Modern large-scale wind farms consist of multiple turbines clustered together, usually in well-structured formations. Clustering has a number of drawbacks during a wind farm's operation, as some of the downstream turbines will inevitably operate in the wake of those upstream, with a significant reduction in power output and an increase in fatigue loads. Wake steering, a control strategy in which upstream wind turbines are misaligned with the wind to redirect their wakes away from downstream turbines, is a promising strategy to mitigate power losses. The purpose of this work is to investigate the sensitivity of open-loop wake steering optimisation in which an internal predictive wake model is used to determine the farm power output as a function of the turbine yaw angles. Three different layouts are investigated with increasing levels of complexity. A simple

Wind energy now plays a central role in meeting world energy requirements, driven by the urgent need to mitigate climate change and a significant recent reduction in its levelised cost. New and ambitious international renewables targets, such as the European Commission’s proposed installation of up to

Wake steering, in which upstream turbines are yawed to deflect their wakes away from downstream machines, is a leading control technique used to mitigate the effects of wake–turbine interactions. Farm power increases ranging from

With a view to enabling robust and predictable wake steering optimisation for large wind farms, this paper investigates the

By optimisation sensitivity we refer to the dependency of wake steering optimisation on (i) the choice of the underlying predictive model; (ii) the choice of the optimisation algorithm and its particular parametric implementation; and (iii) the given operating condition, such as farm layout or atmospheric conditions. Furthermore, we measure optimisation sensitivity in terms of both (i) the optimised farm power and (ii) the optimal decision variables themselves, i.e. the optimal yaw angles obtained from wake steering optimisation. Arguably, the latter is the most important. Of particular interest is to identify situations of high sensitivity: that is, when either the predicted maximised power output or the optimal yaw angles are seen to vary substantially with small changes to the wake model, algorithm, or operating condition. When high sensitivity is identified, we also study potential algorithmic changes to reduce it.

Understanding, quantifying, and reducing cases of high sensitivity are of fundamental importance for robust wake steering. To implement open-loop wake steering, tuning or identification of wake model parameters from field data or high-fidelity simulations is required. As with any tuned model, there can be a mismatch between the predicted and true farm behaviour. A first problem arises when the predicted optimised farm power output is highly sensitive to the wake model parameters or algorithmic implementation. In these cases, small modelling mismatches may cause large deviations between the expected and true farm performance, possibly nullifying any predicted power increase when applied in the field. A second problem occurs in the case of high optimal yaw angle sensitivity. Here, a small parametric change (e.g. to model re-tuning or to atmospheric conditions) may require large yaw angle changes, and it is practicably undesirable to make large control input changes in response to only small operational perturbations. As is discussed in Sect.

In this paper, we do not consider active wake control or closed-loop wake control in which online sensor information is used to dynamically adapt wake steering strategies (see

Due to the large number of underlying modelling, algorithmic, and parametric choices, in this paper we propose a hierarchy of test cases to facilitate the understanding of the different sensitivities present in wake steering optimisation. At the simplest level, a minimal optimisation problem consisting of a

The remainder of the paper is organised as follows. Section

Wind farm modelling in the present work is conducted with version 2.4 (after the resolution of issue number 684) of the open-source FLORIS framework

Although different wake models will logically give different power predictions, in a wake steering optimisation context, it may still be the case that different models give rise to similar optimal yaw angles. Consequently, the question of the modelling fidelity required to enable robust wake steering optimisation may be different to the question of which model is best to capture a given physical wake characteristic. In this regard, we emphasise that the purpose of this study is not to identify which model is best to study a particular wind farm. Rather, we seek to identify situations in which wake steering optimisation is highly sensitive either to the choice of underlying model or to the interaction between the model choice and the particular parametric implementation of commonly used optimisers (see Sect.

We now give a brief overview of FLORIS's wind farm modelling structure. First, an initial condition is defined by specifying atmospheric inflow, wind farm layout, turbine geometry, and operational conditions. Next, the chosen wake model calculates each turbine's steady streamwise velocity deficit. The computation is sequential to allow additional considerations on added turbulence. For yawed turbine cases, a deflection model is employed to determine and apply a cross-stream shift in the streamwise velocity deficit field. Finally, streamwise velocity deficits for each turbine are combined with a superposition model and are applied to the initial flow field. In the current study, all wake models use the sum of squares freestream superposition (SOSFS) model developed by

Streamwise velocity at hub height of a

Farm power calculations are conducted as follows. Turbine operational profiles consist of lookup tables for power and thrust coefficients (

The power

The total farm power is calculated as

Figure

Open-loop wake steering optimisation for power maximisation is conducted on three different farm layouts. The first two are a

For all the optimisation test cases, the incoming flow is fully aligned with the farm columns with a wind speed of

The optimisation problem's objective function in all cases is chosen to be the normalised farm power production (Eq.

The optimisation algorithms used are the sequential least-squares programming (SLSQP) method developed by

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

In this study, the number of initial evaluations generated by the LHS method is set to double the number of optimisation variables

As is subsequently discussed for the

In this study, ftol

Wake and deflection model parameters.

SLSQP and TuRBO parameters.

Power function of a

The second SLSQP parameter to consider is eps

Model comparison for a

We consider the problem of farm power maximisation for a

Figure

Model sensitivity is influenced by both turbine spacing and turbulence intensity

Initialisation sensitivity of a

With a view towards understanding the sensitivity of wake steering optimisation for more complex farms, it is important to highlight the

In summary, this minimal

We consider wake steering optimisation for farm power maximisation on a

In this section, we seek to understand the interaction between the wake model selection, the algorithm choice, and the initial yaw angles used by each optimiser. Insights from the indicative examples given in this section will motivate and clarify the statistical analysis of Sect.

Figure

Arguably more important, from an implementation perspective, is the significant inter-model variation in optimal yaw angles. For the Jensen model, Fig.

To further explore the sensitivity to initial yaw angles between models, Fig.

For the multizone and GCH models, the sensitivity of both the normalised farm power (of up to

In terms of wake model robustness, two key observations deserve attention. Firstly, the Gaussian model demonstrates significantly greater robustness than the multizone model despite both models yielding similar optimal yaw settings (Fig.

Test Case 1 optimisation results of a

To investigate initialisation sensitivity in more detail and to better understand how initial yaw angles affect the optimal yaw angles, we study one additional SLSQP optimisation run – denoted Test Case 1 – for each of the considered models, again initialised by sampling 25 turbine yaw angles from a uniform distribution on

Farm power of a

For the GCH model in Fig.

Figure

These results motivate a statistical comparison of optimisation sensitivity between gradient-based and global optimisation approaches, which is given in Sect.

This section investigates the geometry of the farm power objective function when using different wake models. The aim is to understand the higher initialisation sensitivity observed in gradient-based optimisation and to provide the necessary insights in order to propose a solution to this problem, which is presented in Sect.

Figure

The remaining three models have objective functions with very high gradients, discontinuities, or discontinuous derivatives. This is particularly prominent for the Jensen and multizone models in Fig.

Comparison of the statistics of the objective function

The intrinsically multi-modal, and often rough, nature of farm power objective functions for the

In this section, we compare the performance of the SLSQP and TuRBO optimisers described in Sect.

Figure

Statistics (mean, standard deviation, and minimum and maximum values) of the number of evaluations required for each optimiser are shown in the lower row of Fig.

Comparison of optimisation results of a

Figure

The streamwise velocity flow fields at hub height for the three considered cases are also shown in Fig.

Although initialisation sensitivity is reduced using TuRBO's global optimisation strategy, within-column sign inconsistency of the optimal yaw angles is still possible for all wake models using both SLSQP and TuRBO algorithms, as shown in Fig.

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

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.

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 [

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 [

Optimal yaw configurations for a single SLSQP optimisation with a unique random set of initial conditions in the nominal, C1, and C1

Comparison of the Horns Rev wind farm optimal yaw setting for a single optimisation between the nominal, C1, and C1

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

Table

Comparison of the farm power improvements for a single Horns Rev wind farm optimisation between nominal, C1, and C1

Comparison of the Horns Rev wind farm objective function statistics between the nominal, C1, and C1

Comparison of the Horns Rev wind farm optimal yaw setting statistics between nominal, C1, and C1

Figure

Figure

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

A detailed investigation of the sensitivity of wake steering optimisation for increased power output of wind farms was carried out in this study. Sensitivity to the choice of analytical wake models, the optimisation algorithm, and different operating conditions
was assessed in terms of variability in both optimised farm power and optimal yaw angles. The study was performed with four different analytical wake models, with a gradient-based and a global optimisation algorithm. Three wind farm layouts were investigated: two generic layouts with

For the

In the medium-complexity

Finally, a higher-complexity optimisation was performed with the SLSQP algorithm for the farm layout corresponding to the Horns Rev wind farm (

Future work will look at the sensitivity of wind farm wake steering optimisation for a range of wind directions and inflow speeds and will investigate the impact of atmospheric conditions for more complex optimisation problems, in addition to identifying appropriate optimisation constraints to enable sensitivity mitigation in each setting. Future research will also focus on determining wake steering sensitivities in an online control framework, including state estimation and parameter-tuning algorithms.

The most relevant parameter in analytical wake models is the streamwise velocity

Considering the unsteadiness of the wake phenomenon and its evolution in three-dimensional space, in principle, all terms in Eq. (

The Jensen model

The streamwise velocity deficit induced by a turbine with diameter

The Jensen model is based on a steady description of the wake where turbines are modelled as actuator discs with uniform loading, and no notion of added turbulence intensity due to upstream turbines' operation is included. Moreover, it does not conserve momentum, it is limited to far-wake predictions, and no calculation involving cross-stream and vertical velocity components is included. Despite these limitations, the Jensen model is simple and inexpensive. It can be used for control and optimisation studies, and it can provide valuable insights into power production for large wind farm layouts under normal operating conditions.

The multizone model, developed by

The velocity deficit in the three wake zones is assumed to decay quadratically with the downstream distance rather than being directly related to the wake expansion as in the Jensen model. Due to the presence of different wake zones, a smoother transition from the wake centre to freestream velocity is achieved (see Fig.

The local wake decay coefficient

The multizone wake model is a computationally inexpensive model, suitable for control and optimisation studies, including yaw applications. However, it only describes equilibrium conditions, and it is limited to far-wake predictions. In addition, it does not exhibit any sensitivity to inflow turbulence intensity, and it does not include considerations for added turbulence intensity by upstream turbines. Moreover, it involves many tuned empirical parameters, decreasing the model confidence for a wide range of operating conditions. Finally, it does not explicitly conserve momentum.

The Gaussian wake model was originally developed by

All quantities with the subscript “0” represent wake properties at the far-wake onset (the end of the near wake) and depend on the turbine thrust coefficient

For additional details on the derivations of far-wake onset quantities, refer to

In the Gaussian model formulation,

Although the wake model complexity is increased, the Gaussian wake model is still suitable for control and optimisation applications. Additional considerations include ambient and added turbulence intensity, and the wake model explicitly conserves momentum. The main limitations are the formulation based on a free shear approximation of the Navier–Stokes equations and the inaccurate near-wake predictions. Moreover, it does not compute crosswise and streamwise velocity components, both critical for modelling wake steering effects, and it relies on multiple empirical coefficients, decreasing the range of conditions for suitable predictions.

The Gauss–curl Hybrid (GCH) model

Wake rotation is included by modelling a Lamb–Oseen vortex, where its circulation strength is dependent on the turbine axial induction factor and tip-speed ratio. The counter-rotating vortex system due to turbine yaw misalignment is modelled as two single vortices released at the top and the bottom of the rotor, with the vortex strength dependent on the turbine thrust coefficient and yaw angle. For each vortex described, cross-stream

Vortex dissipation with downstream distance is computed as

As presented in

Cross-stream and vertical velocity fluctuations are defined as the sum of the average cross-stream

Considering yawed upstream turbines, secondary steering refers to the phenomenon of downstream turbines without yaw misalignment experiencing wake deflection and deformation due to the interaction between upstream-generated vortices and downstream turbine wake rotation (see Fig.

Defining

Although crosswise and vertical velocity components are not explicitly modelled, the GCH wake model is capable of capturing, through analytical approximations, the increase in wake recovery due to yaw misalignment and secondary steering effects. The latter is particularly important when considering large wind farms and evaluating wake steering control strategies. However, GCH-added complexity results in 3.5

When a turbine is yawed, the unbalance of thrust on the rotor leads to a cross-stream momentum gain and a deflection of the wake in the direction of yaw. Deflection models aim at quantifying this shift on the streamwise distance

The Jiménez deflection model is based on the empirical formulation proposed by

The Bastankhah deflection model is based on the budget analysis of the continuity and Reynolds-averaged Navier–Stokes equations conducted by

By assuming the validity of Eq. (

Expressions for the symbols in the above equation are provided in Appendix

Wind farm modelling in this study is conducted with version 2.4 of the open-source
FLORIS framework, available at

The data supporting the findings of this study are available from the corresponding author upon reasonable request.

FG: optimisation computations, methodology, validation, data analysis, writing. AW: methodology, data analysis, funding acquisition, supervision, writing. SL: methodology, data analysis, funding acquisition, supervision, writing.

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 in published maps and institutional affiliations.

This research has been supported by a PhD scholarship internally funded by the Department of Aeronautics, Imperial College London, United Kingdom.

This paper was edited by Irene Eguinoa and reviewed by two anonymous referees.