Yaw-Misalignment and its Impact on Wind Turbine Loads and Wind Farm Power Output

To make wind energy cost competitive with traditional resources, wind turbines are commonly placed in groups. Aerodynamic interaction between the turbines causes sub-optimal energy production. A control strategy to mitigate these losses is by redirecting the wake by yaw misalignment. This paper aims to assess the influence of load variations of the rotor due to partial wake overlap and presents a combined optimization of the power and loads using wake redirection. For this purpose, we design a computational framework which computes the wind farm power production and the wind turbine rotor loads based on the yaw settings. The simulation results show that partial wake overlap can significantly increase asymmetric loading of the rotor disk and that yaw misalignment is beneficial in situations where the wake can be sufficiently directed away from the downstream turbine.


Introduction
In recent years, wind energy has evolved as a reliable source among all the renewable energy resources [1][2][3][4][5]. However, the cost of wind generated electricity is generally higher than that of traditional energy resources [6]. This necessitates further research and innovation of the design and operation of wind turbines that can reduce cost of energy [7][8][9][10][11][12][13][14][15].
An effective strategy to cost reduction is to group wind turbines to share the infrastructure. As a consequence when an upwind wind turbine extracts energy from the wind, a speed deficit and increased turbulence occur in it's wake. Therefore, downstream wind turbines produce less power, and they experience altered loads. Hence, the control parameters of upstream turbines are coupled with the power production and loads of downstream turbines. The significance of such coupling manifests for smaller turbine spacing, as the wake has less time to recover. Because traditional control strategies maximize the individual wind turbine, this phenomenon is not addressed from the perspective of the total power of a wind farm. Accordingly, optimal power production of the wind farm is not guaranteed.
This problem can be mitigated using control strategies that take into account the wake effects. One approach is to decrease the power extraction of upstream turbines to improve the Model FLORIS* Computes the effective velocities U ef f,i and velocity distributions U i at hub height at each turbine i CCBlade* Compute the power production P i and the loads {∆M f , ∆M e } i for each turbine i Find Ω Find the optimal rotor speed Optimizer Optimizes the yaw settings γ i of each turbine i in the wind farm Figure 1: Schematic of the optimization framework. The optimizer uses the power and load measurements obtained from the model to optimize the yaw settings of the wind farm. The shorthand notation {ζ i } is used to indicate that {ζ i |i ∈ U} where ζ i corresponds to a property of wind turbine i and U = {1, 2, ..., N } is the set of indices that number all turbines in a wind farm.
at the rotor disk is implemented as described in Section 2.1. Figure 1 shows the framework that consists of a modified version of the FLOw Redirection and Induction in Steady-state (FLORIS) model [24], a modified version of CCBlade [36] and an optimizer 12 . In the FLORIS* module, the effective wind velocities and velocity distributions in the lateral direction at hub height at each wind turbine are calculated. The module 'Find Ω' determines the optimal rotor velocity for every turbine based on the effective wind velocity. The optimal rotor velocity and the velocity distribution are used to compute the power production and the loads at every turbine in the module CCBlade*. The Optimizer subsection closes the loop by including the power production and the loads of the wind farm in one cost function, and is used to find the optimal yaw settings.
2.1. FLORIS* FLORIS* is an extended version of FLORIS, in that it adds an algorithm that computes the velocity distribution at hub height. FLORIS is a data-driven model that predicts the effect of yaw-misalignment on the power production of wind turbines in a wind farm. FLORIS models the wake using a modified version of the Jensen model [37] and characterizes the wake deflection as in [38]. The fidelity of the model is increased by dividing the wake in three zones with individual expansion and velocity properties ( Figure 2). Furthermore, the model is augmented to fit with power measurements obtained through high-fidelity simulations.
FLORIS defines a local system of coordinates (x, y) relative to the global system of coordinates (x , y ) based on the wind direction Φ. The wake deflection of a turbine i is calculated using its yaw γ i . For a wake caused by turbine j, the velocity in each corresponding wakezone z at the x-coordinate of a turbine i is defined by U j,i,z , z ∈ {1, 2, 3}. FLORIS computes the effective wind velocity, U ef f,i , at a downstream turbine i by combining all the overlapping wakes at its rotor disk. This is done by weighting the wake velocities U j,i,z by their overlap of the corresponding wake zones with the rotor disks using the root-sum-square method of [39].
FLORIS* extends this model with an algorithm that computes the velocity distribution at hub height. We introduce the set of indices U = {1, 2, ..., N } that number all turbines in a wind farm. Each turbine j ∈ U| j =i has a velocity distribution over turbine i denoted byŨ j,i (y r,i ) provided by FLORIS. The local y-coordinate on the rotor disk of turbine i at hub height is denoted by y r,i as shown in Figure 3. We define the combined velocity distributionŨ i (y r,i ) over the rotor blade of turbine i with radius R. Now for any y r,i in the range [−R, R], we obtaiñ U i (y r,i ) by applying the root-sum-square method: where U ∞ is the velocity of the incoming wind flow.

Computation of the optimal rotor velocities
Each turbine has a steady-state rotational speed based on the effective wind velocity it experiences which needs to be determined. At below rated wind speeds every turbines internal controller will try to maximize its power production [40,41]. This corresponds to maximizing the power coefficient Cp(λ tip , β), which can be modeled as a function of tip-speed-ratio λ tip and blade pitch β [42,43]. We estimate the optimal steady-state rotor speed Ω * i for turbine i by:  where λ * i is the tip-speed-ratio that maximizes the corresponding power coefficient Cp i (λ * i , β i ), β i is assumed to be 0 which is optimal for below rated wind speeds and U ef f,i is the local effective wind velocity. The Cp-table is obtained using Wt perf [44] which is a simulation tool developed by NREL that uses blade-element momentum theory to predict the performance of wind turbine blades.

CCBlade*
CCBlade* is an extended version of CCBlade, in that it adds an algorithm to compute the load variations due to partial wake overlap. CCBlade is an implementation of a reliable method to solve the blade element momentum (BEM) equations [45] and predicts the aerodynamic power production and loading of wind turbine blades. For each turbine i in the wind farm, the model uses the effective wake velocity U ef f,i and the optimal steady-state rotor speed Ω * i for its computations. These will be used to compute the power (P i ) of each turbine. The model is extended by taking into account the velocity distribution at hub heightŨ i (y r,i ) in order to compute the maximum-to-minimum difference of the flapwise (∆M f,i ) and edgewise (∆M e,i ) bending moments. As the rotor blade will experience these extreme loads during one rotation, these terms will be used analogously to fatigue.
CCBlade will determine how much each section of the blade at each azimuth angle contributes to the total thrust and torque of a wind turbine. These contributions are denoted respectively as Q p (r, Φ) and T p (r, Φ) where r is the radial coordinate of the rotor disk. Now the total thrust force F T and torque Q over one rotor rotation can be determined by: Ψ Wake of turbine j Rotor disc of turbine i  where N b represents the number of blades (N b = 3). The aerodynamic power, assuming an optimal rotor speed, can be computed by: Finally, the maximum-to-minimum difference of the flap and edgewise moments are determined. To be able to detect the influence of partial wake overlap on the loading, CCBlade* is expanded to take into account a velocity distribution instead of the uniform effective velocity distribution U ef f . Because the greatest difference in loading due to partial wake overlap is expected to happen between the loads at Φ = 90 • and Φ = 270 • , we use the velocity distributioñ U i (y r,i ) at hub height. The maximum-to-minimum flapwise and edgewise bending moment differences are computed as follows:

Optimizer
The optimizer utilizes a Game-theoretic (GT) optimization approach [16] to find the set of yawsettings γ = {γ 1 , γ 2 , ..., γ N } that maximizes the power production and minimizes the loads of the wind farm. To achieve the objective, a constrained minimization problem is defined as follows: where γ max is the maximum allowable yaw angle and is chosen to be γ max = 40 • . This constraint is added to limit the search space. The cost functional c(γ) combines the power and the loads of a turbine i ∈ U as follows: where P i , ∆M f,i and ∆M e,i are respectively the normalized power and the maximum variation in the flapwise and edgewise bending moments of turbine i. The tuning parameter of the optimization objective is λ, where λ = 1 corresponds to a single-objective optimization of the power production and λ = 0 of the loads. Any intermediary values would define a multiobjective optimization. Finally, P max , ∆ M f,max and ∆ M e,max are respectively the estimations of the maximum possible values of the power, flapwise bending moment and edgewise bending moment of a turbine at a certain wind speed which are obtained through a series of simulations (see Section 3.1). The GT approach makes random perturbations to the yaw settings and sets these as the new baseline if they yield an improvement over the previous baseline (see [24]). Using this approach, the global minimum is iteratively found. The decision for this algorithm was based on the discovery of many local minima in the objective function which are caused by the discrete nature of the velocities in the different wake zones.

Results
In this section, the results of two simulation scenarios are presented. In the first scenario, a downstream turbine is swept through the wake of an upstream turbine in order to study the effects of partial wake overlap on the power and the loads. The second scenario will consist of various cases in which the yaw settings of an array of two turbines are optimized using a GT-approach. All simulations are done assuming a constant wind velocity of U ∞ = 8 m/s. The results are shown in Figure 5. It can be seen that partial wake overlap results in a significant increase in the loads, which is unfavorable. Furthermore, the loading and power production are not completely symmetric over the range of dY. The maximum loading is higher when the center of turbine 2 is above the wake center. This is likely caused by the effect of the rotational direction of the rotor blade on the wake. Furthermore, several degrees of wake overlap differently affect the power production and the loads. Full symmetric wake overlap (dY=0 m) results in close-to-minimum loading but also sub-optimal power production. Partial wake overlap (i.e. dY=80 m) comes with an increased power production, but also significantly increases the loads. Finally, no wake overlap (i.e. dY=170 m) maximizes both the power and minimizes the loads. In the situation where dY=0, these optima might not be feasible by just yawing turbine 1 as the maximum amount of wake deflection through yawing is limited. This forms motivation for the claim that the potential load reduction while increasing the power production for a given wind farm layout will strongly depend on the wind direction.

Optimization of the power and the loads
In this section, the optimization results are presented for an array of two turbines which corresponds to Figure 4 for dY=0. The optimization is performed using a mixed-objective cost-function (Equation 7) for various Φ at a windspeed of 8 m/s. The optimization was run for 4000 iterations with an exploration factor E e = 0.4. For each windspeed, the following cases will be considered: • Baseline: The yaw-settings are chosen such that both turbines maximize their individual power production. These settings are referred to as greedy. The results are presented in Table 1. The optimized yaw settings are shown in Table 2. A trend can be seen in each case over all wind directions. An optimization of the power (λ = 1) will steer the wake of turbine 1 away from the downstream turbine, striving for zero wake overlap. The minimization of the loads (λ = 0) will set the yaw of turbine 1 so that the wake is evenly   In general, optimizing the power production through yaw-misalignment heavily increases the loads for Φ = 0, and decreases the loads for Φ = 5 • and Φ = 10 • . In the case of Φ = 0 • , there is only a small reduction in loads possible without heavily penalizing the power production. For Φ = 5 • and Φ = 10 • , Case 1-3 will result in both an increase in power, and a great reduction in loading compared to the baseline. This has to do with the fact that partial wake overlap was already present before the optimization. Case 2 and 3 are able to achieve additional reductions in loading opposed to Case 1 at the expense of a small amount of power production.
The Pareto fronts of the cumulative power and differential loading for various wind directions are depicted in Figure 6. It can be seen that for P hi = 0 • , increasing the power comes with an inevitable increase of differential loading. For Φ = 5 • and Φ = 10 • , more of a trade-off can be made between power and loading. The use of a mixed-objective optimization is most beneficial for Φ = 10 • , as a significant loading decrease can be obtained at the expense of a small amount of power production.

Conclusion and Discussion
This paper studies the load variations due to partial wake overlap and the optimal yaw misalignment to maximize the power production while taking into account the loads variations. The power production and loads were computed using an optimization framework that modeled yaw misalignment, wake interaction and partial wake overlap for multi-turbine setups. In the first simulation, the effect of partial wake overlap on the power production and the loads was investigated (Section 3.1). It was found that partial wake-overlap greatly increases the maximum-to-minimum flapwise and edgewise bending moments compared to full symmetrical wake overlap. The preliminary results suggest that yaw misalignment of upstream turbines  Figure 6: Pareto fronts of total power and cumulative differential loading for various wind directions is a desirable solution in situations where the wake can be sufficiently redirected away from downstream turbines. This idea was confirmed in the second simulation, where an array of two turbines was optimized using a GT optimization approach for various cases (Section 3.2). It was shown that optimizing for power will result in a significant increase up to 350% in loading for Φ = 0 • but in a decrease for Φ = 5 • and Φ = 10 • . This suggests that yaw misalignment isn't equally beneficial for all wind directions, and that its benefits depend on the topology and wind direction of a wind farm. Finally, a combined optimization of power and loads is shown to be beneficial as in the previously mentioned situations, a decrease in loading can be realized at a small expense of the power production.
This work presents preliminary results of mixed-objective wind farm optimization using yaw misalignment and is bound to a number of limitations. Our optimization framework uses a steady-state model for computational efficiency, and therefore transitional effects and turbulence are not taken into account. Furthermore, although FLORIS has been validated using SOWFA simulation data, the load computations have not. Finally, the GT optimization approach is not guaranteed to converge on a global optimum as the number of iterations executed is smaller than the total search space due to time constraints.
In future work, a simulation of the proposed combined optimization of power and loads is to be extended to a full-scale wind farm for a complete range of possible wind directions. Furthermore, a dynamic model may be used for simulation purposes as this will allow us to investigate important loading phenomena such as the transition of partial to full wake overlap and the application of more sophisticated algorithms to compute fatigue loading. Finally, the results ought to be validated using high-fidelity simulations, for instance by using SOFWA coupled with FAST [46].