Identification of wind turbine clusters for effective real time yaw control optimization

When a turbine is in the wake of another, the power production can be reduced by 40%–60%.^1–3 Depending on the wind variability at the site, turbulence, and layout of the farm, wind turbine wakes may have a detrimental effect in the annual energy production (AEP) of large wind power plants.⁴ 

In the last decade, control algorithms have been developed to mitigate the wake interaction between turbines and increase the power production of the array. By controlling the pitch angle,⁵ the rotational speed, or the torque gain,⁶ a modest improvement over the baseline can be obtained. A more promising technique consists in imposing a yaw misalignment to the upstream turbine.⁷ The power production of the yawed turbine decreases but its wake is steered away from the downstream turbine.⁸ Previous studies⁹ showed that depending on the separation of the turbines, wind direction, and turbulent intensity, the power production gained on the trailing turbine can overcome the loss of power of the upstream turbine. Thus, the power production of the entire array of turbines can increase. For example, Ciri et al.¹⁰ showed an increase up to 7% for a 2 National Renewable Energy Laboratory (NREL)-5 MW turbines array with a streamwise spacing of 5 diameters and incoming flow with a turbulent intensity of 8%. Bensason et al.¹¹ evaluated the impact of the wake steering control across the U.S. land based wind power plant, showing the potentiality of recovering the baseline wake losses and of reducing turbine spacing in wind farms while keeping the levelized cost of energy constant. Bartl et al.¹² conducted a wind tunnel experiment showing the effect of intentional yaw-misalignment in an array of 2 turbines. The total power production improvement ranged between 3.5% and 11% depending on the inflow turbulence level and the spacing between the turbines. Campagnolo et al.¹³ obtained a power production increase up to 15% for a 3 turbines array with a streamwise spacing of 4 rotor diameters and tested with different yaw misalignment in the wind tunnel. Bastankhah and Porté-Agel¹⁴ experimentally studied the best yaw configuration for 5 aligned turbines, with an inter-turbine spacing of 5 diameters, showing an increase in the cumulative power production up to 17% with respect to the non-yawed condition with an incoming turbulence intensity of 7%. They also pointed out the important role of the incoming turbulence and the length of the turbine array. In particular, they estimated that the wind farm power increase varies linearly with the number of rows until reaching an asymptotic condition. Recent field studies^15–19 corroborated the beneficial effect of this control strategy on the power production of a turbine array.

Because yaw control is effective only when the turbines are aligned with the wind direction, which varies, for instance, during the diurnal cycle, and when the waked turbines are in below-rated wind speed (region II), the increase in AEP is lower than that predicted by ideal cases with fixed wind direction as shown by Ciri et al.²⁰ and Gebraad et al.²¹ who calculated a gain between 3% and 5% when the complete wind variability at the site is considered. Kanev²² pointed out how the variability of the wind direction may challenge the response of yaw control system to track the instantaneous wind direction. They proposed a dynamic wake steering approach that achieves a good balance between maximizing the power production and minimizing the loads on the yaw system. Wake models^9,17,23–26 could be used to predict the wake interaction in real time in closed-loop wake steering optimization algorithms. Uncertainties on the simplified physics of wake models could be reduced with data-driven calibration. For instance, Howland et al.²⁷ developed a closed-loop wake steering methodology for application in transient atmospheric boundary layer (ABL) flows, which does not rely on an open-loop offline yaw misalignment lookup table calculation. They reduced the error on the predicted power gain by using a data-driven state estimation technique to calibrate the wake model and applied this approach to an array of 6 aligned turbines. However, in a real wind farm, because of the variability of the wind direction, the “clusters” of turbines to which the yaw control should be applied continuously change. Broadly speaking, a “cluster” is a group of turbines in wake interaction. Turbines belonging to a cluster are significantly affected by a change in the operating condition of another turbine in the same cluster. On the other hand, a perturbation to a turbine belonging to a cluster, to a good approximation, does not influence turbines belonging to a separate cluster. Thus, a practical coordinated control strategy would require first the identification of turbine “clusters,” i.e., which turbines are in the wake of others, and then coordinating the control algorithm within the turbines of the same cluster only. As the wind direction changes, different clusters of turbines are formed inside the wind farm. Identifying and tracking such turbine clusters is a major goal of our work.

Clustering data based on the relevant features is a technique commonly used in data analysis to identify similarities^28,29 and simplify complex problems.³⁰ In particular, breaking down a wind farm into several clusters has the additional advantage of simplifying the short-term estimation of the wind farm power production^31–35 and the optimization process as pointed out by Siniscalchi-Minna et al.^36,37 For example, for an array of N turbines, an optimization based on a coordinated approach, such as wake steering, where one optimizes the system and not the single turbine, would require exploring the effect of each control variable (for example, the yaw angle of each turbine) on the N turbines. Because of the convection of the wake, when processing the signals, a time delay needs to be considered,³⁸ which is proportional to the distance between the turbines. As a consequence, for large wind farms (large N), finding the operating points in real time maximizing, for example, the power production could become unfeasible. On the other hand, by identifying the clusters, the optimization problem reduces into the optimization of R independent problems, i.e., the number of clusters, with L_i control variables each (i.e., the number of turbines belonging to ith cluster so that $N=∑i=0RLi$⁠), thus reducing converging time.

To apply theoretical control schemes in real wind farms, it is then necessary to calculate in real time the wake direction, identify the clusters, and split the optimization of the system in the optimization of smaller clusters of turbines. Multiple scanning Light Detection And Ranging (LiDAR) sensors or high fidelity simulations could identify the clusters, but the processing required, the cost and the complexity make this solution impractical for real-time control. Knowing the wind speed and direction in real time could be sufficient to determine the clusters. However, in a real wind farm, the accurate wind conditions are known at the met tower, which is usually located far from the wind farm in order to provide measurements not affected by the blockage. Since the wind direction may vary locally, an optimization based on the wind data measured at the met tower may be sub-optimal. While wind velocity and wind direction are also measured at each turbine, Cortina et al.³⁹ pointed out that the identification of the wind direction using the nacelle mounted wind vane can induce an error as high as 22% depending on the atmospheric stability condition and layout of the wind farm. In addition, the dynamic influence of yaw misalignment on the nacelle mounted sensors is mostly unknown.¹⁷ To cope with this issue and reduce the error, a consensus-based approach to predict local wind direction that uses the wind measurements from nearby turbines has been proposed by Annoni et al.⁴⁰ and applied to control large wind farms⁴¹ showing the potentiality to increase the power production of the wind farm⁴² and to forecast the short-term power production.⁴³ However, wind direction and wake direction may not necessarily coincide in the baseline configuration. It is not uncommon for the turbines to present an unintentional yaw misalignment so their wakes do not coincide with the wind direction. A few degrees of yaw error could reduce the performances of the optimization. In addition, even if the wind direction and wake directions are known with high accuracy, yaw control optimization is effective only if the momentum in the wake is significantly smaller than that in the free stream velocity.⁴⁴ This depends mostly on the entrainment of turbulent kinetic energy, thus, on turbulence intensities, and on the integral scale of turbulence,^45–48 which may not be as easy to measure in real time and be assimilated into reduced order models.^49,50 Therefore, while the knowledge of the instantaneous wind conditions is one of the main requirements for the applicability of power optimization control strategy, the availability of local wind measurements is often limited and inaccurate in real wind farms.

In this paper, we propose a method to identify “clusters” of waked turbines in real time only when yaw control within each cluster has the potential to significantly increase the total power production. The method is based on the correlation of the power signal of each turbine. In fact, both numerical simulations^38,51 and wind tunnel experiment⁵² have shown that the velocity and power production of consecutive aligned turbines are strongly correlated. The proposed method, consistently with our previous studies, is model-free, which mitigates model uncertainties due to atmospheric conditions, turbulence, and atmospheric stability. It is quick and easy to implement using the turbine power signal already available in operational conditions. In the present paper, we further corroborate this concept, which we have previously proposed⁵³ for just one wind direction, by testing it for different wind directions and by optimizing the identified clusters. The power increase obtained for the wind farm and for the identified clusters is a measure of the effectiveness of the proposed method as a real time decision method for the application of yaw control.

Section II presents the wind farm setup and the method employed in the numerical simulations; Sec. III explains the approach for the identification of the turbine clusters and presents the results for different wind directions (⁠$0°, 30°, 60°$⁠, and $90°$⁠); Sec. IV shows the application of yaw misalignment to optimize the power production of the clusters of turbines for each wind direction considered; and finally, Sec. V summarizes the findings of the paper.

An ideal wind-farm composed of 16 NREL-5 MW reference turbines,⁵⁴ arranged in 4 rows and 4 columns, is simulated with large eddy simulations (LESs) and rotating actuator disk model. The turbines have a rotor diameter $D=126 m$⁠, rated wind speed $Urated=11.4 m/s$⁠, and rated power $Prated=5 MW$⁠. As shown in Fig. 1, the turbine spacing in the transversal direction (West–East) is 3D, while in the longitudinal direction (South–North) the spacing is 5D.

The average wind speed at the hub height is equal to $Uref=0.8Urated$⁠, while four wind directions have been simulated: $θ=0°,30°,60°$⁠, and $90°$⁠. Given the symmetry of the wind farm, a generalization to the entire wind rose can be performed by simply rotating the domain.

The simulations have been performed with our in-house LES code UTD-WF. The filtered non-dimensional governing equations for incompressible flow are

where U_i is the i^th component of the filtered velocity vector, P is the filtered modified pressure, $Re=UrefD/ν$ is the Reynolds number, ν is the kinematic viscosity, $τijsgs$ is the subgrid stress tensor, and F_i is the body force that accounts for the effects of the turbines on the aerodynamic field. The numerical discretization, described in Orlandi,⁵⁵ consists of a staggered central second-order finite-difference approximation in a Cartesian coordinate system cumulative with a hybrid low-storage third-order Runge–Kutta scheme to advance the equations in time.

The towers and nacelles are simulated using the immersed boundary method (IBM) implemented by Orlandi and Leonardi⁵⁶ and by Santoni et al.^57,58 The forces of the rotor acting on the flow are reproduced using the rotating actuator disk model.^59,60 The actual blades are replaced with the forces they apply to the flow. The local forces on each section of the blade are computed with the local relative velocity, angle of attack, and airfoil lookup tables. A spreading of the forces in the disk area with a Gaussian kernel is then carried out. Hence, the forces are applied in the governing equation through the body force term [F_i in Eq. (1)]. The spreading distribution depends on both the radial distance and the azimuthal distance from the instantaneous position of the blade. The disk rotates in time according to the actual blade motion and accounts for both the thrust and tangential forces; the angular speed is determined according to the rotor dynamics and controlled using a standard region II control law, where the generator torque is taken proportional to the square of the generator speed.^61,62 Specifically, the instantaneous rotational speed of the turbine ω is determined through the angular momentum balance between the aerodynamic torque $Taero$ and the generator torque $Mgen$⁠,

where I is the rotor inertia. The generator torque is determined through the quadratic control law,

where $ωgen=Ngear·ω$ is the high-speed shaft angular velocity, being N_gear the gear ratio (N_gear = 97). In equilibrium (⁠$ω̇=0$⁠), the operating rotational speed is determined by the value of the torque gain $kgen$⁠. To achieve maximum efficiency, the torque gain is taken to be

where $λopt=7.5$ for the NREL 5 MW reference turbine, R is the radius of the turbine, and C_p is the maximum power coefficient of the turbine. We regard the torque gain in Eq. (4) as the ideal design value.

The computational box in the streamwise direction (along the wind direction) and spanwise direction (orthogonal to wind direction) is varied according to the wind direction in order to have weak effects of the periodic boundary conditions at the lateral sides of the domain and to maintain a distance of the first row of turbines from the inlet to a minimum of 4D and a distance of the last row of turbines of at least 5D from the outlet. The vertical size of the domain is kept constant and equal to 10D.

No-slip conditions are applied at the bottom boundary of the domain, on the nacelles and the towers of the turbines. Free-slip conditions are applied at the top boundary of the computational domain. Periodic boundary conditions are imposed at the two spanwise sides of the domain and radiative boundary conditions⁶³ at the outlet. The grid is stretched in the vertical direction in order to have a finer grid resolution in the region with the turbines. The grid resolution in the region of the turbines is uniform in all directions, $Δx=Δz=Δy=0.03D$⁠. Although the resolution is not sufficient to resolve accurately the boundary layer around the tower (as in most LES), the immersed boundary method provides impermeability and mimics the blockage and the overall momentum loss. In fact, a similar grid resolution was adopted in a previous study⁶⁴ where we coupled our in-house code with the Weather Research and Forecasting (WRF) model, thus solving with a good approximation the interaction between a real wind farm in North Texas and the meso-scale. Numerical results of power production and turbulent intensities at the turbines agreed well with SCADA data.

In order to mimic the atmospheric boundary layer at the inlet, turbulence obtained from a precursor simulation is superimposed to a mean velocity profile expressed by the following law:

where y is the vertical coordinate, y_hub is the hub height, and α is the shear exponent set to $α=0.05$⁠. The streamwise component of the wind velocity at height y is denoted by U and U_hub = U_ref is the mean streamwise component of the wind velocity at the hub height. The precursor simulation is run in a computational box with periodic boundary conditions in both streamwise and spanwise directions, no-slip conditions at the bottom, and free-slip conditions at the top. Roughness cubes are placed on the ground of the computational domain to enhance the generation of turbulence.⁶⁵ From the superposition of the mean flow of Eq. (5) and the turbulence from the precursor, the resulting turbulence intensity at the hub height impinging the first row of turbines is equal to 11%.

The most straightforward process to identify clusters of turbines under stationary wind conditions would be to apply a step change to a control variable, for instance, the yaw angle of one turbine, and observe how the power production of the surrounding turbines changes. If one turbine significantly changes its power production, then it would belong to the same cluster. For every different wind condition, this same process should be applied to all the turbines of the wind farm to determine clusters of turbines that actually interact through their wakes. However, this is unfeasible in a real wind farm for several reasons. First, it may induce fatigue loads and reduce performances. In addition, it would be very slow since the process should be repeated for each turbine of the farm and updated as the wind changes direction. Finally, the hypothesis of stationary wind conditions does not usually hold for real wind farms due to the wind variability in time. Hence, it may not even converge before the wind conditions change.

In a previous paper, Bernardoni et al.⁵³ presented a method to identify turbine clusters by using the correlation of the power production fluctuations of the turbines in a wind farm. It was shown to work well for one wind direction. In the present paper, we extend the application of the proposed method to three additional wind directions. For each wind direction, we then carry out a power optimization by imposing yaw misalignment to the identified clusters. Hence, we validate the proposed method by verifying whether the power production of the clusters and of the wind farm increases. In the interest of clarity, we describe here the main hypothesis and elements of the cluster identification method.

We showed that the effect of a change of operating conditions in an upstream turbine strongly affects the closest downstream turbine. Because of ambient turbulence and viscosity, its effect becomes weaker and weaker as the distance from the downstream turbines increases. Hence, despite the equations being elliptic in space, the convective terms dominate over the diffusive terms, lending justification to the dynamic programming of Rotea.⁶⁶ To a close approximation, we can assume that disturbances applied to an upstream turbine affect mainly the consecutive turbine belonging to the same cluster while little effects are experienced by other turbines. From this observation, we developed an approach to identify clusters of turbines, which is based on the correlation of power production of each pair (r, s) of turbines,

where t is the time, τ is the time-lag, N is the total number of time steps considered for computing the time-averaged quantities (represented with an overline), and P_j is the power production of the j^th turbine.

The proposed cluster identification method develops from the observation that the power production of turbines coupled by the wake interaction is strongly correlated. However, large turbulent coherent structures in the ABL may also correlate the power production of a pair of turbines not necessarily in wake interaction. The main features and rationale of the method to filter out turbine pairs coupled by turbulent coherent structures were proposed in Bernardoni et al.⁵³ and recalled here for sake of clarity:

1. Due to the convective properties of the wakes, the power production correlation between two turbines is maximum after a characteristic time delay that depends on the free-stream velocity and the distance between the turbines, as also noted by Stevens and Meneveau³⁸ and by Ciri et al.⁶ We defined the convection velocity, U_c, as the average velocity at which a disturbance travels in the wake from the upstream to the downstream turbine. Using our database of simulations, we found that the ratio $Uc/Uref$ has a mean value equal to $μU=Uc¯/Uref=0.8$ and a standard deviation $σU/Uref=0.09$⁠. This is consistent with Ciri et al.⁶ that found $Uc=0.7Uref$ for an array of three aligned turbines with a hub height velocity of about $0.7Urated$ and a turbulent intensity of 8%. Therefore, the power correlation coefficient between each pair of turbines is computed in the interval of time-lag $τ∈[d/((μU+3σU)Uref),d/((μU−3σU)Uref)]$⁠, where d is the distance between the two turbines. Using this wide range of time-lag, τ allows considering possible variations of the convection velocity U_c due, for example, to different turbulent intensities. We are, however, able to filter out pairs of turbines that are coupled by coherent structures of the incoming atmospheric turbulence that have a time scale different from one of the wakes.

2. In order to better differentiate the pairs of turbines that are coupled by the wake interaction from the pairs coupled by the turbulent coherent structures, we can exploit the variability in time of turbulence. A turbine is in the wake of another until the wind does not change direction while turbulent coherent structures have random orientation, meander, and are not persistent in a particular position as the wakes. As a consequence, the correlation associated with the wake interaction is higher than the one due to the turbulent coherent structures. Exploiting this property, turbine pairs with a correlation coefficient, ρ, larger than a threshold, ρ_th, are considered to be coupled by the wake interaction. The value ρ_th depends on the layout of the wind farm and on the wind direction. A one-time tuning is required for each different wind farm. The tuning can be carried out by performing numerical simulations of the wind farm or using SCADA data of an existing wind farm. Further parametric studies may be needed to determine the dependence of the value ρ_th from wind properties such as turbulence intensity. For the turbine array considered in this study, only the pairs associated with the highest 15% correlation coefficient are considered as indicative of clustered turbines.

3. Turbine wakes are aligned with the wind direction or, in case of yaw misalignment, are within $±10°$⁠. Indeed, even for large yaw misalignment (ψ) of the rotor with respect to the direction of the incoming flow, the wake is deflected with a much smaller angle(⁠$ϕ$⁠). For example, Ciri et al.¹⁰ found a wake deflection of about $ϕ=4°$ when a yaw misalignment of $ψ=30°$ is applied to rotor in the opposite direction. They also showed that imposing larger yaw misalignment (⁠$ψ>30°$⁠) is detrimental for the power production of the turbine array. Thus, we can conservatively consider that the wakes of the turbines will have a consistent direction throughout the array and it is unlikely that the wind direction changes significantly in the neighborhood of the turbine. We use this hypothesis to filter outliers that may be due to very large turbulent structures. For the turbine pairs identified in points 1 and 2, we compute the angle, θ_L, of the line that links the two turbines with respect to the North direction. From the probability density function (PDF) of the empirical angle distribution, we identify the most probable wake direction $θL*$ and neglect all the turbine pairs with an angle $θ<θL*−10°$ and $θ>θL*+10°$⁠.

4. Pairs of linked turbines that are spaced apart for more than 12D can be neglected since for this separation the wake recovery is such to prevent significant gains by yaw control. Indeed, from the data of Ciri et al. (2018), consistently with the $cos2(ψ)$⁶⁷ and $cos3(ψ)$^68–70 power decay models (where ψ is the yaw angle), the power would decrease by approximately 15% for a typical yaw misalignment on the upstream turbine of $20°$⁠. Therefore, for the yaw control strategy to be effective, the increase of power production on the downstream turbine (⁠$ΔPB$⁠) should be larger than 15% of the power production of the leading turbine (P_A), i.e., $ΔPB>0.15PA$⁠. Because the deflection of the wake only partially clears the trailing turbine rotor disk, to some approximation, it can be assumed that only 50% of the loss of power due to the wake is recovered, $ΔPB=0.5(PA−PB)$⁠. Therefore, if the power production of the baseline configuration is $Pbase=PA+PB$⁠, then the power of the optimized configuration is to some approximation $Popt=PA+ΔPA+PB+ΔPB≃Pbase−0.15PA+0.5(PA−PB)$⁠. Under these hypotheses, $Popt/Pbase>1$ if $PB/PA<0.7$ in baseline conditions. Meyers and Meneveau⁷¹ showed that $PB/PA$ grows by increasing the distance, d, between the two turbines. Specifically, $PB/PA>0.7$ for about $d>12D$⁠. Therefore, it is unlikely that yaw misalignment can be beneficial for a spacing larger than 12D. We speculate that 12D spacing is a conservative estimate because wake meandering and entrainment of mean kinetic energy may further increase $Popt/Pbase$⁠.

The method has been demonstrated to work well for $0°$ wind direction.⁵³ As shown in Fig. 2, the links and clusters identified by the method are consistent with the color contours of the velocity. We remind the reader that the flow visualization is used only as a benchmark and not to determine the clusters. The weak asymmetry that can be observed in Fig. 2 is due to the fact that the incoming flow is not symmetric, but it has turbulence. The effect of turbulence would cancel out if the statistics were computed over large temporal windows. However, the method to be effective for practical applications needs to be faster than the wind variability. As a compromise, we choose to calculate the correlations, Eq. (6), with a running average of the last 30 min of operations. This time window can be tuned once for each wind farm, given the wind variability at the specific site. Here, we further validate the proposed approach by applying it to other wind directions: $30°, 60°$⁠, and $90°$⁠. For each different wind direction, the clusters of turbines change and the spacing between turbines in the clusters varies with a consequent different wake recovery. Sections III A–III C assess how the proposed algorithm works in such different wind conditions.

The cells above the diagonal of the matrix in Fig. 3(a) show the correlation coefficients for each pair of turbine when the wind direction is $30°$⁠. To filter out the pairs of turbines correlated by the turbulence, ρ_th is chosen such that the cumulative distribution function is $CDF(ρth)=0.85$⁠. As a result, only the pairs associated with the colored boxes under the diagonal are left. However, some coherent turbulent structures in the incoming wind can have a timescale of the same order of the window, where we compute the statistics and induce correlation in the power production of turbines that do not belong to the same cluster. As a consequence, not all the turbine pairs left under the diagonal of the matrix of Fig. 3(a) are actually coupled by the wake interaction. The link connecting each pair of turbines corresponds to a particular direction, θ_L, which can be easily calculated from the knowledge of the coordinates of the turbines. We counted the number of links in each $10°$ sector and we computed the PDF of the link directions. The PDF of the link directions should indicate the most probable direction in which the pairs of turbines are correlated. The PDF of the directions associated with the pairs of turbines with $ρ>ρth$ has a peak at $θL*=30°$ [Fig. 3(b)]. Hence, the network of coupled turbines is drawn considering only the correlated pairs that are aligned in the direction $θL∈[20°,40°]$⁠. Pairs of correlated turbines in this interval that are spaced more than 12D apart are removed as well. For example, using only the data from Figs. 3(a) and 3(c), T01 would be linked with T06, T11, T16, and T12. However, for yaw control purposes, only the link with T06 is considered relevant. The resulting network, reported in Fig. 3(c), agrees well with color contours of the time averaged streamwise velocity, used as a benchmark and shown in Fig. 3(d). The visualization in Fig. 3(d) suggests that T02-T07 is coupled by the wake interaction even if the link in the network in Fig. 3(c) is missing. This particular case illustrates the sensitivity to the threshold value ρ_th. In this paper, we used the same value of ρ_th as in our previous paper⁵³ for consistency. However, if, for example, ρ_th is reduced such that the pairs with the top 20% correlation coefficients are retained, the pair T02-T07 would be identified. This suggests that one time fine tuning of ρ_th on the wind direction would improve the efficacy of the method even further.

A similar analysis is presented for the case with $90°$ wind direction. The correlation coefficients for each pair of turbines are shown over the main diagonal of Fig. 4(a) while only the pairs with $ρr,s>ρth$ are reported under the main diagonal. The PDF of the directions of the links, shown in Fig. 4(b), exhibits a dominant peak for $θL*=90°$⁠. The network of aerodynamically coupled turbines obtained with the proposed method, shown in Fig. 4(c), agrees very well with flow visualization [Fig. 4(d)], which is used as a benchmark. For this case, the peak in the PDF is higher than that for $30°$ wind direction with very few outliers. This very good performance of the method for this wind direction is due to the small separation of the turbines in the streamwise direction. Thus, the wake of upstream turbines does not recover before impinging on the downstream turbines. This leads to higher correlation coefficients and a better identification with respect to all the other cases.

For this wind direction, the pairs that are aligned with the wind direction are T01-T08, T05-T12, and T09-T16. The distance between these pairs of turbines is 10D, about twice the distance between T01 and T05 for $0°$ wind direction. As a consequence, the wake recovery is enhanced and the correlation between the power productions of two consecutive turbines is smaller. For example. for $90°$ wind direction, the correlation between the first two turbines (T01-T02, T05-T06, T09-T10, and T13-T14) ranges between 0.58 and 0.74. On the other hand, for this wind direction (⁠$60°$⁠), the correlation of the first pair of turbines ranges between 0.24 and 0.56. Figure 5(a) shows in the area over the diagonal the correlation coefficients between pairs of turbines while under the main diagonal only the pairs with $ρr,s>ρth$ are left. Some of the pairs of turbines with $ρr,s>ρth$ are not aligned with the wind direction, but the external region of the wake almost reaches the boundary of the rotor disk (for instance, the wake of T03 is very close to the rotor of T04). Because of the reduced distance between the turbines, their power signals could show higher correlation than those aligned with the wind direction. For instance, the pair T01-T08 is separated by about 10D and has smaller correlation coefficient (⁠$ρ1,8=0.24$⁠) than the pair T03-T04, which is not aligned with the wind direction (⁠$ρ3,4=0.42$⁠). As another example, the color contours of the time averaged streamwise velocity [Fig. 5(b)] show that the wake of T05 does not appear to directly impinge on T06. However, the closeness between the wake of T05 and the turbine T06 causes a strong interaction. Additional pairs of turbines show high peak values of the correlation coefficient even if they are separated by a large distance as, for example, T01-T10, T01-T14, T04-T07, T04-T13, and T07-T13. For this particular wind direction, 10 turbines face the undisturbed wind when compared to 4 for the $90°$ wind direction case. Thus, the fluctuation of the power production of each of those pairs of turbines is due to the inflow turbulence rather than to turbines wakes. To a smaller extent, this happens also for the case with $30°$ wind direction (7 turbines facing the undisturbed wind); however, in that case the PDF of the directions has a clear peak [Fig. 3(b)]. For this wind direction instead, the PDF of the directions associated with pairs of correlated turbines is very broad without a dominant peak [Fig. 5(c)]. As a consequence, the criteria “3” which filters outliers cannot be applied here and the network of turbines does not identify a clear wake direction or clusters [Fig. 5(d), red links highlight pairs of correlated turbines that are rejected from the criterion 3]. Almost all the turbines act as if they were in free-stream conditions as shown in Table I where the power production of each turbine of the wind farm is reported for every wind direction tested. For wind direction at $0°, 30°$⁠, and $90°$⁠, the turbines belonging to a cluster show a significant reduction in the average power production (because in waked conditions). In the case with $60°$ wind direction, only T08, T12, and T16 present a slight reduction of power production, much smaller than those relative to the other wind directions. This suggests that it is not the method used to identify the clusters that fail but instead, for this wind direction there are no well-defined clusters, and thus, yaw control optimization of columns of turbines would not be effective. This speculation is corroborated in Sec. IV D.

Once the clusters are identified, we proceed to the optimization of their power production, as in a real wind farm. We consider the four wind directions $θ=0°, θ=30°, θ=60°$⁠, and $θ=90°$⁠. A yaw misalignment has been imposed to the most upstream turbine of each cluster.

For $θ=0°$⁠, 4 clusters were identified as discussed extensively in Bernardoni et al.⁵³ and briefly reported in Sec. III. Instead of optimizing a system with 16 turbines, we optimize the 4 clusters with 4 turbines each. We recall the reader that the optimization of the clusters can be run in parallel since disturbances (changes in operating conditions) to a turbine in one cluster do not significantly affect turbines in other clusters.

The turbines T01, T02, T03, and T04 are the most upstream of each cluster. A yaw misalignment between the local wind direction and the normal to the rotor plane $ψ∈[−30°,−20°,−10°,10°,20°,30°]$ is applied to these turbines. The average power production of each turbine is then compared with the baseline case (no yaw misalignment $ψ=0°$⁠). The same turbulent inflow (same precursor simulation) is imposed at the inlet of all simulations to have a consistent comparison. The yaw misalignment imposed to T01, T02, T03, and T04 partially deflects their wakes away from T05, T06, T07, and T08, respectively.

Figure 6(a) shows the power production of the pairs T01-T05 (⁠$●$⁠), T02-T06 (⁠$■$⁠) , T03-T07 (⁠$▲$⁠), and T04-T08 (⁠$♦$⁠) for different yaw angles(ψ) normalized by the power production of the baseline. Figures 6(b) and 6(c) report the percentage of the power variation of the first two rows of turbines when a yaw misalignment of $ψ=−20°$ and $ψ=+20°$ is imposed to the most upstream row of turbines. The power production of the first row of turbines, T01, T02, T03, and T04 decreases by approximately 12% in both cases. However, the increase in power production on the second row of turbines overcomes the loss on the first row and the total power increases. The best performance is found for $ψ=−20°$ with the increase in power production ranging between 3% and 7% consistently with the numerical results of Ciri et al.¹⁰ for an inflow with shear. The power production of the entire array increases by about 4% [× in Fig. 6(a)], which is smaller than that for each pair of turbines because averaged out over the entire array (and 2 rows were not optimized). However, even by yawing only the first row of turbines and optimizing the individual clusters instead of the entire system, a significant increase in power production is obtained, which is promising on the effectiveness of the proposed technique.

The increase in power production is not symmetric with respect to the yaw angle. The asymmetry of the cumulative power production with respect to the sign of the yaw angle is a known phenomenon for a single column of aligned turbines.^10,72 However, for the present configuration, the sign of the most beneficial yaw angle depends on the spanwise location of the cluster inside the wind farm. For instance, T05 increases the power production of 60% for $ψT01=−20°$ and 39% when $ψT01=+20°$⁠. On the contrary, T08 experiences a larger power increase when $ψT04=+20°$ (⁠$PT08/Pbase=+47%$⁠) with respect to the case of $ψT04=−20°$ (⁠$PT08/Pbase=+40%$⁠). Thus, the different power increase among the clusters is due to the downstream row of turbines rather than the upstream row. Positive yaw angles are favorable for pairs on the East side of the wind farm, i.e., T03-T07 and T04-T08. Instead, negative yaw angles largely increase the power production of the pairs on the West side of the farm (T01-T05 and T02-T06). The array of turbines can be thought of as a large porous rectangular bluff body to the atmospheric boundary layer flow. The pressure on the first row of turbines is higher than in the surrounding and causes stream of fluids to tilt toward the sides, in the direction of least resistance. The upstream influence of the wind farm on the incoming flow is usually referred to as wind farm blockage.^73,74 A negative spanwise velocity (Westward) is observed in front of turbines T01 and T02, while a positive spanwise velocity (Eastward) is in front of T03 and T04 as shown in Fig. 7 $1.5D$ upstream the first row of turbines. Hence, the blockage effect consistently changes the local wind direction of a few degrees with opposite sign at the two sides of the wind farm even if the prevailing wind direction is uniform upstream of the wind farm. Downstream the yawed turbine, the wake gradually tends to realign itself to the local prevailing flow direction.^75,76 However, if the rotor is yawed consistently with the spanwise velocity induced by the blockage, the wake realigns further downstream; hence, the power production on the trailing turbine is larger. For example, when the turbine is yawed $ψ=+20°$⁠, the wake tilts Eastward. Therefore, the blockage effect is favorable to the wake steering of T03 and T04 (in the same direction). On the other hand, on turbines T01 and T02, the blockage effect induces a flow in the opposite direction to the wake steering. In fact, the wakes of T01 and T02 are only slightly steered away from the downstream turbine in the cluster [Fig. 6(c)]. The opposite occurs for a yaw angle of $ψ=−20°$[Fig. 6(b)], the blockage being beneficial to T01 and T02 and detrimental to T03 and T04. To summarize, the wake has to be steered consistently with the blockage effect, negative yaw on T01 and T02, and positive on T03 and T04. This effect could not be observed before in ideal studies of one single column of turbines.

To assess the potential of yaw control, the entire cluster T03-T07-T11-T15 has been optimized. To isolate the power improvement of this cluster, all the other turbines of the wind farm were operated in the baseline condition, i.e., with no misalignment respect to the wind direction. In the following paragraph, we will proceed with a step by step optimization process by first optimizing the group of turbines T03-T07-T11 and then the entire cluster T03-T07-T11-T15. From the optimization of the pair T03-T07, we have observed a maximum increase in power production when T03 is yawed of $ψT03=+20°$ [Fig. 6(a)]. Therefore, the yaw misalignment of T03 is kept constant and equal to $ψT03=+20°$⁠, and the yaw angle of turbine T07 is varied between $−20°$ and $+20°$⁠, while no misalignment is applied to T11 and T15. Figure 8(a) shows the cumulative power production variation of the turbine group T03-T07-T11 with respect to the baseline. Here, the asymmetry of the power production variation of the group with respect to the sign of ψ is due to the different interaction between the wake of T03 and the turbine T07. When $ψT07=−20°$ [Fig. 8(b)], i.e., opposite to $ψT03=+20°$⁠, the power production of T07 increases by about 65%. However, the power production of T11 decreases by about 24%. In fact, the wake of T07 is steered westward (since $ψT07<0$⁠) while the wake of T03 is deflected eastward (since $ψT03>0$⁠). The wake impinging on T11 is the coalescence on the two wakes of T03 and T07, resulting in a wider low momentum region. Thus, the turbine T11 faces a much less energetic flow with respect to the baseline case. As a consequence, the power production variation of the group T03-T07-T11 with respect to the baseline is almost negligible [$(P03+P07+P11)/Pbase=+0.1%$], despite the significant power increase in T07. When instead $ψT07=+20°$ [Fig. 8(c)], i.e., in the same direction of the yaw misalignment of T03, the turbine T07 increases the power production by about 28% and the power of T11 increases as well by about 41%. Overall, the cumulative power production of the group T03–T07–T11 increases by 9.6%.

From the static map in Fig. 8(a), the best performance of the group T03-T07-T11 is obtained for $ψT07=+20°$⁠. Therefore, for the next step, we kept constant $ψT03=ψT07=+20°$ and imposed different misalignment angles to the turbine T11 in order to optimize the power production of the entire cluster T03-T07-T11-T15. No yaw misalignment is imposed to T15. Figure 9(a) shows the increase in power production $ΔP=P−Pbase$ normalized with respect to the baseline condition. Since the power production varies in time, we also plotted an error bar representing the standard deviation $σΔP$⁠,

where N_t is the total number of time samples and $μΔP$ is the mean power gain represented by the solid dots in the figure. Different upstream turbulent intensities may affect the variability of the power production and, hence, the size of the error bars. The optimal operating condition, that is the one with the largest power production, is the configuration with $ψT03=ψT07=ψT11=+20°$⁠. A yaw misalignment of about $20°$ was also found as best performing by Howland et al.²⁷ for an array on six aligned turbines with a streamwise spacing of $4.2D$⁠. With this configuration, the cluster of the four turbines increases the power production by 9.1%. Bastankhah and Porté-Agel¹⁴ found that applying the same yaw-misalignment along a same cluster is an effective strategy but not the best configuration. They instead suggest to apply a decreasing yaw angle moving downstream in the cluster. In the present case, all turbines were optimal at $ψ=+20°$⁠; however, for $ψT11=+10°$⁠, the increase in power is basically the same as that obtained with $ψT11=+20°$⁠, which could indicate that consistently with Bastankhah and Porté-Agel,¹⁴ the optimal yaw angle could be slightly smaller than $20°$⁠. Extremum seeking control (ESC) could be used to calculate exactly the angle but that is beyond the scope of this paper. As observed before, the best performance is achieved when the wakes are consistently steered in the same direction through the entire cluster [see Fig. 9(b)]. It should be noted that the power gain in Figs. 8(a) and 9(a) should not be compared between each other because they have a different baseline. In fact, the first is referred to a 3-turbines cluster (T03-T07-T11) while the latter to a 4-turbines cluster (T03-T07-T11-T15).

Figure 10(a) shows instead how the cumulative power of the cluster T03-T07-T11-T15 increases when the yaw misalignment is imposed to a different number of turbines. In particular, the variation goes from +4.6% when yaw-misalignment is applied only to T03 (⁠$ψT03=+20°$⁠), to +6.3% when both T03 and T07 are yawed (⁠$ψT03=ψT07=+20°$⁠), and finally to +9.1% when $ψT03=ψT07=ψT11=+20°$⁠. Figure 10(b) shows the power production of each turbine of the cluster in the baseline conditions and in the optimized configuration. Thanks to the wake steering of the upstream turbine, both T07 and T11 increase their power production, despite their misalignment with the wind direction. The power increase in T15 is the largest because it is in the optimal configuration aligned with the wind direction and the wake of T11 is deflected away.

For this case, the clusters of turbines are T01-T06-T11-T16, T02-T07-T12, T03-T08, T05-T10-T15, and T09-T14 [see Figs. 3(c) and 3(d)]. Yaw misalignment has been applied to the upstream turbines of each cluster and the cumulative power production of the first two turbines has been evaluated and shown in Fig. 11(a). Similarly to $0°$ wind direction, the curves are not symmetric. However, in this case, negative yaw angles (⁠$−ψi$⁠) increase the power production more than the corresponding positive yaw angle (⁠$+ψi$⁠) for all the clusters [Fig. 11(a)]. In fact, while for the $0°$ wind direction, the blockage of the entire farm was the main factor affecting how much the yaw misalignment was effective in tilting the wake; for this wind direction, the clusters are much closer and it is the local flow around the turbines to determine the direction of the wake. The spanwise distance between the wakes is, in this case, $3D cos (30°)≃2.6D$⁠. Figure 11(c) shows the color contours of the spanwise velocity when each rotor is perpendicular to the wind (⁠$ψi=0°$⁠). Each turbine induces in front of the rotor a negative spanwise velocity (blue) on the left side and a positive spanwise velocity (yellow) on the right side as a consequence of the thrust of the rotor. For a negative yaw angle, $ψ<0°$⁠, the wake is deflected westward (negative spanwise direction z). If we consider, for example, T01, the closest turbine, T02, induces a pressure gradient and then spanwise velocities that tend to steer the wake in the same direction. Hence, the wake steering of T01 in the westward direction is enhanced by the presence of the induction zone of T02. As a result, the wake is significantly tilted and the rotor of the trailing turbines, T06, is only partially in the wake of T01 [see Fig. 11(b)] with a consequent power increase. A positive yaw angle, $ψ>0°$⁠, on the other hand, tilts the wake eastward (positive z). However, the induction from T02 (that pushes the wake in the negative z direction) tends to realign the wake to the wind direction. As a consequence, the wake is not steered away from T06 [see Fig. 11(d)] and the power does not increase much. It must be also noted that the wake of T01 is very close to T05, which induces positive spanwise velocity, in the opposite direction of the wake steering for $ψ<0°$⁠. The two effects of T02 and T05 on the wake of T01 do not compensate, despite they are in opposite directions. Because T02 is located much upstream, its effect on the wake of T01 is much stronger than the effect of T05. Indeed, T02 induces negative spanwise velocity right after the rotor of T01. The presence of T02 both boosts the wake steering at a very early stage and enhances the wake recovery process. On the other hand, the turbine T05 induces positive spanwise velocity farther downstream in a region where the wake of T01 is already steered and partially recovered. Thus, the presence of the induction zone of T02 helps to steer the wake of T01 toward negative z-direction in much stronger way than the induction region of T05 does in the opposite direction.

The same phenomena apply also for the other turbine pairs. Hence, in this configuration, negative yaw misalignment is beneficial for all the clusters of the wind farm in contrast with the case of $θ=0°$ where the sign of the most beneficial yaw misalignment depended on the blockage effect of the wind farm.

In the case of $θ=90°$⁠, T01, T05, T09, and T13 are the leading turbines of each cluster [see Figs. 4(c) and 4(d)]. Different yaw misalignments to these turbines have been applied to optimize the cumulative power production of the pairs T01-T02, T05-T06, T09-T10, and T13-T14. Figure 12(a) shows that the best performances are obtained when $ψ=±20°/30°$⁠. In this case, the curves are symmetric to some approximation. Indeed, for this wind direction, the blockage effect of the wind farm is much reduced with respect to the case with $θ=0°$ since the distance between the clusters in the direction perpendicular to the wind is much larger (5D instead of 3D in the case with $θ=0°$ and $2.6D$ for $θ=30°$⁠). Therefore, the effect of the sign of the yaw angle in this configuration is almost negligible. In order to test the yaw optimization strategy over the entire wind farm, we imposed to each cluster a yaw misalignment equal to the most beneficial ψ of the respective pair of turbines in Fig. 12(a). The same yaw misalignment of the upstream turbine was applied to all the turbines of each cluster except for the most downstream one since we demonstrated that this is the best strategy to apply when optimizing a cluster for the case $θ=0°$ (Fig. 9). Thus, the wind farm power production was tested by imposing $ψT01=ψT02=ψT03=−20°, ψT05=ψT06=ψT07=30°, ψT09=ψT10=ψT11=−30°$⁠, and $ψT13=ψT14=ψT15=−20°$⁠. The overall power production of the wind farm increases by 10.4% with respect to the baseline conditions in which each turbine is aligned with the wind direction. Figure 12(c) reports the power production increase of each cluster together with the time-averaged velocity color contour. The maximum power increase is of 16.8% for the cluster composed of T05, T06, T07, and T08 while the minimum power increase is of 6.3% for the cluster composed of T01, T02, T03, and T04. Similar results were obtained by Howland et al.¹⁷ in a field campaign with an array of six aligned turbines with a streamwise spacing of about $3.5D$⁠. They found that the cumulative power production of the array increases by about 7% at an average wind speed of 0.5 times the rated speed of the turbines. However, the turbulent intensity during the measurements period for a more accurate comparison is unknown.

For $60°$ wind direction, clusters were not identified because the rather weak correlation between the power signals of the turbines is discussed in Sec. III. However, from the color contours in Fig. 5(b), it could be argued that there are three pairs of turbines are aligned with the wind direction: T01-T08, T05-T12, and T09-T16, where T08, T12, and T16 are in the wake of T01, T05, and T09, respectively, and there is some opportunity for power production optimization. Even if in a real application, a color contour of instantaneous velocity would not be available, and we assessed the power production change of the clusters T01-T08, T05-T12, and T09-T16 by yawing the upstream turbine (⁠$ψ∈[−30°:+30°]$⁠) and comparing the results with the baseline. If the visually identified clusters and the wind farm increased the power production, then our method would have failed in the cluster identification task. The solid symbols of Fig. 13(a) show the percentage variation of the power production of each pair of turbines. The maximum gain (+1.8%) is obtained from the pair T09-T16 for $ψ=−20°$⁠. For the same yaw misalignment, all the other pairs decrease their power production: T01-T08-2.6% and T05-T12-0.9%. In general, the cumulative power of each pair of turbine shows a negligible variation or a decrease with respect to the baseline. The power production of the farm decreases for all yaw angles, given the turbines except for $ψ=10°$ for which the variation is +0.3% and for $ψ=20°$⁠, which basically produces the same power as the baseline.

From Fig. 5(b), it could have appeared that the proposed method to find the clusters failed for this wind direction. However, the method did not fail, but rather indicated that yaw optimization is not effective for this wind direction. In other words, the cumulative power production of any pair of turbines does not increase if yaw misalignment is applied to the most upstream turbines. Indeed, the fact that a group of turbines is aligned with the wind direction does not necessarily imply that yaw control may increase their cumulative power production. In the case with $60°$ wind direction, the larger spacing of the turbine along to the wind direction allows the wake to almost completely recover. Thus, the coordinated yaw misalignment is ineffective at maximizing the power production.

A large turbine array can be divided into smaller clusters of turbines in order to simplify the power production optimization problem. Operational changes to a turbine affect only the power production of turbines belonging to the same cluster. It is then possible to optimize the power production of each cluster separately to maximize the wind farm power production. To identify the clusters, we used a strategy based on the correlation in time of the power production signal of every turbine.⁵³ The proposed method to identify clusters of turbines is thought to be applied by using available operational data (i.e., power) without assuming an a priori wind direction. In the present contribution, we used only the last 30 min of LES-generated data with a sampling frequency of about 2 Hz. In actual operational conditions, the same methodology can be applied in real-time using a running average over the last 30 min of power production data, thus avoiding the need of large data storage. Different wind directions were reproduced in this study to test and validate the methodology under different wind conditions.

Once the clusters are identified, a tailored optimization strategy for each cluster can be applied in order to maximize the power production of the entire wind farm. In this paper, we calculated with LES the static maps from yaw angles to power production for each of the identified clusters. From the static maps, we determine the optimal yaw settings for each cluster and evaluate the total power increase obtained when the wind farm power maximization problem is decomposed into multiple smaller optimization problems at the cluster level. When the algorithm was able to identify the clusters (⁠$θ=0°, 30°$⁠, and $90°$⁠), an actual power production improvement was obtained. In particular, for the case of $θ=0°$⁠, we found that the power production of a single cluster of 4 turbines can be increased up to 9% by applying a consistent yaw-misalignment of $20°$ to all the turbines of the cluster except the most downstream one. Analysis of the flow field of different cases of the static maps showed that the direction of the yaw misalignment is an important factor when optimizing the power production of clusters of turbines belonging to a wind farm. In fact, the overall layout of the wind farm must be considered when deciding the direction of the yaw-misalignment to be applied. The sign of most beneficial yaw-misalignment may be determined by the blockage effect of the entire wind farm, as in the case of $θ=0°$ or by the local induction zone in the case of small separation of the turbines in the transversal direction (as in the case of $θ=30°$⁠). For the wind direction $θ=60°$⁠, the proposed method did not identify any clusters of aerodynamically coupled turbines. For this wind direction, the yaw optimization did not lead to an appreciable increase in power production. This means that the method to identify clusters did not fail, but instead indicated that coordinated intentional misalignment optimization is not effective for this wind direction. Indeed, the fact that a group of turbines is aligned with the wind direction (⁠$±10°$ to account for possible yaw misalignment) is a necessary but not sufficient condition for a coordinated yaw control strategy to be effective in the optimization of the wind farm power production. If the entrainment of mean kinetic energy is such that the wake recovers a momentum close to that of the free stream, the increment of power on the downstream turbine does not overcome the loss on the trailing turbine. Wake recovery due to the entrainment of mean kinetic energy depends on the distance between turbines and on the turbulent intensities. Therefore, yaw control can be effective for some wind direction and ineffective in others.

The proposed model-free method is complementary to the power optimization control problem and identifies in real time the clusters for which yaw optimization is likely to increase power. While in this paper we used set points determined by the static maps, different optimization methods can be coupled with the proposed method. For example, the power maximization of clusters could be done in real time using a model-free extremum seeking control (ESC) method¹⁰ augmented with recent improvement to the ESC algorithm introduced in Refs. 77 and 78 and further validated in Ref. 79.

The cluster identification method is relatively easy to deploy in actual wind farms, as it requires the wind farm layout and operational data (i.e., power) over a 30 min sliding window. The actual calculations to determine the clusters are means and covariances of the data, as well as an estimate of the wake propagation delays. The simplicity of the procedure will make it a useful method to decide which turbines, if any, should be considered for intentional yaw misalignment as a function of dynamic wind conditions.

This research was funded in part by NSF Award No. 1839733.

The data that support the findings of this study are available from the corresponding author upon reasonable request.