# A bin-wise approach to wake steering optimization under fatigue load constraints

Ibtihel Ben Gharbia, Nina Khvoenkova, Paolino Tona, Fabrice Guillemin, Marine Dupoiron, Emilie Quenedey, Alexis Cousin

► To cite this version:

Ibtihel Ben Gharbia, Nina Khvoenkova, Paolino Tona, Fabrice Guillemin, Marine Dupoiron, et al. A bin-wise approach to wake steering optimization under fatigue load constraints. Journal of Physics: Conference Series, 2025, 3025, pp.012017. 10.1088/1742-6596/3025/1/012017. [hal-05214706](https://hal HAL Id: hal-05214706)

https://ifp.hal.science/hal-05214706v1

Submitted on 19 Aug 2025

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# A bin-wise approach to wake steering optimization under fatigue load constraints

Ibtihel Ben Gharbia¹, Nina Khvoenkova¹, Paolino Tona², Fabrice Guillemin¹, Marine Dupoiron³, Emilie Quenedey³ and Alexis Cousin¹

¹IFP Energies nouvelles, Institut Carnot IFPEN Transports Energie, 1-4 Av. du Bois Préau, 92852 Rueil-Malmaison, France

²IFP Energies nouvelles, Institut Carnot IFPEN Transports Energie, Rond-point de l'échangeur de Solaize, 69360, Solaize, France

³GreenWITS, Rond-point de l'échangeur de Solaize, 69360, Solaize, France

E-mail: paolino.tona@ifpen.fr

**Abstract.** When implementing wake steering control, AEP (Annual Energy Production) uplift comes at a cost for the wind farm: the misaligned operating mode has an impact on the mechanical loads and vibrations experienced by the turbines, so there is a trade-off between production and fatigue to be considered. AEP optimization problems under fatigue constraints can be computationally demanding, particularly when they consider cumulative fatigue and require computing multiple damage equivalent loads for various wind conditions. In this study, a simplified but tractable formulation of the AEP optimization problem under multiple fatigue constraints is proposed and solved for a small offshore wind farm. The results show that this pragmatic approach significantly improves AEP compared to the baseline scenario without wake steering, while maintaining fatigue loads within reasonable limits and distributing them more evenly among the turbines.

## 1 Introduction

The impact of wake effects in a wind farm, which is detrimental both to production and turbine components lifetime, can be reduced by implementing a *wind farm flow control* (WFFC) strategy (Andersson et al. 2021), where turbine control setpoints are chosen in a coordinated manner. This approach contrasts with the standard, “greedy” control strategy, which aims to maximize the individual power output of each turbine, disregarding its aerodynamic interactions with the rest of the wind farm. In this study, we focus on *wake steering control*, which consists in voluntarily misaligning the wind turbines with respect to the prevailing wind direction, by modifying the setpoints applied by the turbines’ yaw control system. If the yaw setpoints sent to the turbines are correctly chosen, the turbines furthest upstream will experience a drop in production, which will be more than offset by an increase in production from the downstream turbines, leading to a net gain. However, this could come at a cost: the misaligned operating mode affects the mechanical loads and vibrations experienced by the turbines and a compromise between production and fatigue may need to be considered.

As outlined in (Houck 2022), a comprehensive review on WFFC that dedicates a large section to the effects of wake steering on loads, predicting these effects is not straightforward. Misaligning a turbine can increase some of its loads, but not in all cases. Similarly, while steering wakes away from downstream

<figure><img src="image_2.png" /></figure>

Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
turbines is generally expected to be beneficial, it may have a negative impact on certain loads in cases of
partial overlap.

Despite the possible negative effects on fatigue loads – and consequently on turbine lifetime – most wake steering control strategies focus only on power maximization. Typically, the computation of the power-maximizing yaw setpoints is carried out based on a low-fidelity steady-state wind farm flow model, which predicts the time-averaged effects of a control policy on the power output of the wind farm, under a predefined set of prevailing wind conditions. Yaw optimization is generally carried out offline, with its results used to populate a look-up table to be used for online control; but it can also be directly performed online based on the current operational and ambient conditions. Examples of steady-state wake simulators used in this context include FLORIS, PYWAKE, and FARMSHADOW – the tool on which this study is based.

Fatigue can be included in the optimization problem if a suitable model that relates it to wake effects from yaw misalignments is available, while ensuring computational feasibility. Surrogate models trained on aeroelastic simulations are ideal for this. In (Hulsman et al. 2020), a two-turbine setup uses surrogate models trained on LES and aeroelastic simulations to predict damage equivalent loads (DELs) — flapwise root bending and tower bottom bending moments — based on yaw angles and turbine spacing, without considering free-flow wind variations. These models are coupled with FLORIS (Gebraad et al. 2014), to compute wake steering control policies from the solutions of an optimization problem where the DELs are directly incorporated into the objective function and weighted against power.

Similarly, (Sood et al. 2025) introduces a weighted objective function combining normalized power gain and fatigue load increase, expressed as blade root DELs gains. The fatigue model is a look-up table computed for the DTU 10-MW reference turbine using OPENFAST. Optimal yaw setpoints for the TotalControl 32-turbine plant are determined using a Gaussian wake-merging model and applied in a quasi-static closed-loop LES simulation. Note that the study does not assess whether the cumulative fatigue caused by the optimal control setpoints, computed wind-bin-wise, remains within design limits.

In contrast, (Liew et al. 2024) optimizes yaw setpoints to maximize Annual Energy Production (AEP) while ensuring that the Lifetime Damage Equivalent Load (LDEL) of each turbine remains within constraints. A neural network-based surrogate model predicts DELs for multiple turbine components using a reduced-order rotor wind field representation, free-flow wind conditions, and yaw angles. Trained on aeroelastic simulations, it is coupled with PYWAKE for yaw setpoint optimization, via the derivative-free COBYLA algorithm (Ragonneau 2023). Though the surrogate fatigue model is able to compute DELs for different turbine components, only the constraint on blade flapwise DEL is considered in the application example on a fictitious 9-turbine wind farm.

In this study, a simplified but tractable formulation of the AEP optimization problem under multiple DEL constraints is proposed and solved for a small offshore wind farm, where the closest turbines are spaced about four diameters apart. Wind farm layout and wind turbine characteristics (which cannot be disclosed, for confidentiality reasons), along with typical site conditions from the Basis of Design (BoD), operational and structural health monitoring (SHM) data, provided by the operator, have been used to model the wind farm flow and the turbine fatigue as described in Section 2. Three AEP optimization problems are introduced in Section 3: one without DEL constraints, one with cumulative DEL constraints, and a simplified version with wind-bin-wise DEL constraints. Details on the solvers used are also provided. Section 4 focuses on the results for the first and third problems, since the toolchain linking the wind farm flow model and the surrogate fatigue model makes the AEP optimization problem under cumulative DEL constraints too computationally demanding for the considered case study. Section 5 concludes the paper by summarizing the key findings and outlining potential directions for future research.

## 2 Modeling

### 2.1 Wind farm flow model

Wind farm modeling is based on FARMSHADOW, a steady-state flow solver developed at IFP Energies nouvelles (IFPEN), where the turbines are represented as actuator disks, through their power and thrust coefficient curves, and coupled to a set of state-of-the-art analytical wake models. The wake models compute, among other effects, the velocity deficit, wake-added turbulence, wake deflection, and wake superposition. The most advanced of these models, along with their default tuning parameters, have been validated against several experimental and large-eddy simulation datasets with very good results (Blondel & Cathelain 2020, Blondel 2023). The version of FARMSHADOW used in this study computes a horizontal uniform wind field with vertical wind shear based on power-law exponents. Free-flow turbulence intensity is prescribed – along with the wind speed and direction – and is used as an input to all sub-models.

Figure 1 illustrates how FARMSHADOW, which had been initially developed for wind farm layout optimization, can be used in the context of open-loop WFFC. Its settings are chosen once and for all
<figure><img src="image_3.png" /><figcaption>Figure 1: Use of FARMSHADOW for WFFC via wake steering.</figcaption></figure>

<table><caption>Table 1: Wake modeling choices.</caption><thead><tr><td>Model</td><td>Choice</td></tr></thead><tbody><tr><td>Velocity deficit</td><td>Super-Gaussian (Blondel et al. 2020)</td></tr><tr><td>Added turbulence</td><td>Ishihara (Ishihara & Qian 2018)</td></tr><tr><td>Deflection</td><td>Jiménez (Jiménez et al. 2010) or Qian (Qian & Ishihara 2018)</td></tr><tr><td>Superposition</td><td>Local linear sum (Zong & Porté-Agel 2020, Method C)</td></tr></tbody></table>

with regards to wind farm layout, wind turbine characteristics and wake models. The WFFC optimization problem consists in finding the yaw misalignments which maximize the wind farm power production for a given set of free-wind conditions. For this study, all wind conditions except wind speed and direction are kept fixed during the optimization process. For wake modeling, the options listed in Table 1 were selected.

Two different deflection models were tested: Jiménez (Jiménez et al. 2010) and Qian (Qian & Ishihara 2018). The latter, widely used in wake-steering studies in the literature, was originally designed for a top-hat velocity deficit shape function, as in the Jensen model. As a result, it is not strictly compatible with a super-Gaussian (or Gaussian) velocity deficit shape function. However, since the only output of this deflection model is the altered wake center, it can in practice be used in conjunction with any model. This limitation has been alleviated in (Blondel et al. 2020) by slightly changing the formulation of the Jiménez model. The main reason the Jiménez model is frequently used for WFFC is its ease of parameterization, via a coefficient $k_d$ which is directly related to the amount of deflection produced by misalignment. The value that matches best existing data is $k_d = 0.08$ (based on (Blondel et al. 2020)), which is the default value in FARMSHADOW. The Qian deflection model, derived using a Gaussian velocity deficit profile, considers near wake and far wake region separately and has been derived to match existing data on a large range of thrust coefficient and turbulence intensity. The Jiménez model results in greater deflection than the Qian model when using $k_d = 0.08$. Given the large uncertainty of these deflection models, we decided to conduct the yaw optimization with two models, representing an “optimistic” and a “pessimistic” scenario, as the Jiménez model is expected to yield higher AEP gains.

It would have been appropriate to also include the secondary effects of wake steering in the wind farm flow modeling, as the misalignments required to achieve a given deflection are lower when considering them. However, the implementation of secondary steering in FARMSHADOW has not been finalized yet.

## 2.2 Surrogate fatigue model

The surrogate model considered in the following for load prediction has been developed in the digital-twin study described in (Cadoret et al. 2025) following the methodology outlined in (Mc William et al. 2022).

The first step of this methodology, illustrated in the upper part of Figure 2, involves performing quasi-steady wake simulations using FARMSHADOW (or any similar steady-state flow solver) under varying
<figure><img src="image_4.png" /><figcaption>Figure 2: Wind farm wake effect parameterization using surrogates for local wind estimates.</figcaption></figure>

free-flow wind conditions and for a set of yaw misalignments for each turbine. From these simulations, the vertical plane of the averaged wind speed seen by the rotor of a given wind turbine is extracted. From this vertical plane, a Gaussian function is fitted to model each mean wake effect. A turbulence box is then generated around the turbine using NREL's TURBSIM. At each time step of this turbulence box, the Gaussian functions are added to integrate the wakes. Finally, a simulation is conducted with the turbine OPENFAST model, incorporating this wind field – along with the wave parameters observed in SCADA if an offshore wind farm is considered, as in our case study, – to predict fatigue DELs at locations where load measurements are available. In the second step, depicted in the lower part of Figure 2, a more efficient workflow is designed to replace the computationally demanding TURBSIM and OPENFAST simulations with a set of Kriging – or Gaussian process (GP) regression (Rasmussen & Williams 2005) – metamodels trained on them. The input space of these metamodels is composed of the free-flow wind parameters and the Gaussian parameters describing the wake effects.

For the considered wind farm, the DEL channels provided for each turbine are: the blade root edge-wise bending moment; the blade root flap-wise bending moment; the tower torsional moment; the tower bottom stress at 350° with respect to the North; the tower bottom stress at 80° with respect to the North. In Section 3, they will be referred to as DEL<sub>k</sub>, where k ranges from 1 to 5, in the order of their respective definitions.

The trained surrogate model predicts DELs that closely match the observed ones, as illustrated in Figure 3 for the tower bottom stress DELs, and can be used for digital monitoring of fatigue – the initial purpose of the methodology described in (Mc William et al. 2022) – as well as for wake steering control.

In the latter context, the inputs received by the turbine fatigue metamodels at each optimizer call to FARMSHADOW are:

* The reduced set of incident wake parameters representing the mean wake effect at the rotor, extracted from the simulation for the current wind bin.

* The local wind speed and turbulence intensity, $U$ and $TI$, from the same simulation.

* A prescribed set of, possibly varying, sea conditions, including: the significant wave height, $H_s$, the peak wave period, $T_p$, and the wind-wave misalignment angle, $\beta$.

* The yaw misalignment angle of the turbine rotor with respect to the wind, set by the optimizer during the optimization process.

Note that the computational workflow associated with the current coupling of FARMSHADOW to the surrogate models is significantly more time-consuming than the call to FARMSHADOW itself. As discussed in the next section, this hinders the applicability of AEP optimization under multiple cumulative DEL constraints and justifies the development of an approximate, bin-wise approach.
## 3 AEP optimization approaches

The key mathematical notations used in this section are listed in Table 2. All the optimization approaches presented in the following explore a discrete wind set, encompassing free-flow wind scenarios (or wind bins) defined in terms of possible wind speed and wind directions.

<table><caption>Table 2: Notations</caption><tr><td>N<sub>t</sub></td><td>Number of wind turbines in the wind farm.</td></tr><tr><td>t</td><td>Index of the wind turbine within the farm with t ∈ [1 · · · N<sub>t</sub>].</td></tr><tr><td>w<sub>s</sub></td><td>Random variable representing the free-flow wind speed.</td></tr><tr><td>w<sub>d</sub></td><td>Random variable representing the free-flow wind direction.</td></tr><tr><td>w</td><td>Random free-flow wind variable such that w := (w<sub>s</sub>, w<sub>d</sub>).</td></tr><tr><td>W</td><td>Domain of the random variable w. W = [0, ∞[ × [−π, π].</td></tr><tr><td>γ</td><td>Vector of misalignment setpoints, γ : W ↦→ [−π, π]<sup>N<sub>t</sub></sup>.</td></tr><tr><td>W<sub>s</sub></td><td>Discrete set of possible wind speeds.</td></tr><tr><td>W<sub>d</sub></td><td>Discrete set of possible wind directions.</td></tr><tr><td>N<sub>w</sub></td><td>Number of elements of the discrete wind set (wind bins) W<sub>s</sub> × W<sub>d</sub> with N<sub>w</sub> := card(W<sub>s</sub>)card(W<sub>d</sub>).</td></tr><tr><td>P: [−π, π]<sup>N<sub>t</sub></sup> × W ↦→ R</td><td>Function whose image is the power produced by the farm for the yaw angles and wind configuration provided in argument.</td></tr><tr><td>F: [−π, π]<sup>N<sub>t</sub></sup> × W ↦→ R<sup>N<sub>t</sub></sup></td><td>Function whose image is the DEL for each turbine of the wind farm for the yaw angles and wind configuration provided in argument.</td></tr><tr><td>(.)<sub>i</sub></td><td>i<sup>th</sup> coordinate of the vector in argument.</td></tr><tr><td>DEL</td><td>Damage Equivalent Load of a wind turbine.</td></tr><tr><td>DEL<sub>k,t</sub></td><td>Damage Equivalent Load for fatigue component k of wind turbine t.</td></tr><tr><td>m<sub>k</sub></td><td>Wöhler coefficient for component k, used to compute cumulative DEL.</td></tr><tr><td>AEP</td><td>Annual Energy Production of the wind farm.</td></tr><tr><td>E<sub>w</sub>ϕ</td><td>Expected value of ϕ with respect to the random variable w.</td></tr><tr><td>p<sub>w</sub></td><td>Probability of wind occurrence w.</td></tr></table>

### 3.1 “Pure” AEP optimization

The annual expected production of energy for a wind farm, denoted as AEP, is defined by the equation

$$AEP(\gamma) := E_w P(\gamma(w), w),$$

with the power P being computed on 1 h intervals. E<sub>w</sub>P(γ(w), w) represents the expected power production, where γ(w) is the yaw angle policy for different wind scenarios w.

The AEP optimization problem for wake steering control can be mathematically formulated as follows:

$$\max_{\gamma} E_w P(\gamma(w), w) \qquad (1)$$

<figure><img src="image_5.png" /><figcaption>Figure 3: Comparison between observed and simulated normalized tower bottom stress DELs.</figcaption></figure>
subject to the constraint:

$$ \gamma(w) \in [\gamma^{-}, \gamma^{+}]^{N_t}, \forall w. \qquad (2) $$

Here, $\gamma^{-}$ and $\gamma^{+}$ represent the lower and upper limits of yaw angles, respectively, and $N_t$ denotes the number of turbines.

The optimization problem aims to compute a general optimal policy $\gamma: W \to [\gamma^{-}, \gamma^{+}]^{N_t}$, which means determining the optimal yaw angles for the farm across all wind configurations.

Given that the expected value of the power production is computed by:

$$ \mathbb{E}_w P(\gamma(w), w) = \sum_{i=1}^{N_w} p_{w_i} P(\gamma(w_i), w_i), $$

where $p_{w_i}$ are the probabilities of different wind scenarios, the optimization problem (1)-(2) is separable in terms of $\gamma(w_i)$. The resulting independent optimization problems can be solved separately for each bin, in parallel, facilitating efficient computation.

## 3.2 AEP optimization under cumulative DEL constraints

From a mechanical perspective, it is important to consider the cumulative effects of loading to assess the damage sustained by a component. Damage considerations can be incorporated as constraints within the AEP optimization framework, similarly to (Liew et al. 2024). Thus, the maximization of the farm AEP can be constrained by the requirement that the cumulative optimized DELs for each turbine do not exceed the maximum nominal cumulative DELs observed among all turbines. Here, the “nominal” value refers to the DEL when the turbines are directly facing the wind, without misalignment, i.e., when the yaw angles are zero.

The cumulative DEL for fatigue component $k$ associated with turbine $t$ is defined as follows

$$ \left( \sum_{i=1}^{N_w} p_{w_i} \mathrm{DEL}_{k,t}^{m_k}(\gamma(w_i), w_i) \right)^{1/m_k}, $$

where $m_k$ is the corresponding Wöhler coefficient. Typical values for this coefficient are 10 for the blade root and 3 or 4 for the tower. If the weights $p_{w_i}$ are derived from the long-term wind rose observed at a wind farm site, then this cumulative DEL can be interpreted as the lifetime DEL for component $k$.

The optimization problem, which aims to maximize the annual energy production of a wind farm while considering “global” constraints on cumulative damage equivalent loads, can be formulated as follows:

$$ \max_{\gamma} \mathbb{E}_w P(\gamma(w), w) \qquad (3) $$

under the following constraints

$$ \left( \sum_{i=1}^{N_w} p_{w_i} \mathrm{DEL}_{k,t}^{m_k}(\gamma(w_i), w_i) \right)^{1/m_k} \le \max_{t} \left( \sum_{i=1}^{N_w} p_{w_i} \mathrm{DEL}_{k,t}^{m_k}(0, w_i) \right)^{1/m_k}, \quad \forall k, \forall t, \qquad (4) $$

$$ \gamma(w) \in [\gamma^{-}, \gamma^{+}]^{N_t}, \forall w. \qquad (5) $$

The constraint (5) ensures that the yaw angles are bounded within specified limits. The constraints (4) are defined for each component and each turbine, resulting in a total count of

$$ N_t \times \text{number of DEL components,} $$

Those constraints ensure that, for every component of each turbine, the cumulative DEL does not exceed that observed on the most stressed turbine.

The global constraints (4) can be relaxed and managed using a duality framework. Subsequently, a classical duality-based Uzawa algorithm can be employed to solve the optimization problem constrained by (3)-(4), as proposed in (Ben Gharbia et al. 2023b). Upon convergence, the algorithm computes the optimal yaw angles for the entire wind farm across all wind scenarios. This comprehensive approach is to be favored because the constraints, referred to as “global constraints”, are based on aggregated data from all wind bins.
Unfortunately, unlike “pure” AEP optimization, the resulting problem cannot be decomposed into wind-bin independent subproblems. As a result, the method is computationally demanding, especially when multiple DELs are considered and when relatively long processing times are required to compute them. In the study described in (Ben Gharbia et al. 2023b, Ben Gharbia et al. 2023a), it was successfully applied to a 7-turbine wind farm model – corresponding to the Ablaincourt project in France – coupled with a simple look-up table providing only one DEL: the blade root flap-wise bending moment. For the case study considered here, the method proved to be too time-consuming, due to the current coupling between FARMSHADOW and the surrogate fatigue model.

### 3.3 AEP optimization under bin-wise fatigue constraints

To address the issue of the long computational times associated with the above-defined optimization problem, we propose an alternative approximate approach that involves solving a constrained optimization problem separately for each wind bin. The key idea is to replace the global constraint with a local constraint, which considers only the turbine DEL within the current bin. Finally, the cumulative constraints (4) are verified a posteriori across the entire wind rose.

Such a problem can be expressed for each wind bin $w$ as follows:

$$ \max_{\gamma} P(\gamma(w), w) \qquad (6) $$

under the following constraints

$$ DEL_{k,t}^{m_k}(\gamma(w), w) \leq \max_{t} (DEL_{k,t}^{m_k}(0, w)), \quad \forall k, \forall t, \qquad (7) $$

$$ \gamma(w) \in [\gamma^{-}, \gamma^{+}]^{N_t}. \qquad (8) $$

This approach is effective for rapidly obtaining results for a single wind bin; however, it does not ensure compliance with the global constraints on the cumulative DEL outlined in (4). Although the bin-wise constraints (7) prevent the DEL for each turbine from exceeding the highest DEL in the wind farm, they do not address the global constraint. To verify compliance with (4), we must compute the solution across all wind configurations and perform an a posteriori check, which reveals the inherent limitations of this less conservative method.

With regard to the number of inequality constraints (7), the count remains consistent with that of our initial fatigue-constrained optimization problem (3)–(5).

### 3.4 Optimization solvers

In (Ben Gharbia et al. 2023b, Ben Gharbia et al. 2023a), we had initially tried to solve both problem (1) – (2) (“pure” AEP optimization) and problem (3)–(5) (AEP optimization under cumulative DEL constraints, via the Uzawa algorithm) respectively via the Python solvers COBYQA (Bound Optimization BY Quadratic Approximations) and COBYLA (Constrained Optimization BY Linear Approximations) (Ragonneau 2023). Those DFO solvers were favored because gradient computations were not available for the wind farm flow model, and numerical differentiation would have been computationally inefficient. Since it had been found that both solvers struggle to handle the bound constraints on the yaw angles effectively, these constraints were directly integrated into the objective function by adding a penalty term, implementing the barrier method, an interior-point technique. As a result, the problems were transformed into unconstrained problems and solved using COBYLA.

For this study, to solve both problem (1) – (2) (“pure” AEP optimization) and problem (6)–(8), we instead turned our attention to the COBYQA (Constrained Optimization BY Quadratic Approximations) solver. COBYQA, which was recently introduced in Python, represents a significant advancement in DFO techniques (Ragonneau 2023). This algorithm is particularly well-suited for handling complex optimization problems where traditional derivative-based methods are inadequate due to the absence of gradient information. COBYQA constructs successive quadratic models of the objective function using only function values, thereby eliminating the need for derivative information. It iteratively adjusts those models based on results from previous evaluations, enabling it to effectively navigate through the solution space, even with stringent boundary and inequality constraints.

Given its highly satisfactory performance, COBYQA has become our primary choice for solving AEP optimization problems for wake-steering control, regardless of whether fatigue constraints are considered. A significant difference we observed with the fatigue-constrained problem, is that COBYQA needs to be coupled with a multi-start procedure in this case, whereas it remains quite insensitive to the choice of the
initial yaw angles for the “pure” AEP optimization problem. As expected, convergence is more difficult in the former case.

Notably, an approximate solution to problem (1) – (2) can be obtained via the *SERIAL-REFINE* method, a heuristic algorithm proposed by the NREL for fast wake-steering yaw optimization (Fleming et al. 2022), included in the FLORIS suite. It is based on the serial optimization of yaw misalignments, from upstream to downstream, using a small number of candidate angles. According to (Fleming et al. 2022), *SERIAL-REFINE* can find solutions that are at least as good as conventional optimization approaches, but that require much less computation time. However, comparative results in (Fleming et al. 2022) were only provided with respect to a gradient optimization method, used without multi-start, which is an unsuitable approach to wake-steering AEP optimization when no gradient information is available. In the context of this study, we have also implemented the *SERIAL-REFINE* algorithm for use on a FARMSHADOW wind farm model, to benchmark its performance with respect to DFO methods.

## 4 Application and results

The optimizations have been carried out over 4860 wind bins, obtained by discretizing the wind direction into 2° increments and the wind speed in the operating range of the turbines into 1 m/s increments. The sea conditions were fixed to their average values over the year according to SCADA data to avoid increasing the number of degrees of freedom in the optimization process to five, and consequently, the size of the resulting look-up table of optimal yaw angles. The yaw limits were set to $γ⁻ = deg2rad(-20°)$ and $γ⁺ = deg2rad(20°)$.

Since the inclusion of tower DELs (especially the torsion DEL) considerably worsens the convergence and lengthens the duration of the optimization campaigns, only the two DELs for blade root bending moments ($k = 1, 2$) were eventually included as bin-wise constraints in AEP optimization. Nevertheless, as shown in Table 3, after each optimization and upon obtaining the optimal yaw angles, the cumulative DELs are computed for all five components and the compliance with the global constraints is verified.

<table><caption>Table 3: DEL constraint setup.</caption><thead><tr><th>k</th><th>DEL<sub>k</sub></th><th>Unit</th><th>m<sub>k</sub></th><th>As bin-wise<br>constraints</th><th>As global<br>constraints</th></tr></thead><tbody><tr><td>1</td><td>Blade root egde-wise bending moment</td><td>N m</td><td rowspan="2">10</td><td rowspan="2">Included in<br>optimization</td><td rowspan="5">Verified<br>a posteriori</td></tr><tr><td>2</td><td>Blade root flap-wise bending moment</td><td>N m</td></tr><tr><td>3</td><td>Tower torsional moment</td><td>N m</td><td rowspan="3">4</td><td rowspan="3">Not included in<br>optimization</td></tr><tr><td>4</td><td>Tower bottom stress at 350°</td><td>Pa</td></tr><tr><td>5</td><td>Tower bottom stress at 80°</td><td>Pa</td></tr></tbody></table>

Table 4 displays the percentage gain in annual energy production achieved through AEP optimization, both with and without local DEL constraints, using the derivative-free solver COBYQA on a FARMSHADOW simulator parameterized with the two deflection models. The increase in AEP of the optimization outputs provided by a yaw angles policy $γ$ with respect to the nominal policy ($γ ≡ 0$) is given by the following relation:

$$ \text{AEP Gain}(\gamma) \text{ [\%]} := 100 \left( \text{AEP}(\gamma) - \text{AEP}(0) \right) / \text{AEP}(0). $$

The results illustrate the sensitivity of AEP gains to the choice of deflection models. Notably, the inclusion of fatigue constraints only slightly diminish the AEP uplift. For “pure” AEP optimization, consistent with the conclusions of (Fleming et al. 2022), the *SERIAL-REFINE* algorithm closely approximates the performance of COBYQA, while requiring significantly less time to perform the optimization (about 5 min on a high-end laptop compared to more than 2 h).

<table><caption>Table 4: AEP gains [%] for the different AEP optimization problems.</caption><thead><tr><th rowspan="2">Deflection model</th><th colspan="2">Without DEL constraints</th><th>Under bin-wise DEL constraints</th></tr><tr><th>SERIAL-REFINE</th><th>COBYQA</th><th>COBYQA</th></tr></thead><tbody><tr><td>Jiménez</td><td>3.115</td><td>3.116</td><td>2.741</td></tr><tr><td>Qian</td><td>2.078</td><td>2.084</td><td>1.848</td></tr></tbody></table>
<figure><img src="image_6.png" /><figcaption>(a) For normalized blade root edge-wise moment. The relative difference w.r.t. the maximum is 1.1%.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(b) For blade root flap-wise moment. The relative difference w.r.t. the maximum is 10.7%.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(c) For torsional moment cumulative DEL. The relative difference w.r.t. the maximum is 15.3%.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(d) For tower bottom stress cumulative at 350°. The relative difference w.r.t. the maximum is 4.6%.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(e) For tower bottom stress at 80°. The relative difference w.r.t. the maximum is 5.7%.</figcaption></figure>

Figure 4: Cumulative DELs for a subset of 15 turbines (for the Jiménez deflection model), normalized w.r.t. the minimum (1.0) and the maximum (2.0) values in the baseline case. The cumulative DELs for the baseline case are represented in blue, those for the AEP optimization methods in orange (without DEL constraints) and in green (with bin-wise constraints). The red lines represent the cumulative DEL threshold that must not be exceeded by those methods. The relative differences w.r.t. the maximum value are indicated to illustrate how close the cumulative DELs are to one another for each DEL channel.
To verify compliance with the global constraints of cumulative DELs (5), a verification step is conducted a posteriori after optimizing all wind configurations for the two considered deflection models. For the Jiménez model, Figures 4a and 4b display the normalized cumulative DELs associated to the two blade root moments (edge-wise and flap-wise), for a representative subset of 15 turbines. Each plot presents the cumulative DELs for three distinct scenarios: the baseline case (in blue), and the AEP optimization methods, both without (in orange) and with (in green) the imposition of bin-wise constraints on edge and flap blade root moment DELs. The red line represents the cumulative DEL threshold that must not be exceeded by those methods. The relative differences with respect to the maximum value are indicated to illustrate how close the cumulative DELs are to one another for each DEL channel. For instance, variations in the blade root edge-wise moment are very small, with the minimum value being only 1.1% lower than the maximum value.

AEP optimization has a positive effect on the fatigue of most turbines, with a slight violation of the flap DEL constraint for Turbine T3. The integration of bin-wise constraints on the blade root DELs significantly reduces the cumulative values for all turbines, including those not represented in the figures. Although the DELs associated with the tower are not directly accounted for, the plots in Figure 4c, 4d and 4e clearly demonstrate that the overarching constraints for those components are satisfied within the AEP optimization framework, with a significant reduction of cumulative DELs for most turbines. When bin-wise constraints on the blade root DELs are incorporated into the optimization problem, the cumulative tower DELs do not change much, which makes sense given that the correspondent local DELs are not accounted for in the optimization process.

Note that Turbines T3, T6, T9, and T12 are the most downstream turbines with respect to the main wind direction, leading to a slightly higher cumulative DEL for the blade root flap-wise moment. Turbines T5, T8, T10, and T12 are located inside the farm, experiencing higher tower-bottom stress.

Similar conclusions are reached regarding the cumulative damage equivalent loads when using the Qian deflection model.

## 5 Conclusions

Structural load considerations are rarely incorporated into the design of wake steering control strategies, likely due to the challenge of computing accurate solutions in an efficient way. Taking into account cumulative fatigue damage in a rigorous manner inevitably leads to complex optimization problems that must be solved across a large number of wind bin conditions and various load cases. The computational efficiency of surrogate fatigue models is a crucial factor in this context. Indeed, the AEP optimization approach under cumulative DEL constraints we intended to apply to our case study – a small offshore wind farm, for which a fatigue surrogate model accurately predicting five representative DELs had been trained – turned out to be too demanding in terms of computation.

This motivated us to develop an approach for AEP optimization via wake steering under fatigue load constraints, defined and solved bin-by-bin, where the cumulative DELs are verified a posteriori. In spite of the simplified formulation, for the considered case study, this pragmatic approach significantly improved AEP compared to the baseline scenario without wake steering, while maintaining fatigue loads within reasonable limits and distributing them more evenly among the turbines. However, these findings may be site-specific, and the global constraints may not always be verified a posteriori. In such case, a different method would be needed.

A redesign of the computational workflow behind the coupling of the wind farm flow model and the surrogate fatigue model is in progress and has already resulted in a speed improvement of several orders of magnitude. We are confident that this will let us run optimization campaigns of reasonable duration to assess the performance of the AEP optimization under cumulative DEL constraints solved via the Uzawa algorithm.

One key challenge remains the handling of global fatigue constraints in a way that enables fast and practical bin-wise optimization. Addressing this issue would not only streamline offline computations of look-up tables but also pave the way for its implementation online (in real-time). This would offer several advantages even in an open-loop context, such as the ability to account for the operational status of the turbines when computing yaw setpoints.

Finally, for all the considered AEP optimization approaches, further work is envisioned to use more advanced analytical wake modeling configurations, namely by including secondary steering and momentum-conserving superposition.

## 6 Acknowledgments

This study has been carried out as part of the CAP project, supported under the France 2030 investment plan operated by ADEME, the French Agency for Ecological Transition.
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