Controls-Oriented Model for Secondary Effects of Wake Steering

This paper presents a model to incorporate the secondary effects of wake steering in large arrays of turbines. Previous models have focused on the aerodynamic interaction of wake steering between two turbines. The model proposed in this paper builds on these models to include yaw-induced wake recovery and secondary steering seen in large arrays of turbines when wake steering is performed. Turbines operating in yaw misaligned conditions generate counter-rotating vortices that entrain momentum and contribute to the deformation and deflection of the wake at downstream turbines. Rows of turbines can 5 compound the effects of wake steering that benefit turbines far downstream. This model quantifies these effects and demonstrates that wake steering has greater potential to increase the performance of a wind farm due to these counter-rotating vortices especially for large rows of turbines. This is validated using numerous large eddy simulations for two-turbine, three-turbine, five-turbine, and wind farm scenarios.

the remainder of the paper. The Gaussian model was included as a wake model within the FLORIS tool (NREL (2019)). It has been used to design a controller for a field campaign in Fleming et al. (2019a), study wake steering robustness (Simley et al. (2019)), and has been validated with lidar measurements (Annoni et al. (2018)). The Gaussian model is also used in wind farm design optimization in Stanley and Ning (2019).
One of the main issues observed with the Gaussian model in FLORIS is that the model tends to under-predict gains in power 5 downstream with respect to LES and field data. In addition, Fleming et al. (2016) and Schottler et al. (2016) show wake steering is asymmetrical, i.e., clockwise and counter-clockwise yaw rotations do not produce equal benefits at the downstream turbine. Fleming et al. (2018a) investigates the importance of considering explicitly the counter-rotating vortices generated in wake steering (Medici and Alfredsson (2006); Howland et al. (2016); Vollmer et al. (2016)) to fully describe wake steering in engineering models. These vortices deflect and deform the wake at the downstream turbine. It is also noted in Fleming et al. 10 (2018a) that these vortices persist farther downstream and impact turbines that are third, fourth, etc. in the row. This is known as secondary steering. It was proposed that modeling the counter-rotating vortices generated in wake steering could provide a means to model this process and how wake steering will function when dealing with larger turbine arrays. Further, Ciri et al. (2018), has shown that modeling/accounting for the size of these vortices versus the length-scales in the atmospheric boundary layer explain variations of performance of wake steering for differently sized rotors. 15 Martínez-Tossas et al. (2019) provides a wake model, known as the curl model, which explicitly models these vortices. The paper shows that modeling the vortices can predict the deflection of the wake in misaligned conditions as well as the change in wake shape and cross-stream flows observed in Medici and Alfredsson (2006); Howland et al. (2016); Vollmer et al. (2016); Fleming et al. (2018a). However, the RANS-like implementation of the curl model, and flow-marching simulation solution significantly increases the computation complexity (around 1000x). 20 This paper presents a hybrid wake model, which modifies the Gaussian model Porté-Agel (2014, 2016); Niayifar and Porté-Agel (2015)), with analytic approximations made of the curl model in ).
This hybrid model will be referred to as the Gauss-Curl Hybrid, or GCH model. We propose it as a compromise which maintains the many advantages of the Gaussian model, while incorporating corrections to address the following three important discrepancies: 25 1. Vortices drive a process of added yaw-based wake recovery, which increases the gain from wake steering to match LES and field results.
2. The interaction of the counter-rotating vortices with the atmospheric boundary layer shear layer and wake rotation induces wake asymmetry naturally.
3. By modeling of the vortices, secondary-steering and related multi-turbine effects are included.

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In this paper, we will introduce the analytical modifications made to the Gaussian model in Section 2. We will use numerous LES simulations to show that the the improvements made in GCH resolves the discrepancies identified above. This model will demonstrate how it compares with LES of two-turbines (Section 3), three turbines (Section 4), five turbines (Section 5), and a 38-turbine wind farm (Section 6). In addition to these simulations, the proposed model is also validated using the results of a wake steering field campaign at a commercial wind farm (Fleming et al. (2019b)).

Controls-Oriented Model
This section briefly describes the Gaussian model used to describe the velocity deficit and the effects of wake steering in a wind farm. The proposed model builds upon the Gaussian model introduced in Bastankhah and Porté-Agel (2016); Abkar and Porté-Agel (2014); Abkar and Port-Agel (2015); Niayifar and Porté-Agel (2015) by including entrainment, asymmetry, and secondary wake steering effects seen in LES as well as field results.

Velocity Deficit Model
The wind turbine wake model used to characterize the velocity deficit behind a turbine in normal operation in a wind farm was introduced by several recent papers including Bastankhah and Porté-Agel (2016); Abkar and Port-Agel (2015); Niayifar and Porté-Agel (2015); Bastankhah and Porté-Agel (2014). The velocity deficit of the wake is computed by assuming a Gaussian wake, which is based on self-similarity theory often used in free shear flows (Pope (2001)). An analytical expression for the three-dimensional velocity, u G , behind a turbine is computed as: where C is the velocity deficit at the wake center, U ∞ is the freestream velocity, δ is the wake deflection (see Section 2.1.1), z h is the hub height of the turbine, σ y defines the wake width in the y direction, and sigma z defines the wake width in the z direction. Each of these parameters are defined with respect to each turbine. The subscript "0" refers to the initial values at the start of the far wake, which is dependent on ambient turbulence intensity, I 0 , and the thrust coefficient, C T . For additional details on near-wake calculations, the reader is referred to Bastankhah and Porté-Agel (2016). Abkar and Port-Agel (2015) 25 demonstrate that σ y and σ z grow at different rates based on lateral wake meandering (y direction) and vertical wake meandering (z direction). The velocity distributions σ z and σ y are defined as: where D is the rotor diameter, u R is the velocity at the rotor, u 0 is the velocity behind the rotor, k y defines the wake expansion in the lateral direction, and k z defines the wake expansion in the vertical direction. For this study, k y and k z are set to be equal and the wake expands at the same rate in the lateral and vertical directions. The wakes are combined using the traditional sum 5 of squares method (Katić et al. (1986)), although alternate methods are proposed in Niayifar and Porté-Agel (2015).

Wake Deflection
In addition to the velocity deficit, a wake deflection model is used to describe the turbine behavior in yaw misaligned conditions which occur when performing wake steering and is also implemented based on Bastankhah and Porté-Agel (2016). The angle of wake deflection, θ, due to yaw misalignment is defined as: The initial wake deflection, δ 0 , is then defined as: where x 0 indicates the length of the near wake, which is typically on the order of 3 rotor diameters. This can be computed analytically based on Bastankhah and Porté-Agel (2016). 15 The total deflection of the wake due to yaw misalignment is defined as: where E 0 = C 2 0 − 3e 1 12 C 0 + 3e 1 3 . See Bastankhah and Porté-Agel (2016) for details on the derivation.

Spanwise and Vertical Velocity Components
The spanwise and vertical velocity components are currently not computed in the Gaussian model, but they are a critical Wake rotation is included by modeling the effects of rotation using a Lamb-Oseen vortex to de-singularize the behavior near the center of the rotor. The circulation strength for the wake rotation vortex is now: The vertical and spanwise velocities can then be computed using the strength of the vortex, Γ, by: where y 0 is the spanwise position of the turbine, represents the size of the vortex core. In this paper, = 0.3D.
In addition to the wake rotation, when a turbine is operating in yaw-misaligned conditions, the turbine generates counter- where ρ is the air density.
As is done with wake rotation, The spanwise V and vertical W velocity components can be computed based on the strength of the wake rotation and yaw misalignment of a turbine. The spanwise velocity can be computed as: where R is the turbine radius and V top and V bottom are the spanwise velocities generated at the top and bottom of the rotor.
The spanwise and vertical velocities are combined using a linear combination at downstream turbines as is done in Martínez-Tossas et al. (2015); Bay et al. (2019). The total spanwise velocity is: Similarly, the vertical velocity can be written as: 5 https://doi.org/10.5194/wes-2020-3 Preprint. Discussion started: 11 February 2020 c Author(s) 2020. CC BY 4.0 License.
The total vertical velocity can be computed as: Note, ground effects are included by adding mirrored vortices below the ground as is done in Martínez-Tossas et al. (2019).
Finally, the vortices generated by the turbines decay as they move downstream. The dissipation of these vortices is described in Bay et al. (2019) and can be computed as: where ν T is the turbulent viscosity, which is defined using a mixing length model: where l m = κz 1+κz/λ T , κ = 0.41, and λ T = D/8. λ T is the value of the mixing length in the free atmosphere (Martínez-Tossas et al. (2019)).

Added Wake Recovery due to Yaw Misalignment
The streamwise velocity and the wake deflection are influenced by these velocity components, V and W . First, the wake recovers more when the turbine is operating in misaligned conditions due to the large-scale entrainment of flow into the wind 15 farm domain. In this paper, we include added wake recovery that is assumed to be primarily due to added entrainment from the presence of vertical velocity, W . Using a control volume approach for momentum conservation, the modified wake velocity, u, can be computed by adding an additional wake recovery component to the Gaussian model: added velocity from yaw (23) where u G is computed by (1), W is computed by (21), i.e., the vertical velocity generated by wake rotation and the counter-20 rotating vortices present when a turbine is operating in misaligned conditions, α r is a tuning parameter that dictates how much the entrainment affects the wake recovery. For this paper, that term is set to α r = 0.03. The larger α r is the smaller the effect of entrainment is on the streamwise velocity component.

Secondary Steering through Effective Yaw
In addition to added wake recovery, the model proposed in this paper is able to predict secondary steering. The wake deflection 25 model described in Section 2.1.1 can be used to describe the deflection of the wake for a two turbine case. However, additional 6 https://doi.org/10.5194/wes-2020-3 Preprint. Discussion started: 11 February 2020 c Author(s) 2020. CC BY 4.0 License.
information is needed to describe the impact of yaw misalignment on turbines in large wind farms as is shown in Fleming et al. (2018a).
Specifically, the vortices described in the previous section propagate far downstream, dissipate, and affect all turbines directly downstream of the turbine that generated the vortices. When they reach a downstream turbine, they impact the wake of the downstream turbine in a phenomenon called secondary steering (Fleming et al. (2018b)). The spanwise and vertical velocities 5 generated by the counter-rotating vortices act like an effective yaw angle at the next turbine. In other words, the spanwise and vertical velocities of upstream turbines affect the deformation and deflection of a wake downstream as if the downstream turbine were implementing wake steering even when it is aligned with the flow. This can be approximated as an apparent or effective yaw angle. To model secondary steering, an effective yaw angle is computed to describe the effect of the vortices generated at the upstream turbine on the downstream turbine wake. The effective yaw angle is computed using the spanwise 10 velocity, V , present at the turbine rotor, which contributes to an effective circulation that is responsible for deflecting and deforming the wake. At a downstream turbine, the strength of the effective circulation, Γ eff is calculated by taking the inverse of (20): where V is spanwise velocities inside the rotor area and N is the number of points in the rotor area. The effective yaw angle, 15 γ eff , is then computed using Γ eff and solving for γ in (13). The total wake deflection can be computed using (9) where the total yaw angle, γ is: where γ turb is the amount of yaw offset the turbine is actually applying. This γ is used in (9) to compute the lateral deflection of the wake. 20 Due to the presence of the effective yaw angle, downstream turbines generally do not have to yaw as much as upstream turbines to produce large gains. This phenomenon was observed in a wind tunnel study (Bastankhah and Porté-Agel (2019)).
In this section, SOWFA simulations were run for a variety of two-turbine scenarios. Each two-turbine scenario is run using both the Gaussian and the GCH model. Note, the effects of secondary steering will have no effect on two-turbine scenarios, and therefore GCH is equivalent to only the yaw-added recovery (YAR) effect addressed in Section 2.3. Therefore, we will refer to the model as YAR in this section. This analysis focuses on a wind speed of 8 m/s with turbulence intensities of 6% and 5 10% TI, where spacing between the turbines is fixed at 7D. It should be noted that the Gaussian model and the GCH model share the same tuning parameters and have been tuned to the same value, which is consistent with what has previously been published in literature. Only turbulence intensity and freestream velocity are changed between simulations to match large eddy simulation. Mean wind speeds vary slightly between low and high turbluence scenarios, e.g. U ∞ = 8.34 m/s for low turbulence and U ∞ = 8.38 m/s for high turbulence cases.

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The first turbine is fixed at one location, while its yaw angle is varied through a range of positive (CCW) and negative (CW) yaw offsets. The second turbine is always aligned the flow, but is moved between -R,0,+R offset laterally from the upstream turbine. The SOWFA simulations with the front turbines yawed +20 are illustrated in Fig. 1 (right column).  The results are analyzed by considering the percentage gain (or loss) in power of the second turbine from yawing the front turbine in each case either +20 • (CCW) or -20 • (CW). These results are shown in Fig. 2. Most of the improvements in GCH are expected to mainly benefit larger arrays of turbine scenarios where secondary steering plays a role. However, the suggested improvements in YAR can be seen in the asymmetry introduced by wake rotation and shear. This can be seen most clearly in the aligned 0R cases (middle row of Fig. 2). The initial Gaussian model assumes the gain on the second turbine is equivalent  Further, considering the case in which positive yaw is harmful, -R, the Gaussian consistently over-estimates the loss, while the 5 YAR model more closely matches SOWFA by producing less negative results. However, it is noted that YAR does not perform as well as the Gaussian model when predicting the gains of negative yaw when the second turbine is offset by -R. This will be subject to future research. It should also be noted that a loads study of wake steering suggested that positive yaw angles are less harmful to turbines and thus mainly positive angles are used for wake steering (Damiani et al. (2017)).

Three-Turbine Analysis
Next, the Gaussian model, GCH model, and SOWFA are compared in three-turbine array simulations. The three-turbine array demonstrates the benefits of yaw-added recovery (YAR) effect as well as secondary steering (SS). The following plots show the contributions of YAR and SS compared with the Gaussian model and the full GCH model, which contains both YAR and SS.
The YAR and SS models are computed by disabling the model produced in Section 2.3 and Section 2.3.1 respectively within 5 the GCH model to isolate the two effects. The three turbine scenario was simulated at 8 m/s with 6% and 10% turbulence intensities and spaced 7D apart in the streamwise direction. Fig. 3 shows the flow fields from the SOWFA simulations for baseline (top row), the first turbine yawed    The Gaussian model is not able to capture the secondary effects of wake recovery and secondary steering. The Gaussian model is able to capture gains in low turbulence (6%) conditions, see Fig. 5. However, it does not see any gains when turbulence intensity is higher as shown Fig. 6. It is also important to note that for Turbine 3 and the total gains, the Gaussian model forecasts a change in power which is symmetrical about changes in Turbine 2, whereas SS and the GCH model predict that a -10 • yaw on a turbine, which is behind a turbine yawed +20 • is counter-productive, while a complementary +10 • yaw is more valuable The combined GCH model is most like LES in both low and high turbulence scenarios. i.e., Turbine 1 is set to +20 • where the wind speed was 8 m/s and the turbulence intensity was 6%. The results of the change in power of the third turbine, as well as the overall total of all three turbines, reflect the importance that the two yaw offsets are in the same direction. i.e., Turbine 1 is set to +20 • where the wind speed was 8 m/s and the turbulence intensity was 10%.  Table 1.

Five-Turbine Analysis
Next, five turbines were simulated in SOWFA, the Gaussian model, and the GCH model for different combinations of yaw angles, starting with all aligned, the first turbine yawed 25 • , the first and second turbine yawed 25 • , and the first three turbines yawed 25 • . Fig. 7 shows the flow field for GCH in baseline conditions (top) and optimized conditions (bottom). The five turbine array was simulated with a wind speed of 8 m/s and turbulence intensities of 6% (labeled as low turbulence) and 10% (labeled 5 as high turbulence). The turbines are spaced 6D in the streamwise direction. Fig. 8 shows the absolute powers of each turbine in the five turbine array, excluding the first turbine, for low turbulence conditions. The GCH model is able to most closely capture the trends seen in SOWFA especially when evaluating total turbine power. The power gains for each turbine are shown in Fig. 9. The Gaussian model is pessimistic about the potential gains for the five turbine case. YAR and SS both contribute significantly to the total gains seen in the five turbine case. It should be noted 10 that all models have a difficult time predicting the absolute power and the power gain of the last turbine. This may be resolved with a more rigorous "turbulence" model than the one used in this model, see Annoni et al. (2018). In addition, this model does not directly account for deep array effects and this may also be a source of error and is a subject of ongoing research. Fig. 10 shows the same five turbine analysis for the high turbulence scenario (10% turbulence intensity). Again, the GCH model most closely follows the trends seen in SOWFA. The most notable difference between GCH and the Gaussian model 15 is that the Gaussian model is extremely pessimistic about wake steering in high turbulence, i.e., there are no gains to be realized under these conditions. However, according to SOWFA, large gains are still expected from wake steering even in high turbulence. The GCH model is able to capture the power gains seen in SOWFA at high turbulence intensities although GCH is still slightly under-predicting the potential gains of wake steering.

Optimization of Five Turbine Array
Engineering wake models in FLORIS are often used to determine optimal setpoints for wake steering and assess the performance of these setpoints. The results of optimizing the Gaussian and GCH models are compared in this section. Specifically, the Gaussian and GCH models were optimized individually for the five turbine case under low and high turbulence conditions. These yaw angles from each optimization were simulated in SOWFA. The power predicted in SOWFA, the Gaussian model, 5 and the GCH model, for each set of yaw angles, are compared in Table 1. The results are compared to 1) a baseline where the yaw angles of all turbines in the five turbine array are zero, and 2) a naive strategy of simply maximizing yaw offsets, subject to an upper bound of 25 • to limit structural loads, for all turbines except the last.
In both low and high turbulence cases, the GCH optimized yaw angles produced higher power gains in SOWFA compared with the Gaussian model, and also outperformed simply operating all turbines (except the last turbine) at a maximum yaw angle 10 of 25 • . Similarly to results observed in a wind tunnel study in Bastankhah and Porté-Agel (2019), GCH produces decreasing yaw angles at farther downstream turbines indicating that GCH is taking advantage of the effective yaw angle produced by the counter-rotating vortices generated by upstream turbines. Note that GCH more closely predicts the gain observed in SOWFA versus the Gaussian in all cases. Finally, a full wind farm analysis was performed to quantify the potential of wake steering when effects such as yaw-added recovery and secondary steering are included. For this analysis, we used a 38-turbine wind farm used as in Thomas et al. (2019). The flow field for the baseline case is shown on the left in Fig. 12 where the wind direction is 270 • and all turbines are aligned with the flow. The middle plot shows the flow field with yaw angles from the optimized Gaussian model and the 5 right plot shows the flow field with the yaw angles from the optimized GCH model. The analysis was performed for two wind directions, 95 • and 270 • , and will be referred to as Case 1 and Case 2 respectively.   The results of the optimization are shown in Table 2. The optimized yaw angles from the Gaussian model and the GCH model were tested in SOWFA. As with the five turbine case, the yaw angles produced in the optimization with the GCH model 5 had the largest gain in SOWFA. In addition, the gains computed by the GCH model are closer to the gains in the SOWFA results than the Gaussian model indicating that the GCH model is better able to capture the secondary effects of the large-scale flow structures generated by misaligned turbines.
Lastly, a full optimization over a wind rose was run for the wind farm in low and high turbulence conditions. The wind rose is shown in Fig. 15 to compute annual energy production (AEP). The Gaussian model and the GCH model were optimized for 10 wake steering over this wind rose and the AEP gains are reported in . Table 3. The Gaussian model predictions of AEP gains are less than half of the gains predicted by the GCH model under both low and high turbulence conditions. This is a promising result for understanding the full potential of wake steering in large wind farms. By taking advantage of these large-scale flow structures, there is more potential for increasing the power production in a wind farm.

Annual Energy Production Results
Model Low Turbulence, TI = 6.5% High Turbulence, TI = 9% Gauss 1.3% 0.7% GCH 2.8% 2.1% This paper introduces an analytical model that better captures the secondary effects of wake steering in a large wind farm. These secondary effects include yaw-added wake recovery as well as secondary wake steering that significantly boosts the impact of wake steering. The results of this model were compared with LES for two, three, and five turbine arrays as well as a 38-turbine wind farm. The model compared well with results from LES and outperformed the Gaussian model in most cases. Furthermore, 5 this paper demonstrated the possible gains in a large wind farm when considering these large-scale flow structures. Controllers can be developed in the future to manipulate these flow structures to significantly improve the performance of a wind farm.
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