A single wind turbine reaches its maximum power output when fully aligned with the wind. When the wind direction changes, a turbine uses its yaw to rotate its nacelle on a horizontal plane. By using active yaw control, a wind turbine can keep track of the changes in the wind direction and ensure maximum energy production over time by minimizing its misalignment with the wind. It corresponds to greedy control, where a wind turbine solely tries to maximize its power output .

In the space immediately behind a turbine, the wind speed is slower and more turbulent. Such a phenomenon is called the “wake effect” and is the natural consequence of wind power extraction by the machine. When a wind turbine is located in the wake of another, its power output is reduced (because of a slower wind speed) and its fatigue increased (because of the turbulence). Within a wind farm, depending on the wind direction and the farm layout, most of the turbines can be affected by the wake of others.

Because of wake effects, greedy control can be suboptimal within a farm. Therefore, instead of keeping every turbine aligned with the wind, yaw control can also be used to voluntarily misalign some turbines in relation to the direction of the wind . When a turbine is misaligned with the wind, its wake effect is steered. By intelligently yawing the turbines and steering the wake effects, the wind flow across the turbines can be optimized. Such a method is known as WFFC . A simple example of a two-turbine wind farm is given in Fig. .

Current implemented wake steering strategies usually involve lookup tables (LUTs) . Wake steering strategies are computed for a finite set of different wind conditions prior to the farm operation. The yaw angles of each turbine are computed with steady-state models, regardless of the wind and turbine dynamics. Because a wake steering strategy creates misalignment with the wind, it is highly dependent on variations in the direction of the wind. The wind direction can change over time, and yaw control is constrained by the limited rotational speed of the nacelles. If the wind varies in directions and frequencies that the yaw actuators cannot easily track, computing adequate wake steering strategies over time can be a challenging task.

The study of wind direction dynamics is gaining interest within the research community. Wind direction dynamics can be broken down into large-scale drifts and small-scale fluctuations and can be observed on different scales: the synoptic scale describes long distances and extended time periods, the mesoscale depicts the farm level and time periods from days to weeks, and the microscale corresponds to the turbine level and variations from seconds to minutes. The wind direction is fundamentally nonstationary, and there is incomplete knowledge regarding the physical and statistical characteristics of wind direction fluctuations across specific length scales and timescales that are essential for effective WFFC .

As the farm operates, the wind direction varies in both time (at the farm level) and space (at the turbine level). The study by found that spatial wind direction changes relevant to the operation of wind farm clusters in the German Bight exceed 11° in 50 % of cases. In this present work, numerical simulations are run with steady-state wake models. Therefore, only variations in the wind at the farm level are studied. When the direction varies over time, this work considers it to affect the whole wind farm.

WFFC is most beneficial at low wind speeds because this is where small changes in the wind speeds can lead to important power output variations. The same wake steering strategy will lead to higher power gains at low speeds compared to higher wind speed. Because the wind direction variability is higher for low wind speeds , the study of dependence between wind direction variations and yaw control is important. Also, because the impact of climate change on wind dynamics is unknown, designing robust controllers is necessary for long-term operation.

As tracking wind direction is essential for wind turbines, the literature is rich in studies seeking better wind direction tracking mechanisms. developed a model predictive control (MPC)-based controller on a finite control set to track the wind directions. designed a yaw controller based on very short-term wind predictions. But performing WFFC and wake steering is a more complex optimization problem.

LUTs can be adapted for dynamic control with different methods. Usually, a low-pass filter is used to apply control only for high variations of the direction. A sampling method can be used to adjust the yaw control frequency, and hysteresis mechanisms avoid unnecessary yaw control and restrict the yaw actuators . improved a traditional LUT by anticipating the wind direction changes ahead of upstream turbines. performed WFFC with a receding horizon using gradient-based optimization and ran tests in large-eddy simulations under realistic variations in wind direction and speed. But the wake steering strategies of an LUT fundamentally do not consider the wind dynamics; only their implementation does.

Regarding machine learning (ML) methods, and more particularly reinforcement learning (RL), which is becoming a source of great interest to the scientific community, wind direction variations are often overlooked. The importance of the wind direction dynamics is clearly pointed out by and , but most of the studies carried out later only consider static or quasi-static wind directions. Some recent works have started to consider time-varying wind directions in WFFC optimization .

The remainder of this paper is structured to mirror the three main contributions. Each contribution forms the basis of an individual section, and Sect.  concludes. The contributions and their corresponding sections are as follows.

This work proposes a discretized formalization of the WFFC problem over time as the succession of multiple steady-state optimization problems interconnected by the rotational constraints of the turbines and the evolution of the wind. Due to the discretization hypothesis and the yaw actuation constraints, the important hypotheses regarding the transition between one steady state and the next are formulated. This formalization is conducted in Sect. .

At a time step t, the exogenous variables are a global incoming wind direction Kt∈[0,360] [°] and a global incoming wind speed Vt∈[νmin,νmax] [m s-1]. The wind data can be measured or predicted. Then, a controller does not have access to Kt and Vt directly but to Kt′ [°] and Vt′ [m s-1], with Kt′ a noisy wind direction defined as Kt′=Kt+ϵK,t, and Vt′ is a noisy wind speed defined as Vt′=Vt+ϵV,t. The random noises ϵK,t and ϵV,t can come from either measurement imprecisions or prediction errors.

Because the turbines alter the wind flow inside the farm, the wind in front of a turbine can be different from the global incoming wind. Then, at a time step t, in front of a turbine i, the local wind direction is denoted as κti [°] and the local wind speed is denoted as νti [m s-1]. The computation of the local velocities is based on complex fluid mechanics and is subject to numerous uncertainties. This work conducts numerical experiments where such computations are done using a low-fidelity, steady-state simulator: FLOw Redirection and Induction in Steady State (FLORIS) . The local wind directions stay equal to the global wind direction for any time step, i.e., κti=Kt∀i∈{0,1,…,N-1}, as is common for low-fidelity, steady-state simulations.

At a time step t, a turbine i is characterized by its absolute angular position βti∈[0,360] [°] and its relative orientation or yaw (often used to compute the power output) αti=fyaw(Kt,βti)∈[-180,180] [°] such that 1fyaw(Kt,βti)=(Kt-βti+180)mod360-180. Adding and subtracting by 180 ensures that the yaw stays in the range [-180,180]. As illustrated in Fig. , the yaw corresponds to the rotational movement going from the absolute angular position βti to the wind direction Kt such that βti+αtimod360=Kt. Positive values of the yaw indicate that the turbine is rotated anticlockwise from the wind direction, and negative values of the yaw indicate that the turbine is rotated clockwise from the wind direction.

At a time step t, the yaw setting uti∈[umin,umax] [°] of a turbine i corresponds to the rotational movement of the turbine between time steps t and t+1. Because of mechanical constraints related to the yaw actuator of the nacelle, the yaw setting is bounded between two consecutive time steps. As illustrated in Fig. , the setting is used to update the orientation of the turbine βt+1i=fcontrol(βti,uti) such that 2fcontrol(βti,uti)=(βti+uti)mod360.

The power curve Φ of a turbine gives the theoretical power output (megawatts, MW; y axis) of the machine as a function of the wind speed ν [m s-1] (x axis), considering no yaw misalignment, such that 3Φ(ν)=12⋅ρ⋅A⋅ν3⋅CP(ν), with ρ [kg m-3] the air density, A [m2] the rotor blade area, and CP the power coefficient of the turbine. The theoretical power output is strictly positive if the wind speed is within certain bounds [νcut-in,νcut-out] [m s-1].

At a time step t, the power output of a turbine i considering yaw misalignment Pti=fpower(νti,αti) [MW] is computed from the power curve and the yaw angle such that 4fpower(νti,αti)=Φ(νti)⋅cos⁡p(αti)⋅𝟙{αcut-in⩽αti⩽αcut-out}, with p a parameter accounting for power losses due to misalignment and [αcut-in,αcut-out] [°] a safety bound for the yaw taken into account by the indicator function. Because too much misalignment with the wind can damage the machine, if the yaw is too great, the turbine is shut down and its power output is null.

A policy π is a function returning the yaw settings (ut0,ut1,…,utN-1) of all the turbines at a time step t given a state st. Each wind farm controller is associated with a specific policy. In this work, the state st may be composed of

the current orientations of the turbines {βti}i∈{0,1,…,N-1},

An episode is defined by H time steps during which turbines are controlled via their yaw. An episode is characterized by time series for the wind directions and the wind speeds as well as initial positions for the nacelles. During an episode, it is assumed that all the states belong to the same class (defined in Table ) and the policy is presumed to be stationary (it does not change over time).

The full evolution of an episode is described in Algorithm (). At each time step, the policy returns the yaw settings based on the current state, the system is updated, and the power output of the farm is computed. The yaw setting of a turbine i at the end of time step t is indexed t+1 (because it has been updated) and it is the one used for the power computation of time step t.

At a time step t, to compute the power output of each turbine, local wind velocities are needed. Such computations rely on complex fluid mechanics, depending on the incoming wind and the updated yaw angles of each turbine. For optimization, performing such complex computations is computationally expensive. Therefore, in this work, these computations are carried out by a steady-state simulator νti=fsimulationi(Kt,Vt,{αt+1j}j∈{0,1,…,N-1}), which is used as a substitute for real-life measurements.

The simulation is said to be steady-state because it only depends on the current global wind data and the updated yaw angles. It does not consider previous wind data, previous yaw angles, or time delays in the wake propagation. The evolution of an episode over time is constrained by the rotational bounds of the turbines and the variations of the wind.

At each time step, during the “control policy” operation, a controller knows the evolution mechanisms of the system; i.e., it can conduct any computations with the fcontrol,fyaw,fsimulationi, and fpower functions but based on the wind data provided by the state. Because such data can be noisy, all the computed values can be inexact. For example, at a time step t, if a controller computes the yaw of a turbine i based on its updated orientation βt+1i, it would be equal to αt+1i′=fyaw(Kt′,βt+1i). Because the observed wind direction Kt′ can be different from the true wind direction Kt, the yaw αt+1i′ estimated by a controller can be different from its true value αt+1i.

At a time step t, for a turbine i, and due to the steady-state nature of the simulation, the WFFC problem thus formalized considers a single power output Pti. In reality, during a time step, the wind is time-varying and a turbine takes time to rotate because of mechanical constraints. Therefore, the discretization of the continuous control problem results in the loss of some information and possibly less inaccurate power outputs. To ensure that the discretized power outputs are good approximations, from one time step to another, a turbine is assumed to rotate immediately and the wind is assumed to be quasi-constant.

The duration of a time step is always considered constant during an episode. At a time step t, when a setting uti is applied to a turbine i, the rotational time Tr (minutes) for the turbine to go from its current orientation βti to its next orientation βt+1i is always considered largely inferior to the duration of the time step, i.e., Tr≪Δt for all uti∈[umin,umax]. A turbine always rapidly reaches its target position before the end of the time step duration. But during a time step, no other control will be applied to the turbine. For this reason, the rotational constraints [umin,umax] need to be consistent with the duration of a time step Δt.

The coherence time Tc (minutes) of a wind variable (either the direction or the speed) is the maximum duration during which the variable is quasi-constant. If the coherence time of the wind direction is strictly smaller than the time step duration, a discretized value Kt would stretch too far away from its corresponding continuous signal. The same goes for the speed. Therefore, in this work, the coherence time is always equal to the time step duration, i.e., Tc=Δt, for both the direction and the speed.

The naive controller always tries to keep turbines aligned with the current wind direction as much as possible. It is a weak baseline as it does not conduct any wake steering optimization. It runs with no foresight (i.e., L=0), as it is only concerned with the current observed wind direction Kt′. Therefore, the state st is reduced to {Kt′,{βti}i∈{0,1,…,N-1}}. It consists of a greedy control (no wake steering) where the objective function at a time step t minimizes the amplitude of the yaws, i.e., 7fobj(st,ut)=-∑i=0N-1|αt+1i′|,8withαt+1i′=fyaw(Kt′,βt+1i),9withβt+1i=fcontrol(βti,uti). At a time step t, the rotational movement required for a turbine i to stay aligned with the observed wind direction is equal to fyaw(Kt′,βti). Because of the rotational constraints, this movement is clipped such that it is always an acceptable setting with regards to the yaw actuator, giving a closed-form expression for the solution, defined as 10πnaive=(ut0,ut1,…,utN-1),11such thatuti=clip(fyaw(Kt′,βti),umin,umax),∀i∈{0,1,…,N-1}.

Compared to naive control, wake steering is used to optimize the power output of the farm. In this work, two distinct wake steering strategies are used. One is based only on the instantaneous wind data, and one is based on instantaneous and predicted wind data. The instantaneous controller searches for the yaw settings maximizing the instantaneous power output of the farm. The prediction-based controller maximizes the instantaneous and future power outputs. At each time step t, the same Gauss–Seidel (GS) method is used for both controllers, but with different objective functions. In this work, optimization is conducted with a GS method (described in Algorithm ). A similar approach was first proposed by with a serial-refine algorithm.

The GS method works as follows. A first solution is initialized from the naive controller, where each initial yaw setting keeps its turbine aligned as much as possible with the wind. Then, the GS method iterates over each turbine, from upstream to downstream ones. At each iteration, it solves the optimization problem for the current turbine, considering the yaw settings of all others fixed, by conducting a grid search over a discretized solution space S={umin+l⋅umax-uminny-1} for all l∈{0,1,…,ny-1}, with ny being a precision parameter. Once optimized, the setting of the current turbine is updated, and it goes to the next one.

To have an upper bound in terms of performance (power output) of a wake steering strategy, the rotational constraints are relaxed. It means that in Eq. (), the variables umin and umax are equal to -180 and 180°, respectively. Between two consecutive time steps, each turbine is assumed to be capable of reaching any orientation.

From a different point of view, the upper bound corresponds to the wake steering instantaneous controller, but for a complete steady-state version of the system evolution presented in Algorithm (). All time steps are entirely independent from each other, as there are no longer any rotational constraints for the turbines.

The same objective function of the instantaneous controller, presented in Sect. , is used. It always runs under no foresight (i.e., L=0), as it performs wake steering for the current wind data only. Therefore, the state st is reduced to {Kt′,Vt′,{βti}i∈{0,1,…,N-1}}. The yaw settings computed by the upper bound would not be admissible in reality if the corresponding targeted orientations are too far away from the current ones.

The wind data time series are artificially generated with custom Wiener processes. The wind directions {Kt}t∈{0,1,…,H+L-1} are computed with Algorithm (). The wind speeds {Vt}t∈{0,1,…,H+L-1} are computed with Algorithm (). To generate the time series, an initial value is cumulatively incremented at each time step by variable mt. Each increment mt is independently sampled from a normal distribution of mean 0 and standard deviation σt such that 32σt=τ×δtX,∀t∈{0,1,…,H+L-1}, with τ a normalization variable with regard to the number and range of the generated values and δtX a variation parameter for the wind variable X (either the direction or the speed).

To maintain the wind directions in the range of valid values, i.e., [0,360] [°], the modulo operation is sufficient. To maintain the wind speeds in the range of valid values, i.e., [νmin,νmax] [m s-1], a mirrored function as explained in Fig.  is proposed. The generated values inside the wind speed bounds are not modified. The generated values outside the bounds are recursively mirrored inside the bounds.

The variable δtX defines the level of variation of the wind variable X time series (either the direction or the speed). When equal to 0, the signal is constant. As δtX increases, the absolute value of the increments increases on average. At each time step, δtX is sampled from a uniform distribution defined between δminX and δmaxX. When δminX and δmaxX are equal, all increments {mt}t∈{0,1,…,H+L-1} are independently sampled from the same distribution: the generated time series is stationary with regard to the increments. When δminX<δmaxX, increments are independently sampled from different distributions: the generated time series is nonstationary with regard to the increments. In Fig.  the impact of the δtK variable is shown for the wind direction.

The function fsimulationi(Kt,Vt,{αt+1j}j∈{0,1,…,N-1}) computes the local wind speed in front of a turbine i at a time step t given wind data Kt,Vt, and the yaw of each turbine {αt+1j}j∈{0,1,…,N-1}. This function, introduced in Sect. , is ensured by the low-fidelity, steady-state simulator FLORIS . FLORIS is used with a Gauss curl hybrid wake model. The Gaussian velocity model is implemented based on and . To compute the deflection of the wakes depending on the yaws, the models described by and are used. The turbulence model described by is used. The optional wake modeling options “secondary steering”, “yaw added recovery”, and “transverse velocities”, provided by FLORIS and giving additional features to the fsimulationi function, are enabled.

A wind farm of 34 International Energy Agency (IEA) identical 15 MW wind turbines is used. It has cut-in and cut-out speeds of νcut-in=3 m s-1 and νcut-out=25 m s-1, respectively. Each wind turbine has a rotor diameter of 242.24 m, i.e., a rotor area of 46 087 m2. The air density is ρ=1.225 kg m-3 and the tunable parameter accounting for the power losses due to misalignment is p=1.88. WFFC strategies are sensitive to the distances between turbines. To make the numerical simulations more robust to the distances between turbines, a diamond shape is used for the layout. With a diamond shape, there is an identical distance between each machine and its surrounding turbines. Using 34 machines creates a sufficiently large wind farm for wake steering to be impactful and is sufficiently small for optimization to converge quickly. A FLORIS illustration of the layout used is given in Fig. .

The limits for the wind speed are νmin=4 m s-1 and νmax=10 m s-1. The interval [4,10] m s-1 approximately corresponds to the ascending part of the power curve, where wake steering is the most beneficial for the farm. For wind speeds of [10,25] m s-1, the power output is constant; if the wind speed is reduced because of wake effects, there will be no power deficit. Because this work conducts a sensitivity analysis of yaw control, the wind speed is kept in the range of [4,10] m s-1.

The horizon size is H=144 and the length of the foresight for the prediction-based controller is L=10. The initial wind values are Kinit=270° and Vinit=8 m s-1. The discount factor used for the heuristic Ht is γ=0.99. The precision parameter for the GS methods is ny=120, giving the grid search method good precision.

More technical details regarding the simulations and numerical instabilities are given in Appendix . The time step duration Δt is intentionally undefined, as it will be explained in Sect. . Depending on the time step duration value, different interpretations of the same results will be made. For example, if Δt corresponds to 5 min, then the horizon L=10 means that the prediction-based controller has access to a prediction of the wind of 50 min. In Table , a summary of the experimental setting used in this work is given.

To empirically demonstrate the importance of optimizing yaw control over a long-term time horizon, numerical simulations are performed with perfect and imperfect (noisy) wind predictions. In the graphs, for each curve, the centerline corresponds to the mean and the colored (shaded) area corresponds to the standard deviation of the results obtained through 11 Monte Carlo trials.

For one episode, the total farm power output of a controller C given by Algorithm () is denoted as PC=∑t=0H-1∑i=0N-1Pti. The metric to benchmark a controller C is the power gain [%] between the total farm power output of C and the total farm power output of the naive controller. The power gain is equal to 100⋅(PC-Pnaive)Pnaive.