Wake redirection at higher axial induction

The energy produced by wind plants can be increased by mitigating the negative effects of turbine-wakes interactions. In this context, axial induction control and wake redirection, obtained by intentionally yawing or tilting the rotor axis away from the mean wind direction, have been the subject of extensive investigations. We have recently shown that the combination of static tilt control with static axial over-induction results in significant power gains. However, these early results were based on idealized turbine models where wake-rotation effects, radial force distributions and realistic turbine controller effects 5 were neglected. In this study we therefore compute power gains that can be obtained by operating tilted rotors at higher axial induction for the more realistic native NREL 5-MW turbine model implemented in SOWFA. We then extend this approach to the case of yaw control. We show that power gains obtained by standard wake redirection based on yaw or tilt control are highly enhanced when the yawed or tilted turbines are operated at higher axial induction. These results confirm our early findings for the case of tilt control and extend them to the case of yaw control suggesting an high potential for the practical application of 10 overinductive wake redirection.

In axial induction control the induction factors of selected (usually upwind) turbines are steered away from the greedy operation mode in order to increase the power production of other (usually downwind) turbines. While static axial induction 25 control has not demonstrated significant power gains in realistic settings (Knudsen et al., 2015;Annoni et al., 2016), dynamic axial induction control has shown promise for significant power gains (Goit and Meyers, 2015;Munters and Meyers, 2017). In wake redirection control the intentional misalignment of rotor axes from the wind direction is used to deflect turbine wakes in the horizontal or in the vertical direction by acting on yaw or tilt angles respectively with a documented increase of the global power produced by the wind farm (Dahlberg and Medici, 2003;Medici and Alfredsson, 2006;Jiménez et al., 2010;Fleming 30 et al., 2014Fleming 30 et al., , 2015Campagnolo et al., 2016;Howland et al., 2016;Bastankhah and Porté-Agel, 2016).
In two recent studies (Cossu, 2020a, b) we have shown that an appropriate combination of (static) tilt and (static) axial induction control results in a significant enhancement of the global power gains obtained in spanwise-periodic wind-turbine arrays. In particular, power gains were observed to be highly enhanced (up to a factor of 2 or 3) when the turbines with rotor tilted by the optimal angle (ϕ ≈ 30 o ) were operated at disk-based thrust coefficient C T = 3 higher than in the baseline case 35 (C T = 1.5).
These early results (Cossu, 2020a, b) were obtained with an actuator-disk model where wake-rotation and the radial distribution of actuator-disk forces were neglected and the turbines were assumed to operate at constant given C T . This highly idealized setting, used in many previous investigations (e.g. Calaf et al., 2010;Goit and Meyers, 2015;Munters and Meyers, 2017), has been instrumental in obtaining general results not depending on the specific turbine control law and blade design 40 but calls for confirmation on more realistic turbine models. Hence, a first goal of this study is to determine the power gains that can be obtained with high-induction (overinductive) tilt control when realistic turbine models are used that take into due account blade-design, wake-rotation and controller specificities. This goal is addressed in the first part of this study, by making use of SOWFA's (Churchfield et al., 2012) native actuator disk model for the NREL 5-MW turbine. In this implementation of the turbine model the radial dependence of the actuator disk force as well as wake rotation and C T are computed from turbine 45 blades properties and NREL 5-MW's five-region realistic controller (Jonkman et al., 2009) is used.
In the second part of the study we address the case of yaw control. Indeed, the increased power gains obtained by operating tilted turbines at higher thrust coefficients mostly result from the increase of wake deviations obtained without a penalization of the power production of the tilted turbine. Overinductive wake deflection could therefore be beneficial also in the case of yaw-control where it is known that higher thrust coefficients also result in larger wake deviations (Jiménez et al., 2010;50 The potential of oveinductive wake redirection will be investigated by computing power gains that can be obtained in a wind-turbine array composed of two spanwise-periodic rows of wind-aligned turbines where the same control is applied to all 60 upwind-row turbines while downwind-row turbines are left in default operation mode. This idealized configuration, which is an extension to the spanwise-periodic case of the two-turbine configuration considered by Fleming et al. (2015), is chosen in order to keep simple the physical interpretation of the results by isolating the effects of tilt or yaw angle and axial induction of the upwind turbines without entering the problem of the optimization of these parameters in multi-row configurations. Importantly, the relevance of these power gains will be tested without excessive assumptions by means of large-eddy simulations in the 65 atmospheric boundary layer using a turbine model which includes the effects of wake-rotation, radial force distribution and a realistic turbine controller.
We anticipate that substantial enhancements (up to a factor of 3) of the power gains induced by wake redirection are found when operating the tilted or yawed turbines at higher axial induction.
The formulation of the problem at hand is introduced in §2. Results are reported in §3 and further discussed in §4. Additional 70 details on used methods are provided in the appendix.

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NREL 5-MW turbines (Jonkman et al., 2009) are considered, which are modeled with SOWFA's native actuator disk method where wake rotation, the radial distribution of aerodynamic forces and the thrust coefficient are all computed from blade properties providing a reliable descriptions of the wake structure except in the near-wake region. We also make use of SOWFA's native implementation of NREL 5-MW's realistic five-region turbine controller based on generator-torque control in the Region-II regime corresponding to the mean wind speeds considered in the following; in this regime we modify axial induction by chang-80 ing the rotor-collective blade pitch angle β. Higher axial inductions are obtained by enforcing negative values of β, resulting in higher local thrust coefficients C T = 2T /(πρu 2 n A), where T is the thrust magnitude and u n is the disk-averaged wind velocity component normal to the disk of area A = πD 2 /4. The incoming flow, generated by means of a precursor simulation in a 3km x 3km domain in the absence of turbines, has a 100m-thick capping-inversion layer centered at H=750m separating the neutral boundary layer with constant poten-85 tial temperature (θ=300K) from the geostrophic region above where the vertical potential temperature gradient is positive (dθ/dz) G = 0.03K/m. In the capping-inversion layer this gradient is (dθ/dz) CI = 0.03K/m. In the precursor simulation, the ABL is driven by a pressure gradient adjusted to maintain an horizontally-averaged mean of 8m/s from the west at z=100m (a few meters above hub height z h =89m). In the region spanned by the turbines (z <152m) the streamwise mean velocity is well approximated by the logarithmic law and the vertical wind veer is less than 4 o (see Cossu, 2020b, where the same ABL 90 has been already considered). Simulations in the presence of wind turbines are repeated in the same 3km x 3km domain starting from the solution of the precursor simulation at t 0 =20000s, corresponding to a well developed ABL, up to t 1 =30000s. Statistics are computed starting from t=24000s, when turbine wakes are fully developed. The pressure gradient issued from the precursor simulation is enforced during the simulation with turbines and the (previously stored) ABL solution at x=0 (west boundary) is used as 95 inflow boundary condition.
In each (spanwise-periodic) row, turbines are spaced by 4D in the spanwise direction (where D=126m is the rotor diameter) and the two rows, are spaced by 7D in the streamwise direction with corresponding turbines of each row aligned with respect to the mean-wind direction (see Fig. 1). Downwind-row turbines are always operated in default mode with the rotor axis at zero yaw angle γ = 0 o (aligned with the mean wind), tilt angle ϕ = −5 o (to prevent rotor-tower hits) and rotor-collective blade 100 pitch angle β = 0 o . In the baseline (reference) case upwind-row turbines are also operated in default mode. The baseline case is then compared to a set of controlled cases where all the turbines of the upwind row are operated at the same non-zero tilt or yaw angle and, possibly, non-zero pitch angle.

Effect of overinduction on tilt control 105
In the baseline case (all turbines operated with γ = 0 o , ϕ = −5 o , β = 0 o ), the usual situation is found where the turbines of the downwind row see a strongly reduced mean wind (see Fig. 1a) therefore producing only ≈30% of the total power, i.e. ≈40% of the produced by the upwind row of turbines (see Fig. 2b). In the following, power gains will be computed with respect to the mean power P Ref produced in this baseline case.
We then consider the case where upwind-row turbines are tilted by ϕ = 30 o , an angle in the range where best power gains 110 have been found in previous studies (Fleming et al., 2014(Fleming et al., , 2015Cossu, 2020a, b), while keeping their rotor-collective blade pitch angle at the default value β = 0 o . In this case, the wakes of the upwind turbines are pushed down by the tilt-induced downwash increasing the mean wind available to downwind turbines (see Fig. 1b). The tilt-induced decrease of power produced by upwind-row turbines is compensated by the increase of the power produced by downwind-row turbines resulting in an ≈5% global power gain for ϕ = 30 o tilt angles (see Fig. 2b).

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In a further step, the rotor-collective blade pitch angle of the tilted upwind-row turbines is changed. Enforcing increasingly negative values of β (i.e. increasing the mean angle of attack of all rotor blades) results in increased thrust coefficients (increased axial induction) which, starting from C T = 1.5 in the baseline case (β = 0 o ), attain C T = 3 for β = −5 o in turbines tilted by ϕ = 30 o (see Fig. 2a). The effect of the increased thrust is twofold: it reinforces the downwash, further increasing the available wind and extracted power in downwind turbines despite the higher wake deficit of upwind turbines (compare Fig. 1c to Fig. 1b) 120 but it also (more slightly) increases the power produced by tilted turbines (Fig. 2b). The combination of these two effects results in highly enhanced (almost tripled) power gains. This is probably related to the fact that for a selected fixed value of β, the local thrust coefficient C T remains almost constant with the tilt angle up to ϕ ≈ 30 o but increases for higher values of ϕ (see Fig. 3b) leading to a stronger downwash which more efficiently exploits the vertical velocity gradient. Smaller values of the optimal tilt angle ϕ would probably be found if C T was held constant instead of β (as in Cossu, 2020b).

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Contrary to a first intuition, increasing the local thrust coefficient C T is not an issue for mean turbine loads because turbines are tilted. Indeed, when turbines are tilted at the default β = 0 o , their mean load (essentially the thrust force) decreases because of the reduced incoming mean wind u n normal to the rotor. For the considered cases, the increase of C T obtained with negative values of β counteracts this thrust decrease but the thrust magnitude remains almost unchanged (within 5%) with respect to the baseline case for ϕ 30 o when optimal β values are used for each ϕ and is reduced for larger ϕ values (not shown). Thrust 135 magnitudes exceed that of the baseline case by more than 5% only when a (suboptimal) excessive induction is enforced for ϕ 20 o . Note also, that in tilted turbines an important part of the thrust force is directed along the (positive) vertical direction compensating the gravity force; the remaining horizontal part of the thrust force is therefore always reduced in turbines operated at the optimal β.  Increasing the local thrust coefficient C T by means of increasingly negative blade-pitch angles in yawed turbines (see Fig. 5a) has effects similar to those observed for the tilt-control case: an increase of wake deficits in upwind-row turbine wakes 150 but also their higher deviation away from downwind turbines (see Fig. 4c) resulting in an increase of the mean power produced by all turbines (Fig. 5b)

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Similarly to the tilt-control case, at higher induction the optimal yaw angles are higher (γ ≈ 30 o ) than in the standard yawcontrol case (γ ≈ 20 o for β = 0 o ). However, differently from the tilt control case, the optimal pitch angle is not very sensitive to the yaw angle, β = −4 o being the optimal value for all considered tilt angles and the increase of C T observed for γ 30 o (see Fig. 6b) does not result in a shift of optimal yaw angles to values larger than 30 o even for the highest considered β = −5 o . This can be probably explained by observing that the vertical wind shear is not exploited by yaw control and thus, in the yaw-control 160 case very large wake deviations are less beneficial than in the tilt-control case.
The main goal of this study was to assess the magnitude of global power gains that can be obtained in wind turbine arrays by combining static wake redirection control and static axial induction control operating tilted or yawed turbines at higher axial induction (overinduction). Results have been obtained by means of large-eddy simulations of a two-rows array of NREL 5MW 165 turbines in a neutral atmospheric boundary layer.
In a first part of the study we have considered the effect of higher induction on tilt-control by using an actuator disk model less idealized than the one used in our previous studies of this approach. The results confirm that, also with this more realistic turbine model, power gains can be highly increased by operating tilted turbines at higher induction (power gains above 15% are found, to be compared to ≈5% obtained with default induction, for the considered set of parameters). This substantial 170 enhancement of power gains is consistent with those found in our previous studies (the absolute level of power gains was, however, larger in Cossu (2020a, b) where three rows of turbines were considered instead of the two rows considered here).
It is also found that rotor-collective blade pitch angles β (and therefore local thrust coefficients C T ) maximizing global power gains do increase with the rotor tilt angle ϕ suggesting that an optimized law β(ϕ) depending on the specific turbine design should be used in tilt-control operation. Our result also indicate that the use of such an optimized law would also guarantee 175 that the thrust magnitude in overinductive tilt-control does not exceed the one of the baseline case by more than 5%.
In the second part of the study we have ascertained if the overinductive wake redirection approach results in power gain enhancement also in the case of yaw control. To this end, we have first considered the standard case where yaw turbines are operated at the standard β = 0 finding power gains of the order of 5%, similar to those found in numerous previous studies (e.g. Fleming et al., 2015, for the two-turbines case). We then show that a very significant increase of power gains (almost threefold, 180 up to ≈15% for the cases considered) is obtained by operating yawed turbines at higher induction.
The findings concerning overinductive yaw control are probably the most relevant of this study. They could, indeed, provide an explanation for the high power gains found by Park and Law (2015) and Munters and Meyers (2018)  layer with NREL 5-MW turbines simulated with SOWFA). Furthermore, yaw control can be tested and applied in most of currently installed wind farms, unlike tilt control which is promising for specifically designed future generation downwindoriented and/or floating turbines (Bay et al., 2019;Nanos et al., 2020).
Additional investigations are, however, necessary to further refine, in many directions, the conclusions of the present study.
Quantitative refinements based on higher-fidelity simulations making use of the actuator line method, and necessarily more 190 refined grids, would be welcome, especially for the largest considered values of the yaw, tilt and pitch angles where the nearand middle-wake structure is probably more sensitive to details of the turbine model.
Another issue is the wind direction. The present study is limited to the wind-aligned case, but it is, of course, important to evaluate power gains also in non-aligned configurations. In this context, an improvement of existing simplified wake models obtained with overinductive yaw or tilt control for realistic wind roses and wind farm configurations.
Finally, it would be very interesting to ascertain if additional power gain enhancements could come from the simultaneous activation of tilt, yaw and axial induction control. It might indeed be possible that, as a consequence of the symmetry breaking associated to wake rotation effects and Coriolis acceleration, optimal power gains are obtained with "hybrid" yaw-tilt rotor-axis rotations even in wind-aligned configurations. This is the subject of current intense research effort. The solution domain extends 1km in the vertical direction and 3km x 3km along the x and y axes and is discretized with cells extending 15m x 15m in the x and y directions and 7m (near the ground) to 21m (near the top boundary) in the vertical direction. ∆t = 0.8s time steps are used to advance the solution. These parameters keep manageable the amount of data stored in the precursor simulation.

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The aerodynamics forces on NREL 5-MW turbines, having a D=126m rotor diameter and z h =89m hub height (Jonkman et al., 2009), are modeled with SOWFA's native actuator disk method where aerodynamic forces are computed from the characteristics of NREL 5-MW blade profiles, rotational speed and the resolved wind velocity. A Gaussian projection of the discretized body forces with a smoothing parameter ε = 20m is also used to avoid numerical instabilities (Martínez-Tossas and Leonardi, 2013). The NREL 5-MW five-region controller implemented in SOWFA is used to control the turbines rotational 225 speed and axial induction. In the Region II regime, the one accessed in the presented simulation, the turbine is driven to the design point (tip-speed ratio and thrust coefficient corresponding to the maximum power coefficient for an isolated turbine)