Maximising the power production of large wind farms is key to the transition towards net zero. The overarching goal of this paper is to propose a computational method to maximise the power production of wind farms with two practical design strategies. First, we propose a gradient-free method to optimise the wind farm power production with high-fidelity surrogate models based on large-eddy simulations and a Bayesian framework. Second, we apply the proposed method to maximise wind farm power production by both micro-siting (layout optimisation) and wake steering (yaw angle optimisation). Third, we compare the optimisation results with the optimisation achieved with low-fidelity wake models. Finally, we propose a simple multi-fidelity strategy by combining the inexpensive wake models with the high-fidelity framework. The proposed gradient-free method can effectively maximise wind farm power production. Performance improvements relative to wake-model optimisation strategies can be attained, particularly in scenarios of increased flow complexity, such as in the wake steering problem, in which some of the assumptions in the simplified flow models become less accurate. The optimisation with high-fidelity methods takes into account nonlinear and unsteady fluid mechanical phenomena, which are leveraged by the proposed framework to increase the farm output. This paper opens up opportunities for wind farm optimisation with high-fidelity methods and without adjoint solvers.

The transition towards a net-zero world is intertwined with a continually growing demand for renewable energy sources. Wind power has emerged as the leading contributor to renewable electricity in numerous countries

One key issue is that wind farms that consist of many turbine rows are typically less efficient than less deep wind farms (i.e. downstream rows produce less power than upstream ones)

The long-established strategy for mitigating wake losses is to optimally position the wind turbines across the available land. This approach is most commonly referred to as

More recently, an approach called

In the vast majority of relevant studies published in the literature, the optimisation of the farm layout or the turbine yaw angles is carried out with low-fidelity flow solvers, commonly known as

On the other hand, higher-fidelity flow models have been employed in a limited number of layout optimisation studies

By “long term” we mean after many Lyapunov times.

time average of a chaotic turbulent flow is ill posedIn this work, we propose a data-driven framework for optimising the output of a wind farm based on a gradient-free Bayesian approach and high-fidelity large-eddy simulations of the wind farm flow. The structure of the paper is as follows. The proposed optimisation strategy is outlined in Sect.

The methodology is presented in two parts. First, we introduce the optimisation framework in Sect.

Bayesian optimisation (BO) is a gradient-free global optimisation strategy, which is particularly attractive for optimising complex functions (as in large codes for which the adjoint algorithm is cumbersome). Its effectiveness has been demonstrated in various fluid mechanical applications (e.g.

A Gaussian process (GP) model is a non-parametric probabilistic model that is defined by a prior mean function,

The acquisition function

Bayesian optimisation

Acquire a set of initial objective function observations

Initialise the variables holding the best solution,

Train the GP model given

Append

In the current work, Bayesian optimisation is implemented with the GPyOpt library

To predict the wind farm flow and power output, we employ the wind farm simulator WInc3D

In this work, the Smagorinsky sub-filter-scale model

In order to realistically simulate the interaction of the wind farm with the turbulent atmospheric flow, we perform precursor simulations of pressure-gradient-driven fully developed neutral atmospheric boundary layers. A free-slip condition is applied at the top boundary, and a no-slip condition with a wall model is used at the ground. The precursor atmospheric flow simulations are run until statistical convergence is reached, after which we begin storing two-dimensional planes of the flow field, normal to the streamwise direction, at every time step. These planes are subsequently used as inflow conditions in the wind farm simulations.

In layout optimisation, the objective is to maximise the power production of a wind farm with

In reality, accurate and robust estimation of the wind direction poses a significant challenge

In a preliminary proof-of-concept study

However, modern wind farms are composed of several more turbines. The layout optimisation problem therefore becomes high-dimensional, with the design space being

Here, we propose a way to reduce the dimensionality of the layout optimisation problem. It is based on the assumption that, in most cases, the turbines in a wind farm are identical. In that case, the cost function is invariant under permutations of the turbine labelling, and there are

We consider a wind farm of

Mean streamwise velocity

The wind farm is embedded in a computational domain of size

Figure

The optimisation starts by evaluating the farm power output of a large set of layouts sampled from a Latin hypercube, along with two user-designed layouts. These are a uniform

Efficiency of designs evaluated during the layout optimisation process. The dashed line shows the evolution of the best-performing design.

At the end of the optimisation, the best-performing design has

Mean flow fields for the best-performing design proposed by the large-eddy simulation Bayesian optimisation framework (LES-BO). The flow fields show the same layout exposed to different wind directions. In all cases, the wind is shown as blowing from left to right, with the wind farms rotated to match the wind direction shown on the inset wind roses. The reference non-rotated layout is that of the westerly wind case (top left).

In parallel with the training of the surrogate model, we can also learn what constitutes good modelling practices. By means of validation on subsequent test points, we observe that the rational quadratic (RQ) kernel outperforms both the widely used Matérn and the squared exponential (RBF) kernels. Figure

GP regression with different covariance functions on a standardised training dataset of 600 points.

In order to assess the quality of the best-performing design obtained with the proposed framework (hereafter referred to as LES-BO), we perform a series of optimisations with the FLOw Redirection and Induction in Steady State (FLORIS) software

The efficiencies of the 100 optimal designs outputted by FLORIS are then evaluated using WInc3D (the designs suggested by the low-fidelity model and evaluated with LES will be referred to as LF/LES). The LF/LES results, ordered from best to worst, are presented in Fig.

Efficiency of optimal designs outputted by FLORIS, evaluated with both FLORIS and LES (LF/LES). The dashed line shows the optimal design suggested by the LES-BO framework.

Figure

However, several layouts suggested by the wake-model-based optimisation outperform those suggested by LES-BO. This finding supports the practice of using wake models to design wind farms in the wind energy industry, as they demonstrate excellent performance at only a fraction of the cost (see also

Nevertheless, the efficiency predictions of FLORIS are lower compared to those of LES. As shown in Fig.

FLORIS-suggested best design.

Figure

FLORIS-suggested best design. LES predictions for the

Wake visualisation in the FLORIS-suggested best design case by means of transport of a passive scalar shed from the turbine rotors.

Section

In order to show the benefits of adopting the LES-BO approach, we propose a simple multi-fidelity strategy. We train a surrogate model with a dataset composed of only the first 300 samples from our original investigation (see Fig.

Within just 30 iterations, EI-LES-BO manages to improve upon the FLORIS-suggested best design by

Efficiency of best-performing LF/LES and EI-LES-BO layouts for each direction in the wind rose.

In wake steering, the objective is to maximise the power output of a wind farm by adjusting the angle

The problem we consider draws on the Horns Rev I wind farm, which is located in the North Sea and consists of eighty turbines with diameter

We assume that the turbines operate with a constant local thrust coefficient,

The domain of size

The initial training dataset comprises 50 designs. These include yaw angle combinations sampled with the Latin hypercube method, the non-yawed condition shown in Fig.

Optimal yaw angles suggested by LES-BO (solid line) and FLORIS (dashed line). Individual turbine efficiencies are shown as bars with solid fill (LES-BO) and bars with hatched pattern (LF/LES).

Qualitatively, both frameworks suggest a decreasing trend in yaw angles as we move downstream in the farm, similar to results reported in

As actuator disk theory tends to overestimate the power output of wind turbines at large yaw angles

Figure

Mean streamwise flow at the turbine hub height.

This work introduces an optimisation framework aimed at enhancing the efficiency of power production in wind farms. The proposed method follows a Bayesian approach and utilises surrogate models based on high-fidelity large-eddy simulations of wind farm flows. As part of an extensive computational campaign involving around 5000 large-eddy simulations, the framework was effectively used to mitigate losses caused by wake effects through two distinct strategies: layout optimisation (also known as micro-siting) and wake steering through yaw angle optimisation. The achieved optimisation outcomes were also compared with results obtained from low-fidelity wake-model-based optimisation. Finally, a simple strategy was proposed to combine both large-eddy simulations and wake models in a multi-fidelity approach.

In the layout optimisation problem, the best layouts found had

Simulations were performed using

Low-order farm modelling is conducted using version 3.4 of FLORIS

In terms of optimisation, the algorithm of choice is the FLORIS-default gradient-based sequential least squares programming (SLSQP) method

The data that support this study are available upon reasonable request.

NB: conceptualisation, methodology, investigation, and writing (original draft). FG: investigation and writing (review and editing). AW: writing (review and editing). SL: writing (review and editing) and funding acquisition. LM: conceptualisation, writing (review and editing), and funding acquisition.

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 would like to thank Christian Jané-Ippel for his work in the development of the flow solver. We also acknowledge use of the Cambridge Service for Data Driven Discovery (CSD3) system. Filippo Gori would like to acknowledge the Department of Aeronautics, Imperial College London, for supporting this work with a doctoral studentship.

Nikolaos Bempedelis has been supported by EPSRC grant no. EP/W026686/1. Sylvain Laizet has been supported by EPSRC grant nos. EP/W026686/1 and EP/Y005619/1. Luca Magri has been supported by grant nos. EP/W026686/1 and EP/Y005619/1, as well as ERC Starting Grant PhyCo (grant no. 949388) and the EU-PNRR Young Researcher TWIN ERC-PI_0000005. This research has also been supported by the UK Turbulence Consortium (grant nos. EP/R029326/1 and EP/X035484/1), which provided access to ARCHER2.

This paper was edited by Cristina Archer and reviewed by two anonymous referees.