Wake steering is an emerging wind power plant control strategy where upstream turbines are intentionally yawed out of perpendicular alignment with the incoming wind, thereby “steering” wakes away from downstream turbines. However, trade-offs between the gains in power production and fatigue loads induced by this control strategy are the subject of continuing investigation. In this study, we present a multifidelity multiobjective optimization approach for exploring the Pareto front of trade-offs between power and loading during wake steering. A large eddy simulation is used as the high-fidelity model, where an actuator line representation is used to model wind turbine blades and a rainflow-counting algorithm is used to compute damage equivalent loads. A coarser simulation with a simpler loads model is employed as a supplementary low-fidelity model. Multifidelity Bayesian optimization is performed to iteratively learn both a surrogate of the low-fidelity model and an additive discrepancy function, which maps the low-fidelity model to the high-fidelity model. Each optimization uses the expected hypervolume improvement acquisition function, weighted by the total cost of a proposed model evaluation in the multifidelity case. The multifidelity approach is able to capture the logit function shape of the Pareto frontier at a computational cost only 30 % that of the single-fidelity approach. Additionally, we provide physical insights into the vortical structures in the wake that contribute to the Pareto front shape.

As wind energy systems have matured, plant-level control has emerged as a new paradigm, where groups of turbines are controlled in coordination to maximize collective power production.
This is in contrast to more traditional control strategies, where individual turbines are controlled to maximize their own power production.
A potentially promising form of such plant-level control is “wake steering”, where upstream wind turbine yaw positions are intentionally misaligned from the incoming wind, “steering” the wake away from downstream turbines. A counter-rotating pair of vortices is generated by the lateral thrust of the wind turbine rotor, which is determined by the yaw offset direction

It is speculated that wake steering may produce more power while inducing less total fatigue on all turbines when compared to the baseline strategy of aligning each turbine with the incoming wind

Despite its promise, plant-level control via wake steering involves complex physics and is challenging to model. Engineering wake models have dubious accuracy when predicting fatigue loading, which higher-fidelity models predict more accurately

Multifidelity optimization exploits the correlation between low- and high-fidelity models to reduce the overall computational cost of optimization.
For instance,

The novelty of the present study is the application of this multifidelity technique to wind energy systems, resulting in new insights into wake-steering flow physics.
The present approach uses the low-fidelity model to first explore the full parameter space and then iteratively builds the low- and high-fidelity model surrogates to gain the most improvement in the Pareto front per model evaluation costs.
While this framework is similar to that presented by

A Bayesian framework for multiobjective multifidelity optimization is presented. Throughout this section, we assume that minimization of functions is the objective of the optimization procedure (as opposed to maximization).

This study employs GP models to approximate power and loading dynamics.
A GP is a collection of random variables, any finite number of which have a joint Gaussian distribution

We perform Bayesian inference on functions by conditioning the GP on a set of observed input–output pairs

The kernel covariance function encodes prior knowledge about structural properties of the underlying signal, such as smoothness, periodicity, and stationarity. In this study, we employ an anisotropic radial basis function kernel

In Bayesian optimization, an acquisition function is defined to maximize a metric representing both exploration and exploitation

There is a range of potential outcomes from sampling a new point, and the GP framework conveniently estimates this uncertainty. These uncertainties are used to compute the expected value of the improvement function. It is important that the improvement function contains the maximum function; otherwise, there would be no exploration of regions of larger uncertainty.
Other acquisition functions available include the knowledge gradient

The expected improvement may be extended to a multiobjective context.
This is done by introducing a hypervolume function,

The hypervolume,

Although these ideas may also be extended to more objectives, assuming two objectives simplifies the problem, in this case, the observed Pareto set is defined as

Once the EHVI is estimated, it must be maximized. This is not necessarily trivial, as the EHVI computation is complicated and difficult to vectorize, and the cost of the optimization grows exponentially with the number of design variables. The EHVI optimum may be determined using a grid search, random sampling, direct optimization, or surrogate-based optimization. While a grid search is the most comprehensive option, the latter approaches are more computationally efficient for high-dimensional design inputs.

The multifidelity approach introduces computationally cheaper but lower fidelity representations of the high-fidelity model, which allow for greater control between exploration and exploitation in the Bayesian optimization.
Samples of the low-fidelity model are adaptively refined throughout the optimization as a cheap means for exploration of the high-fidelity function space. Throughout this section, we assume a known hierarchy of model fidelities,

The lowest-fidelity model,

New GPs are defined to extend the EHVI to a multifidelity context. No matter which fidelity is to be sampled next, the ultimate goal is to minimize the highest fidelity functions, so each GP is constructed to predict the high-fidelity outputs. However, GPs associated with lower-fidelity models should not take into account uncertainties associated with higher-fidelity models, as these uncertainties will not be collapsed if the lower-fidelity model is sampled. Sampling the highest-fidelity model must take all sources of surrogate uncertainty into account, as a high-fidelity model evaluation will be associated with sampling all lower-fidelity models at the same point.
So, new GP models,

In this study, we examine the bifidelity case (

Workflow visualization for the bifidelity optimization case.

This section outlines the numerical approaches used in this study. Section

We use the WindSE framework

The simulations solve the filtered conservation of mass and Navier–Stokes equations given by

The turbine forcing is computed as

Low- and high-fidelity models were developed for this study using the WindSE framework. A Cartesian discretization of the computation domain is specified, where the grid is refined twice in the wake region as well as near the turbine rotors. Each high-fidelity simulation is run to 1200 s using Taylor–Hood elements

Time-averaged velocity magnitude fields at the turbine hub height associated with the low- and high-fidelity models. Panel

The objective of the optimization is to minimize negative power,

Power and loading are quantified using the actuator line model results, discarding an initial transient period. Power is computed as the average total power after the initial transient period. While there are several ways to quantify loading, this study provides a demonstration of the optimization framework by summarizing the time history of the flapwise bending moment of one blade in the front and back turbines after the same initial transient period. Using the high-fidelity model, loading is computed as the sum of damage equivalent loads (DELs)

The average power production of each turbine is computed as

Here we use a simple optimization approach for simplicity of demonstration.
In each iteration, the

Initial sampling points are selected using a heuristic approach, where an assumed kernel is used to progressively minimize the standard deviation of the predictor. The simplest approach to initializing the optimization procedure is to randomly sample the low-fidelity and discrepancy functions. Random initial sampling may affect the optimization results, so a deterministic and symmetric sampling strategy is used as a test case.
An isotropic kernel is used with a correlation scale of 10

We considered several different low-fidelity model forms for loading and selected the one with the highest correlation to the high-fidelity model, as that is known to result in the best multifidelity performance. We used 100 samples obtained using the heuristic sampling method described in Sect.

Correlations observed between high-fidelity DEL and different loading proxies of the low-fidelity model using 100 heuristic samples.

The convergence of the single-fidelity and multifidelity optimization approaches is compared in Fig.

Convergence history of the single-fidelity and multifidelity approaches. The left plots show the EHVI, hypervolume, and best-observed power and loading. The right plots show the yaw configurations associated with the best-observed power and loading.

Figure

Sampled inputs and outputs associated with power greater than 3 MW and loads less than 3 MN m. Points associated with the single-fidelity approach are shown using blue square markers, and points associated with the multifidelity approach are shown using green circular markers. A Pareto set constructed from the single-fidelity and multifidelity results is highlighted with hollow circles, where darker (magenta) circles correspond to Pareto points with lower loads and lower powers.

Although the primary goal of the present study is to develop and demonstrate a multifidelity multiobjective optimization framework, once several of the points in the Pareto set are identified the set can be further refined using a grid search. As a demonstration of this additional step, the Pareto set resulting from the combination of the single-fidelity and multifidelity approaches was interpolated to create refinement samples using B-spline interpolation

The Pareto set resulting from these additional refinement samples is visualized in Fig.

Sampled inputs and outputs associated with power greater than 3 MW and loads less than 3 MN m. Points associated with the single-fidelity approach are shown using blue square markers, and points associated with the multifidelity approach are shown using green circular markers. Refinement points are shown as black triangles. A Pareto set constructed from the single-fidelity, multifidelity, and refinement samples is highlighted with hollow circles, where darker (magenta) circles correspond to Pareto points with lower loads and lower powers.

The multifidelity approach was successful in quantifying the trade-offs between loading and power and was shown to be more efficient than its single-fidelity counterpart. From the presented results, we find that loading may be reduced by 4 % while only reducing the optimal power by 0.3 %. The accuracy of the single-fidelity and multifidelity GP models is quantified using a leave-one-out analysis in the Appendix

Observed power and loads for various yaw configurations.

Figure

Flow fields associated with the extreme and neutral offsets in the front turbine, viewed from upstream. The

Time-averaged flow fields associated with the optimal power,

Time-averaged flow fields associated with the optimal power (left) and loading (right) found by the optimization, viewed from upstream, six rotor diameters away from the front turbine and one rotor diameter upstream of the downstream turbine. Brighter colors indicate faster velocity magnitudes. In each plot, the vertical location and cross-flow location of the back turbine are shown as a white ellipse.

Figure

Loading histories associated with different yaw offset strategies (values of

This paper has demonstrated a multifidelity multiobjective optimization approach for wake-steering strategies. Actuator line simulations were carried out using the WindSE tool and using a coarser simulation as the low-fidelity model. The high-fidelity loading was characterized as the sum of flapwise bending moment DELs on blades on the front and back turbines. Due to oscillations in the low-fidelity simulations, characterizing the low-fidelity loading with a DEL resulted in a relatively low correlation between the low- and high-fidelity loading predictions, so a different low-fidelity surrogate was developed with a higher correlation.

The multifidelity multiobjective optimization approach was effective in exploring the trade-offs between loading and power when developing a wake-steering design. Convergence was achieved in the multifidelity optimization case after approximately 30 % as many equivalent high-fidelity model evaluations as in the single-fidelity case. Future work should apply this approach and a low-fidelity loading function to more complex wind plant layouts to confirm their effectiveness. Exploring the solutions in the final Pareto sets guided insights into the fundamental flow physics. Given the specified turbine spacing and atmospheric conditions, a positive front turbine yaw offset is more effective at reducing loading and increasing power than a negative yaw offset because the counter-rotating vortices associated with the negative front turbine yaw offset produce a greater velocity deficit in the downstream wake. The boundary layer is convected by the counter-rotating vortices, adversely affecting loading, and this may be avoided using less-extreme front turbine yaw offsets. Slightly modifying the back turbine yaw offset reduced loading by 4 % and only reduced power by 0.3 %. Greater offsets in the back turbine also led to less overall loading, with significantly less power generation.

It is well known that yaw position errors can adversely affect the performance of wake-steering strategies. This is especially true when it comes to turbine loading. A 30

A drawback of the presented approach is that it requires sequential high-fidelity model evaluations. In practice, it is often feasible to evaluate a high-fidelity model several times in parallel, and the greatest expense is the time needed to run the optimization. This framework may be extended to allow for parallel function evaluations. A simple approach is to use predictions of the GP as stand-ins for future model evaluations, iteratively using these points to construct the next iteration of the GP and the associated EHVI

In future work, this framework can be applied to a larger array of turbines using more realistic control strategies with different turbine spacings and atmospheric conditions. While considering more turbines presents additional complications in maximizing the EHVI, we anticipate there will be even greater cost savings from the multifidelity approach as the number of turbines increases. Additionally, the framework can be extended to allow for optimization under uncertainty, as it is not realistic to assume perfect control of wind turbine yaw positions. Finally, the framework can incorporate more lower-fidelity models and be combined with layout optimization to realize the full benefits of multifidelity multiobjective wake-steering optimization.

This study employed the WindSE software, commit hash f217bd0986325d86ee0ffac083a71fd4ead91b63. The repository is available at

The data set associated with our analysis can be accessed at

GB, PEH, RNK, and JQ designed the experiments. RNK, GB, and JQ developed the problem formulation. JQ and PEH performed the simulations. JQ, RNK, and PEH prepared the manuscript with contributions from all co-authors.

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 work was authored in part by the National Renewable Energy Laboratory, operated by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under contract no. DE-AC36-08GO28308. Funding was provided by the U.S. Department of Energy Wind Energy Technologies Office. The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for U.S. Government purposes. This research was performed using computational resources sponsored by the Department of Energy's Office of Energy Efficiency and Renewable Energy and located at the National Renewable Energy Laboratory.

This research has been supported by the U.S. Department of Energy (grant no. DE-AC36-08GO28308).

This paper was edited by Johan Meyers and reviewed by Ishaan Sood and one anonymous referee.