This paper aims to develop fast and reliable surrogate models for yaw-based wind farm control. The surrogates, based on polynomial chaos expansion (PCE), are built using high-fidelity flow simulations coupled with aeroelastic simulations of the turbine performance and loads.
Developing a model for wind farm control is a challenging control problem due to the time-varying dynamics of the wake.
The wind farm control strategy is optimized for both the power output and the loading of the turbines.
The optimization performed using two Vestas V27 turbines in a row for a specific atmospheric condition suggests that a power gain of almost

Wind turbines operating in the wake of upstream turbine(s) experience power losses and increased fatigue loads. Accordingly, wind farm control aims at improving the power production and potentially decreasing structural loads by optimizing the collective operation of the wind farm, as well as providing better integration of wind power into the grid. Intentionally yawing the turbine has been the focus of recent studies as one of the methods to mitigate the wake effects. The objective is to intentionally change the trajectory of the wake in order to increase the power output of the downstream turbine(s) and possibly reduce the fatigue loads.

However, in order to apply yaw-based wind farm control, a thorough understanding of the wake and the loading on the wind turbine components is necessary. The earliest studies investigating the performance of a propeller in yaw were conducted by

However, these numerical studies are computationally demanding, while dynamic wind farm control requires fast and reliable wake models in the optimization loop. Therefore, engineering models which capture the effects of wake steering are necessary.

This study aims to develop and validate yaw-based wind farm control strategies, based on surrogate models through the use of the high-fidelity flow solver EllipSys3D LES (large eddy simulation) and the aeroelastic tool Flex5. A surrogate model essentially constructs response surfaces based on the input and output of the determined domain. Both the power output and the turbine loading are included in the optimization and assessment of different wind farm control settings. The analysis also highlights the advantage of using the surrogate models as compared to the conventional analytical models in the optimization.

The turbulent wake behind a wind turbine is simulated using EllipSys3D and the actuator line method to represent the turbine, while the turbine performance and response are calculated using the aeroelastic tool Flex5. These tools and methods are briefly described in the following; for a
more comprehensive description, see

Large eddy simulations (LESs) have been performed with EllipSys3D, which was developed at the Technical University of Denmark by

The atmospheric boundary layer is modelled by using body forces (

Turbulence is introduced to the Navier–Stokes equation by imposing body forces (

The wind turbine is modelled using the actuator line method (AL), as developed by

The numerical domain used for the simulations is

Cyclic boundary conditions have been applied on the lateral boundaries, no-slip on the bottom, and far-field conditions on the top boundary. The prescribed atmospheric boundary layer profile is shown in Fig.

Prescribed boundary layer in EllipSys3D. The shaded red area indicates the wind profile

The Mann box has been generated using the following parameters:

For the simulations in EllipSys3D, a Flex5 model of the V27, which is based on the V27 turbines located at the SWiFT facility as described by

Main parameters of the Vestas V27 given in

Calibration of the V27 model based on the experimental data from

FLORIS

The python code of the FLORIS model was obtained from GitHub and version 0.1.1 was used (

The input parameters for FLORIS are determined from the EllipSys3D setup. Furthermore, the turbine characteristics, shown in Table

The main purpose of a surrogate model is to build a model that defines the relationship between the input and the output of a given data set in a fast and accurate manner. In the recent study of

The surrogate model was built using the python software toolbox Chaospy (

The training data for the surrogate models for the upstream and the downstream turbine are obtained from simulations performed with the aeroelastic tool Flex5. These simulations have been run using detailed turbulent inflow generated via EllipSys3D with a single turbine operating at different yaw misalignment. The angle between the rotor axis and the free-stream velocity is denoted by

Table

Overview of the cases simulated with the coupling between EllipSys3D LES and Flex5 and the cases simulated with the extracted flow field in Flex5.

Surrogate models have been created for the power and the damage equivalent loads (DEL) for both the upstream and the downstream ghost turbine pair. Together with the produced power, only the DEL of the flapwise root bending moment and the total bottom tower bending moment are used to assess the effect of the yaw angle. This was done in order to simplify the model and to include the most important DEL influenced by yaw steering. The models are constructed using three input parameters: the upstream yaw angle (

Three unique surrogate models for the upstream turbine were created for the power as well as the DEL for the flapwise root and the tower bending moments. Note that these upstream surrogate models only depend on the upstream yaw angle (

In addition, the stability of the surrogate models are improved by increasing the number of data points. Multiple realizations are extracted from the time series of Flex5 using a moving average with a window of 10 min shifted every 30 s. Therefore, there are 23 individual points for each unique configuration. This results in

Overview of the created surrogate models for the upstream turbine and the downstream turbine. The distribution of the upstream yaw angle (

The example data set and the script for the surrogate model training can be found at the project git (

First, the turbine performance, wake deflection and velocity deficit is analysed for different yaw angles and downstream distances. Then the surrogate models for the upstream and downstream turbines are created, analysed, and used to optimize the operation of the turbine pair, which is examined in terms of uncertainty and model errors.

The power output of the upstream turbine for the yaw cases

Time series of the power output of the upstream turbine determined with the aeroelastic tool Flex5 at

As the turbine is yawed, the inflow is no longer aligned with the rotor axis. The misalignment leads to a difference in the axial induction and thus an asymmetric loading on the rotor blades; see

The deflected wake extracted from EllipSys3D is shown in Fig.

Time averaged horizontal plane of the averaged relative velocity (

Figure

The cumulative power output increases for increasing turbine spacing as expected since the wake recovers.

Normalized cumulative power of the upstream and downstream turbine determined with the numerical simulation performed in EllipSys3D coupled with Flex5 at

An increase in the cumulative power production for larger turbine spacing is also seen in the FLORIS results (not shown for brevity). However, as the FLORIS model does not capture the asymmetric behaviour of the wake deflection, the cumulative power at

The results of EllipSys3D presented in Sect.

Here, the generated surrogate models are response surfaces based on multivariate polynomials. These response surfaces are highly dependent on the selected polynomial order and the available training data. To visualize the dependency of the generated PCE surrogates to the polynomial order and input data, an example case is used. The example case is based on the power of the downstream turbine at

The response of the surrogate model is illustrated in Fig.

Power output of the surrogate model of the downstream turbine at

The over-fitting, which results in large errors in the estimates, can be reduced by populating the training data set further. Figure

This also indicates that in order to develop a surrogate model based on simulation data, a trade-off needs to be made between an acceptable error, the polynomial order, and the cost of high-fidelity simulations. The surrogate models built with the entire data set (as illustrated in Fig.

The polynomial order for the PCE approach also has to be determined. Therefore, the relative error of each surrogate model with different polynomial orders is examined. The relative error is computed for each turbine spacing (

The relative error is the difference between the output of the surrogate model (

Box plots of the relative error,

As discussed previously, there is a trade-off between acceptable error and polynomial order, but no decisive conclusion can be drawn with regards to the ideal polynomial order to build the surrogate models. The error distribution of all polynomial surrogates are comparable, and the median relative errors are generally decreased continuously from 3 % at

Box plots of the relative error of the power output of the downstream turbine between the FLORIS model and the results from the simulations performed with Flex5, where the flow field is extracted and used as an input to the aeroelastic tool. The distributions corresponds to all the available combinations of the optimization variables

For comparison, the relative error between the simulations performed with Flex5 and the results obtained from the FLORIS model is shown in Fig.

The optimization of wind farm control strategies is performed by applying a weight factor assigned for each surrogate model depending on the objective of the optimization. The aim of this study is to showcase how to develop a control strategy, which considers both the loads and the power output through the use of surrogate models. The weighting used for the optimization is shown in Eq. (

Note that the weights of the surrogate models for the equivalent loads are subtracted from 1 since the aim is to reduce the fatigue loads. The optimal point is determined by calculating the power and the equivalent loads for the upstream and downstream turbines using the trained surrogates with input parameters of

a power-driven optimization

a combined load and power optimization with a large weight attributed to the power

a combined load and power optimization with a small weight attributed to the power

The power-driven optimization is performed using Eq. (

Figure

It is observed that the power gain is the largest at a turbine spacing of

Optimal yaw configuration and power gain for a power-driven optimization. Each bar indicates the cumulative power gain (

A negative yaw angle for the upstream turbine is expected as it yields a larger wake deflection (as shown in Sect.

As shown in Fig.

Since the dynamic Flex5 simulation gives different results for each 10 min realization (as can be seen in, for example, Fig.

Surrogate model error and estimated power gain likelihood for the optimized yaw settings

Figure

The surrogate model error is subsequently used to assess the power gain likelihood,

First and foremost, Fig.

The uncertainty is partly due to the natural variability of the flow and the turbulent wake and partly due to the uncertainty associated with the surrogate models. The former is inherent and difficult to reduce, while the latter can be reduced by adding more high-fidelity data as shown in Sect.

It should also be noted that wider error distributions and higher bias (seen in Fig.

The loads can be included in the optimization by changing the weight factors (

The power-driven optimization in the previous section showed that the polynomial order has some influence on the expected power gain, but that it can potentially be corrected for. The DEL of the flapwise root bending moment (FlapM) and the tower bottom bending moment (TBBM) for the upstream and the downstream turbines are surrogated. It should be noted that the total tower bottom bending moment have been used here, i.e. the total length of the tower bending moment in the streamwise and lateral directions.

The combined optimization for both power and loads is conducted using polynomial of minimum order,

Here, the main aim is to visualize the effect of including loads in the optimization process and how it changes the optimum operational condition. But it should be noted that the reduction (or increase) in loads under optimum control strategies is not as critical as the power production, because the “business case” of load reduction is not as straightforward. For additional information on lifetime extension with regards to load management, see

As previously shown, weight factors of [1.0, 0.0, 0.0, 0.0, 0.0] would only optimize for the power, while a combined scenario of power and load optimization is generated with two additional weighting factors, namely [0.60, 0.10, 0.10, 0.10, 0.10] and [0.40, 0.15, 0.15, 0.15, 0.15]. Figure

Figure

Figure

Figure

However, this optimization is of course based on an equal weighting of the flapwise root and tower bottom bending moment. An additional optimization was tested with weights of [0.40, 0.30, 0.30, 0.00, 0.00] and [0.40, 0.00, 0.00, 0.30, 0.30] in order to isolate the effects of only including the flapwise root bending moment and the tower bottom bending moment in the optimization, respectively, as opposed to previous settings where the tower bottom bending moment dominated the optimization space. The results show that it is no longer possible to increase the power production for such a severe weighting of the loads; see Fig.

Combined load- and power-based optimization with respect to the DEL of the flapwise root bending moment (FlapM) and the DEL of the combined tower bending moment (TBBM) for the upstream and the downstream turbine. The weighting is given as follows: [

As any simplified model, the surrogates include model errors and uncertainties as previously mentioned and quantified. Figure

Contour plots of the cumulative power output dependant on the upstream and the downstream yaw angle at

The validation is performed as a blind test via additional simulations in Flex5 with the optimum yaw settings

The resulting power gain likelihood presented here can be compared to reported gains in literature, where

The equivalent load of the flapwise root bending moment determined with the surrogate model also yields very comparable results with the equivalent loads obtained directly from the Flex5 simulations. There is only a minor difference in the flapwise root bending moments, while the surrogates underestimate the tower bottom bending moment by less than 5 %, which should be accounted for by including the model error.

Comparison between the power output and the equivalent loads of the surrogate model for the optimal yaw settings (

EllipSys3D has been used to perform large eddy simulations of a V27 turbine operating at different yaw angles to investigate wake steering. The turbine has been modelled using actuator lines, which are fully coupled to the aeroelastic tool Flex5. The full flow field is extracted at different downstream distances and used as turbulent input to Flex5 to mimic a downstream turbine operating in the deflected wake of an upstream turbine. The upstream turbine has also been modelled with turbulent inflow in Flex5. The performance and response of the two turbines are used to construct surrogate models of different orders based on polynomial chaos expansion.

It is shown how the accuracy of the surrogate models depends on the amount of training data, and how the choice of order for the polynomials needs to be considered to capture more complexity but also to avoid overfitting. The constructed surrogate models consistently yield median errors for a variety of control inputs, i.e. the yaw angles of the upstream and downstream turbines at different turbine spacings. Considering the entire domain of the optimization, the surrogate models consistently overestimate the power output of the downstream turbine by approximately 2 % for most turbine spacings. The performance of FLORIS is also compared to the high-fidelity results for different control settings. FLORIS yields very large relative errors for close turbine spacings and moderately wide, biased distributions for larger turbine spacings, with median errors of 5 % and 3 % standard deviation for the investigated configuration.

Due to their higher accuracy, the surrogate models are used to optimize the power production. The two surrogate models of order

A combination of surrogate models has also been used to include the DEL in the optimization. The results showed that it is possible to reduce the tower bottom bending moments for both turbines by sacrificing some of the power gain. On the other hand, it is generally not possible to reduce the flapwise root bending moments. The combined power and load optimization also generally converge to normal operation with no yawing of the two turbines for larger spacings.

Finally, the optimization results were compared and validated against additional Flex5 simulation at the optimum yaw angles predicted by the surrogates. The validation confirmed the power gain likelihood assessment and provided estimates of the DEL of both flapwise root and tower bottom bending moments, which were underpredicted by less than 5 %.

The surrogate approach used in this study could be extended in several ways. To be generally applicable it should include different flow cases, e.g. wind speed, turbulence intensity, shear, and atmospheric stability. The surrogates can also be expanded by including field measurements when available. Additional surrogate models can be constructed for other turbine models. The true performance test of the presented optimization procedure should be conducted in a wind farm environment, where the flow complexity would increase and hence also the requirements on the model corrections and uncertainty estimations.

Averaged horizontal scan of the flow angle obtained with the coupling between EllipSys3D and Flex5 at

Combined load- and power-based optimization with respect to the DEL of the flapwise root bending moment (FlapM) and the DEL of the combined tower bending moment (TBBM) for the upstream and the downstream turbines. The weighting is given as follows: [

The input data and surrogate model data are available at

LES was performed by SJA, while the initial analysis was done by PH. All authors contributed to the extended analysis and reporting.

The authors declare that they have no conflict of interest.

The authors also wish to thank Nikolay Krasimirov Dimitrov for fruitful discussions on the polynomial chaos expansion theory. The study is partially supported by the CONCERT Project (Project no. 2016-1-12396), funded by Energinet.dk under the Public Service Obligation (PSO) and the CCA on Virtual Atmosphere.

This research has been supported by the Energinet.dk (grant no. 2016-1-12396).

This paper was edited by Athanasios Kolios and reviewed by two anonymous referees.