Wake steering is a technique that optimizes the energy production of a wind farm by employing yaw control to misalign upstream turbines with the incoming wind direction. This work highlights the important dependence between wind direction variations and wake steering optimization. The problem is formalized over time as the succession of multiple steady-state yaw control problems interconnected by the rotational constraints of the turbines and the evolution of the wind. Then, this work proposes a reformulation of the yaw optimization problem of each time step by augmenting the objective function by a new heuristic based on a wind prediction. The heuristic acts as a penalization for the optimization, encouraging solutions that will guarantee future energy production. Finally, a synthetic sensitivity analysis of the wind direction variations and wake steering optimization is conducted. Because of the rotational constraints of the turbines, as the magnitude of the wind direction fluctuations increases, the importance of considering wind prediction in a steady-state optimization is empirically demonstrated. The heuristic proposed in this work greatly improves the performance of controllers and significantly reduces the complexity of the original sequential decision problem by decreasing the number of decision variables.

As global energy consumption increases, there is a strong willingness and necessity to decarbonize electricity production. Hence, renewable energies are becoming increasingly important

In the context of global warming, designing more efficient wind farms is essential. Wake steering is the subject of growing interest within the community to optimize the energy production of wind farms. However, most research regarding wind farm control technologies disregards the relevance of the wind direction variation. This work is motivated by a central question: from what magnitude of wind direction fluctuations is it necessary to consider the wind evolution in a wake steering optimization? To answer this question, this work proposes a new controller based on wind predictions and conducts a synthetic sensitivity analysis of wake steering and wind evolution using steady-state models and artificial wind data.

In wind farm optimization, the use of low-fidelity models (usually based on steady-state models) is favored over higher-fidelity models (usually based on computational fluid dynamics and real-time wake interaction) due to the complexity and computational load associated with solving dynamic equations for every turbine in the farm. Some recent works such as

In wind farm flow control (WFFC), developing effective closed-loop controllers is essential for scaling to larger wind farms and dealing with unpredictable wind conditions. These controllers dynamically adapt their strategies in real time using continuous sensor feedback to guide their decisions. Model-based, closed-loop controllers, in particular, rely on simulators of the environment to conduct continuous optimization while the farm is in operation. Fast and computationally efficient simulation is crucial for these controllers to quickly react to wind and turbines changes. This work focuses on the optimization process itself, adhering to community standards by using widely accepted, open-source, low-fidelity simulators.

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

Example of WFFC on a two-turbine wind farm with the wind coming from the west. The first (upstream) turbine is misaligned and its wake effect is steered away from the second (downstream) turbine. By letting the wind flow more freely to the second turbine, the misalignment of the first turbine increases the total power output of the farm.

Current implemented wake steering strategies usually involve lookup tables (LUTs)

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

As the farm operates, the wind direction varies in both time (at the farm level) and space (at the turbine level). The study by

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

As tracking wind direction is essential for wind turbines, the literature is rich in studies seeking better wind direction tracking mechanisms.

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

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

The remainder of this paper is structured to mirror the three main contributions. Each contribution forms the basis of an individual section, and Sect.

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.

To develop a prediction-based controller, this work presents a reformulation of the instantaneous, steady-state original sequential decision problem over a future time window. The default objective function is augmented by a new heuristic, computed on a prediction of the wind. The proposed heuristic acts as a penalization for the optimization without increasing its dimension and encourages solutions that will guarantee future energy production. The heuristic and the other studied controllers are detailed in Sect.

This work conducts a sensitivity analysis of the wind direction variations and wake steering optimization. It empirically demonstrates the importance of a wind-prediction-based control when the magnitude of the wind direction fluctuations becomes large. The new proposed heuristic greatly improves the performance of a traditional steady-state wake steering optimization when the variations of the wind direction are important. Numerical simulations using synthetic wind data are conducted in Sect.

The environment is composed of a wind farm and some exogenous variables related to the wind direction and the wind speed. The wind farm consists of

At a time step

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

At a time step

At a time step

Example of a wind turbine

The power curve

At a time step

A policy

the current orientations of the turbines

an observation of the current wind

a prediction of the wind at time step

a prediction of the wind at time step

until time step

Four different classes of states based on two distinct properties, with

An episode is defined by

The full evolution of an episode is described in Algorithm (

Full episode evolution over time.

At a time step

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

At a time step

The duration of a time step is always considered constant during an episode. At a time step

The coherence time

At a time step

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.,

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

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

The instantaneous controller searches for the yaw settings maximizing the immediate power output of the farm. It always runs under no foresight (i.e.,

A traditional prediction-based controller searches for the yaw settings of time steps

The optimization problem thus described multiplies the number of decision variables by

The first term of Eq. (

The complexity brought by the prediction-based controller comes from the fact that Eq. (

Each local velocity

Each updated yaw angle

The cosine function at power

The indicator function is removed so that there is no discontinuity. Even if a yaw is too great, it can be of some interest for the optimization to know about the potential power output. The more a turbine is misaligned, the less likely it will be to produce energy and the more it will be penalized.

For each time step, the overall expression is multiplied by a discounted factor

Because this new proposed heuristic depends on neither the future optimized yaw settings (naive control) nor the future local velocities (no simulation), it does not increase the number of optimization variables. The heuristic is a scalar acting as a penalization for the optimization. The final objective function of the prediction-based controller can finally be written as

The heuristic is the discounted weighted sum of the future theoretical power outputs. By choosing certain optimized yaw settings

For example, if the future expected power outputs are high, the heuristic will encourage yaw settings that will put the turbines in good orientations for the future. The heuristic will penalize the objective function for yaw settings that will prevent turbines from keeping track of the wind. An illustration of the heuristic is given in Fig.

Illustration of the heuristic for a turbine

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. (

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 (

The same objective function of the instantaneous controller, presented in Sect.

In Sect.

The wind data time series are artificially generated with custom Wiener processes. The wind directions

Wind direction generator.

Wind speed generator.

To maintain the wind directions in the range of valid values, i.e.,

Toy example of the mirrored function used to keep the generated wind speeds inside specific bounds. Raw data are generated thanks to a process described by Algorithm (

The variable

The function

A wind farm of 34 International Energy Agency (IEA) identical 15 MW

Layout in the form of a diamond shape. The farm comprises 34 identical IEA 15 MW wind turbines. There is an identical space equivalent to the diameter of four turbines between a machine and its adjacent turbines. A distance of four turbine diameters is sufficiently small to create detrimental wake effects for the farm, and therefore the optimization is pertinent and sufficiently large for the design to be realistic. Here the direction is 287.4°, the wind speed is 8.4 m s

The limits for the wind speed are

The horizon size is

More technical details regarding the simulations and numerical instabilities are given in Appendix

Detail of the variables and their values used across the simulations. This configuration is shared by all the numerical simulations. The foresight length is equal to 10 only for the prediction-based controller. Otherwise, it is equal to 0. The yaw rotational constraints will vary across the simulations, but

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

The first set of simulations explores the performance of each controller over increasing variations of wind directions using perfect predictions. Each state comprises perfect information, i.e.,

Numerical simulations are run on 21 different values of

The objective here is to study the impact of the wind direction variations on yaw control. The greater the

In Fig.

As the variations of the wind direction increase, the performance of each controller diverges from the others. For small variations of the wind direction, both the instantaneous controller and the prediction-based controller give similar results. When the variations of the wind direction become large, the instantaneous controller struggles to maintain good performance. The heuristic of the prediction-based controller manages to find better yaw control strategies. The gap between the performance of the upper bound with the other controllers shows how strong wind direction variations, in relation to the rotational constraints of each machine, impact yaw control.

Based on the results given in Fig.

For wind turbines that can rotate from

For wind turbines that can rotate from

In the second set of simulations, the robustness to noisy predictions of each controller is tested. The yaw limits are

Plot of the time series

In Fig. 10, the noise for the wind direction is increasing, i.e.,

Only the noise applied to the wind directions strongly impacts the different policies. The prediction-based controller results in a poorer performance than a naive controller from a noise of 8°. The wind speed noise insignificantly affects the performance of the algorithms. This corroborates the fact that yaw control mainly depends on the wind directions. Because the prediction-based controller uses more wind data points, it is more robust than the instantaneous controller.

As WFFC is becoming more important to increase the energy production of wind farms, this work studies wake steering as a steady-state optimization problem over time. The yaw control problem is formalized as successive multiple steady-state optimization problems interconnected by the rotational constraints of the turbines and the evolution of the wind. Because the function computing the power outputs is steady-state, only the dynamics of a homogeneous global wind and the rotational constraints of the machines are captured. Low-fidelity, steady-state simulators are used because they are not time-consuming and they are suitable for optimization. But future works should perform the same studies with continuous and higher-fidelity simulators such as HAWC2Farm

Traditionally, yaw control is optimized in a steady-state manner. Yaw settings are computed so that they maximize the instantaneous power output of the farm. To optimize wake steering over a long-term time horizon, an MPC method is usually used. Such an approach increases the complexity of the optimization problem, making it harder to solve. To overcome such complexity, a reformulation of the steady-state optimization problem is proposed in this work to consider future wind data. The traditional objective function is augmented by a new heuristic estimating the future expected theoretical power outputs of the farm, weighted by how far the turbines will be from the wind if they are controlled by a naive approach. The new prediction-based controller proposed in this paper has the same number of decision variables as an instantaneous optimization.

Lastly, this work conducts a sensitivity analysis of yaw control and the variations of the wind direction. It demonstrates the importance of optimizing yaw control over future wind data when the variations of the wind directions become large. For strong wind variations, the new prediction-based controller greatly improves the performance without increasing complexity. This work shows, for example, that if deploying wind turbines that can rotate from

This study is conducted on synthetic wind data so future works should explore the same question of dependence between the wind variations and yaw control over real wind data. Because the hypotheses regarding the transition regime may be far from reality, the proposed heuristic could be combined with low-pass filters and hysteresis mechanisms for more realistic implementations. Future works should incorporate the fatigue in the optimization process, as WFFC can have a major impact on the lifetime of each turbine. For example, the objective function of the prediction-based controller could be augmented by a heuristic taking into account the magnitude of the yaw actuations. However, the results provided by this work also suggest that with wake steering strategies more robust to wind direction variations, it would be possible to reach the same level of performance with fewer yaw actuations.

GS method.

The GS method iterates over each turbine in the direction of the wind, one by one, from upstream turbines to downstream ones. The turbines' default coordinates

First, some modifications have been made to FLORIS in order to shut down turbines misaligned too much with the wind during a simulation. At a time step

Round out the power output computed by the simulator.

Take the yaw setting

Find all the yaw settings giving a performance close to the maximum.

Among these selected settings, keep the one closest to the setting corresponding to the naive controller.

To compute the solutions of the upper-bound controller described in Sect.

Detailed results of the simulations conducted on perfect predictions in Sect.

Detailed results of the simulations conducted on perfect predictions in Sect.

The code related to the simulation is open-source and is publicly accessible at

The farm and turbine data are publicly accessible at

EK contributed to the original idea of the heuristic, implemented the codes, conducted all the simulations, and wrote most of the paper. PB, FC, PC, and DE contributed to the writing and review of the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The authors are grateful for the support of the R&D Wind program of TotalEnergies OneTech, especially Cédric Eneau for his support.

This paper was edited by Johan Meyers and reviewed by two anonymous referees.