The optimisation algorithms used are the sequential least-squares programming (SLSQP) method developed by and the trust region Bayesian optimisation (TuRBO) approach by . The gradient-based optimiser SLSQP is the standard algorithm used in the FLORIS framework. In contrast, TuRBO uses a probabilistic approach for the stochastic global optimisation of large-scale high-dimensional problems, and we now give a brief overview of this alternative algorithm. The first step is to take an initial sample of design space via Latin hypercube sampling (LHS). This generates near-random samples of the objective function from a multi-dimensional distribution. Two subsequent stages then follow. First, the available samples are fitted with a Gaussian process, a probabilistic surrogate model, to give a posterior distribution which both approximates the objective function and determines the approximation's uncertainty. Second, an acquisition function, Thompson sampling (TS) in the case of TuRBO, is minimised to determine the next sample point at which to evaluate the objective function. The role of the acquisition function is to find a balance between minimising the objective function and reducing the uncertainty in the fitted surrogate model. The two optimisation steps are iterated until pre-defined stopping criteria are met.

The global statistical nature of Bayesian optimisation has been found to outperform local, gradient-based algorithms when applied to multi-modal or discontinuous objective functions . At each iteration, TuRBO's global approach can potentially sample from anywhere in the design space, while gradient-based SLSQP typically takes a small step computed using local gradients. A known drawback of Bayesian optimisation is the higher computational complexity involved in high-dimensional problems which require a large number of objective function evaluations. To address this issue, TuRBO uses a trust region approach in which multiple Gaussian process surrogate models are fitted in evolving promising areas of the design space, referred to as trust regions. The allocation of new samples to evaluate across the models is achieved via an implicit bandit approach. Further explanation of Bayesian optimisation is given via an illustrative example in Appendix .

In this study, the number of initial evaluations generated by the LHS method is set to double the number of optimisation variables N (here the number of turbines). Moreover, the number of trust regions is set to one. To explain this choice, note that if multiple trust regions are used, TuRBO will perform a multiple-starting-point search and fit a Gaussian process in each trust region, thus conducting multiple sub-searches in parallel. A single TuRBO optimisation run would then be comparable, in complexity, to multiple SLSQP ones. To ensure a fair comparison between the two methods, a single TuRBO trust region is used, initialised with the same conditions as those for each SLSQP optimisation. The parameters used for SLSQP and TuRBO optimisers are shown in Table .

As is subsequently discussed for the 5×5 case (see e.g. Fig. ), the considered objective functions G(γ) typically exhibit multi-modality, flat regions, and discontinuities. Multi-modality refers to the presence of multiple and distinct maxima. These features present a fundamental challenge when seeking to robustly identify global maxima. For the TuRBO algorithm, preliminary investigations indicated only a low parametric influence on optimisation results, with the recommended parameter values in identified as the most sensible choice. These findings are consistent with the stochastic nature of this global optimiser. In the case of SLSQP, a sensible choice of optimisation parameters is required to address this complexity and to essentially regularise the optimisation problem. The parameter to consider is the precision goal for the value of the objective function in the stopping criteria, called ftol. It corresponds to the achievement of equal objective function values in the evaluations performed for the numerical approximation of the gradient. Large values of ftol may cause the optimiser to terminate in flat regions, significantly increasing the variability of the objective function and the optimal decision variables to the choice of optimisation initialisation. However, a small ftol typically increases the number of objective function evaluations required for convergence.

In this study, ftol =1×10-16 for the 5×5 case is identified as a suitable trade-off between these two behaviours, based on preliminary investigations. For the Horns Rev case, the value ftol =1×10-12 is used due to the higher dimensionality of the optimisation problem, i.e. its larger computational requirements.

The second SLSQP parameter to consider is eps ∈[0,1], corresponding to the step size (in this case equal to eps ×[25∘-(-25∘)]) used for the finite-difference gradient approximation. Large eps values lead to inaccurate gradient approximations. However, as shown in Fig. , the objective functions considered in this study may have very high gradients, discontinuities, or discontinuous derivatives, especially when using the Jensen and multizone models. This “rough” objective function behaviour, in conjunction with small eps values, may cause the SLSQP optimiser to identify optimal decision variables in local maxima. A 2.5∘ step size (eps =0.05) is identified to be a suitable value since this relatively large value of eps has the effect of artificially smoothing the rough objective function while limiting the loss of information about the local gradient.

We consider the problem of farm power maximisation for a 2×1 wind farm layout with NREL 5 MW turbines. The only optimisation variable is the yaw angle γ1∈[-25∘,25∘] of the upstream turbine. The optimal yaw angle of the furthest downstream turbine is necessarily zero since no other turbine would benefit from a deflection in its wake, and a non-zero yaw angle reduces the power output of this turbine. Optimisation is therefore performed via a simple parameter sweep of γ1. Different streamwise turbine spacings and ambient turbulence intensity are considered to provide insights into the sensitivity of both the optimised farm power and the optimal yaw angle, denoted γ1∗, to operational conditions. Observations made for this minimal study are used to explain and analyse the sensitivity and performance of wake steering optimisation for more complex farm layouts in Sect.  and .

Figure a, b show the extracted power PWF(γ1) of the entire 2×1 farm, PT1(γ1) of the upstream turbine, and PT2(γ1) of the downstream turbine. Power production PT1(γ1) of the upstream machine is the same for all models. A clear sensitivity to model choice is observed in Fig. a with the Jensen model's optimal yaw angle at γ1∗=0∘, while, in contrast, all other models have optimal yaw angles close to constraint boundaries with γ1∗>20∘. Model sensitivity is caused by the flatness of the Jensen model's downstream power curve PT2(γ1) apparent in Fig. b; that is, ddγ1PT2(γ1) is significantly smaller for Jensen than for other models. Flatness arises due to the uniform, or “top-hat”, profile of the Jensen distribution (see Fig. ), which results in a lack of sensitivity of the streamwise velocity deficit to moderate yaw perturbations. In contrast, it can also be seen in Fig.  that all other models predict a streamwise velocity deficit which is both larger at the centreline and decreases more rapidly in the transverse direction. For the multizone model, this decrease in the streamwise velocity deficit is discontinuous due to the model's definition of splitting the wake into various zones, with sharp interfaces between each zone. Overall, the Gaussian, GCH, and multizone models give downstream power curves PT2(γ1), which are more sensitive to changes in γ1 than the Jensen model.

Model sensitivity is influenced by both turbine spacing and turbulence intensity I0. Figure c, d show that the optimal yaw angle for the Gaussian, GCH, and multizone models decreases with increased turbine spacing. Although not shown, γ1∗≈0∘ for all models for spacing greater than 32D. Similar behaviour can be observed in Fig. e, f as I0 increases, with γ1∗≈0∘ whenever I0≥15% for all models which incorporate this parameter (the multizone model does not depend on I0). From an optimisation perspective, increased spacing and turbulence intensity have the analogous effect of increasing wake recovery and effective wake diameter at the downstream turbine. This reduces the sensitivity of PT2(γ1) to γ1 and renders the wake steering optimisation problem trivial. Despite this observation, it is clear from Fig.  that for any practically relevant turbine spacing of up to 10D, and for turbulence intensities of I0≤10% (at a typical 7D spacing), it is still the case that substantial model sensitivity is present in the 2×1 farm power maximisation problem.

With a view towards understanding the sensitivity of wake steering optimisation for more complex farms, it is important to highlight the multi-modality of the farm power curves PWF(γ1) for the Gaussian, GCH, and multizone models. Multi-modality can be seen for all considered spacings in Fig. a, c, d and for I0≤10% in Fig. a, e. For the Gaussian model, this arises simply due to the symmetry of PT2(γ1) for the aligned inflow conditions considered here. Multizone and GCH models are weakly asymmetric due to the incorporation of wake rotation effects, with PT2(γ1)>PT2(-γ1), for γ1>0, in the case considered here of a clockwise spinning rotor. However, the farm power PWF(γ1) curves for both models are still clearly multi-modal.

In summary, this minimal 2×1 case demonstrates that wake steering optimisation can exhibit model sensitivity due to the flatness of the Jensen model, while the Gaussian, GCH, and multizone models exhibit multi-modal objective functions. Both observations are of importance to understanding the behaviour and sensitivity of wake steering optimisation for the following cases of more complex farm layouts.

We consider wake steering optimisation for farm power maximisation on a 5×5 wind farm layout with NREL 5 MW turbines. Although the Jensen model was seen to perform poorly in Sect. , we still report its performance for this example due to the Jensen model's ongoing use in the wind energy community for wake steering applications . The objective of this medium-complexity case is to understand the sensitivity of wake steering performance to both the choice of wake model and the choice and particular implementation of an optimisation algorithm. Due to the high-dimensional and multi-modal nature of the underlying optimisation problem in this case, statistical analysis is required to quantify optimisation sensitivity (see Sect. ). However, we first study several indicative cases, which will make the statistical results transparent.

In this section, the results for farm power maximisation via wake steering optimisation are presented for the well known Horns Rev wind farm. To the best of our knowledge, only a small number of studies, see for instance , , and , have performed wake steering optimisation on this farm layout. In , sign-coherent optimal yaw settings are obtained using gradient-based optimisation. However, in this study, explicit knowledge of the optimal solution was used to define initial yaw angles to ensure optimiser convergence. In contrast, used an evolutionary optimisation algorithm and found optimal yaw angles whose signs exhibit negative values and high column inconsistency, reflecting the observations made for the 5×5 layout considered in Sect. .

The first objective of this high-complexity optimisation case is to confirm whether wake steering sensitivities uncovered in the low- and medium-complexity cases transfer to a realistic setup. The second objective is to add optimisation constraints to try to mitigate initial yaw angle sensitivity and to achieve coherent and interpretable optimal yaw settings that could eventually be implementable in the field. In large-scale optimisation problems with many design variables and required evaluations, the TuRBO algorithm becomes computationally demanding for real-time control applications due to the tuning of the Gaussian processes. For the Horn Rev case, TuRBO's computational complexity is about 2 orders of magnitude higher than SLSQP's. For this reason, and the observation from Sect.  that SLSQP can exhibit rapid convergence if the initialisation sensitivity is appropriately mitigated, we only employ the SLSQP optimisation algorithm in this section, using the parameters in Table . Note that the Jensen model is not included in this section due to the poor wake steering performance observed in the previous sections.

For sensitivity mitigation, we consider two additional sets of optimisation constraints. The constraint “C1” refers to the case in which the yaw angles of every turbine are constrained to be positive. The aim is to improve optimiser performance by avoiding the maxima corresponding to within-column yaw angles with alternating signs. Under aligned conditions, constraining the yaw angles to be either positive or negative is equivalent for symmetric models like the Jensen and the Gaussian. However, when considering wake models that incorporate wake rotation effects, such as the multizone and the GCH models, positive yaw angles lead to higher turbine power production. Implementation of the C1 constraint involves modifying the variable bounds from [-25∘,+25∘] to [0∘,+25∘]. To maintain a step size of 2.5∘ for gradient approximations, the SLSQP parameter eps is adjusted to 0.1, as outlined in Table . A second set of constraints, “C2”, corresponds to the case where the yaw angles of each turbine (excluding turbines in the most upstream row) are forced to be equal to or lower than the yaw angles of the upstream turbines in the same column and is implemented in SLSQP using linear inequality constraints. It should be noted that these additional sets of constraints can be easily adapted to other operating and atmospheric conditions; e.g. for a different wind direction, a simple permutation of the turbines' labelling is required to specify columns which correspond to the new downstream direction.

The investigated cases are limited to a fully aligned layout, as it is the predominant condition and the one holding the largest potential for the implementation of wake steering strategies. The analysis performed is statistical, where each of the 50 optimisation cases per wake model is initialised with an independent set of random initial conditions sampled from a uniform distribution on [-25∘,+25∘]. For clarity throughout the section, optimisation performed using only the original yaw angle bounds of [-25∘,+25∘], without applying either the C1 or the C2 constraint, is referred to as “nominal”. Optimisation cases with additional constraints are indicated as “constrained”. Note that for the additional constrained Horns Rev cases, a 2.5∘ step size corresponds to eps being equal to 0.1 due to the modified variable bounds [0∘,+25∘], as specified in Table .

Optimal yaw configurations for a single SLSQP optimisation with a unique random set of initial conditions in the nominal, C1, and C1 + C2 optimisation cases are presented in Fig. . In line with the results of , the first row of subfigures shows that, in the nominal case, the optimal yaw settings obtained for the three wake models are not practical, with frequent switches between large positive and negative yaw angles within any given row or column of the wind array. The only common trend is that the final downstream row of turbines has optimal yaw angles close to zero. The key message is that, without a sensitivity mitigation strategy, the proposed yaw angle settings cannot be implemented in real life due to obvious operational constraints.

The introduction of additional constraints clearly improves the identification of the optimal variables. For all models, optimal yaw angles represent a more interpretable and practical solution. Overall, a similar performance is achieved with the two different constrained approaches, C1 and C1 + C2. This successfully proves how the use of simple constraints, motivated by an understanding of the optimisation problem, can lead to considerable improvements in optimisation performance. The C1 constraint forces the optimiser to avoid the maxima corresponding to within-column yaw angles with negative and alternating signs, which is of particular importance when such maxima are sub-optimal for models with power asymmetry due to yaw (multizone and GCH). The C2 constraint, limited to C1 + C2 cases, effectively decreases initialisation sensitivity by imposing column-equal or column-monotonically decreasing optimal yaw angle solutions. For the GCH model, which captures secondary steering effects, introducing constraints leads to a smoother decrease in optimal decision variables from row to row and to the identification of an optimal decrease rate.

Table  provides the farm power improvements obtained from the single SLSQP optimisation outlined in Fig.  for the nominal, C1, and C1 + C2 optimisation cases. Across all three wake models, it is evident that the introduction of additional constraints enhances the performance of the optimiser. Compared with the nominal case, the constrained cases yield a 7%, 1%, and 4% increase in the objective function for the multizone, Gaussian, and GCH models, respectively. When comparing the C1 and C1 + C2 optimisation cases, minimal or negligible differences in farm power improvement are observed. This indicates that the introduction of the C1 constraint alone already leads to column-equal or column-monotonically decreasing optimal yaw angles in the cases presented (as depicted in Fig. ). However, to consistently ensure this behaviour across various optimisation conditions, the implementation of the C2 constraint is necessary. While examining the results of a single SLSQP optimisation is informative, a statistical analysis is required to evaluate the benefits of implementing additional constraints to mitigate initialisation sensitivity.

Figure  presents the statistical results for farm power improvement in the nominal, C1, and C1 + C2 optimisation cases. For all wake models, a significant reduction in initialisation sensitivity can clearly be observed when constraints are imposed, in terms of both standard deviation and minimum and maximum values. For the GCH model, dependency on initial conditions in the nominal case leads to sub-optimal solutions with potential farm power losses of up to 10% (the difference between the maximum and minimum power increase values). With the introduction of constraints, these losses are below 0.1%. Further, a higher mean farm power improvement is experienced in the constrained cases for all models. In the multizone case, avoiding sub-optimal solutions due to constraints leads to a 12% increase in mean farm power improvement. Finally, consistent with the conclusions for the medium-complexity 5×5 case, an overall strong dependency of wake steering strategies on wake model choice is observed, exhibiting significant discrepancies in farm power improvement. To further explore model sensitivity, an inter-model cross-validation is performed with the multizone and Gaussian models' performances evaluated with the optimal decision variables computed using the GCH model. As expected, both models exhibit lower normalised farm power increases when using the optimal GCH yaw angles (24% and 6% lower for the multizone and Gaussian models, respectively). These results suggest that the row-monotonic decrease in optimal yaw angles, attributed to the GCH model capturing secondary steering effects , represents a better solution to the wake optimisation problem than is possible when using either the Gaussian or the multizone model. This observation is in agreement with several recent studies, including , , , and .

Figure  presents the statistical results for the optimal yaw settings in the nominal, C1, and C1 + C2 optimisation cases. For each row in the farm layout, a probability density function of the optimal yaw angles is shown in this figure. It is defined by averaging the probability density functions for the single turbines in the row. The aim is to express the distributions of optimal yaw settings per farm row and see how they compare for different optimisation cases. When examining Fig. d, e, f and g, h, i for the C1 and C1 + C2 constrained cases, respectively, it becomes evident that the probability density functions exhibit higher magnitudes compared with the nominal cases shown in Fig. a, b, c. With the exception of the last turbine row, the probability density function values, which are approximately 0.25 in the nominal cases, are at least twice higher when the additional constraints are introduced in the optimisation. This behaviour is observed across all wake models and indicates a more consistent distribution of optimal yaw angles for different initialisations, thereby highlighting improved optimisation robustness and reduced dependence on initial conditions for the constrained cases. For the multizone and Gaussian models, optimal yaw settings in the nominal case tend to converge at the optimisation bounds, as shown in Fig.  and confirmed by bi-modal probability density functions in Fig. a and b. Caused by initialisation sensitivity and lack of constraints, this undesirable behaviour of bi-modality is easily removed in the C1 and C1 + C2 optimisation cases. For the GCH model, bi-modal probability density functions can also be identified in the nominal optimisation results. However, the peaks of the modes exhibit a more complex pattern depending on the farm row (Fig. c). As for the previous models, bi-modality is removed by the introduction of constraints. Finally, consistently with the discussion for the 5×5 farm layout in Sect. , significant overall discrepancies between wake models can be identified in terms of optimal yaw angles.

The presented results indicate that the simple strategy proposed for SLSQP sensitivity mitigation is effective for this realistic and complex wake steering optimisation problem. The enforcement of constraints successfully decreases the impact of initial conditions in terms of both the objective function and the optimal decision variables while achieving generally higher farm power improvements and more consistent optimal yaw settings. We note, finally, that the constrained gradient-based approach can simply be adapted to misaligned wind directions. This can be achieved by permuting the turbine labelling to create columns aligned with the incoming wind direction. Either constraint C1 or constraint C1 + C2 can then be applied analogously in this situation.