulation environment. The latter relates to the efficiency loss the upstream turbine experiences when applying the helix. With AMR-Wind, a pitch amplitude of 4° results in a power loss of 2 %, whereas the same pitch amplitude results in a power loss of 5 % when simulated with standalone OpenFAST. This difference is attributed to the different aerodynamic modeling approaches underlying each simulation environment. Where AMR-Wind resolves blade loading directly from the LES flow field via the actuator line method, standalone OpenFAST relies on blade-element momentum theory with a dynamic inflow correction. As the surrogate model for generator power is trained on standalone OpenFAST data, this modeling difference propagates directly into its power loss predictions for T1.