A total of 18 high-fidelity simulations of large wind farms have been performed by three different institutions using various inflow conditions and simulation setups. The setups differ in how the atmospheric turbulence, wind shear and wind turbine rotors are modeled, encompassing a wide range of commonly used modeling methods within the large eddy simulation (LES) framework. Various turbine spacings, atmospheric turbulence intensity levels and incoming wind velocities are considered. The work performed is part of the International Energy Agency (IEA) wind task Wakebench and is a continuation of previously published results on the subject. This work aims at providing a methodology for studying the general flow behavior in large wind farms in a systematic way. It seeks to investigate and further understand the global trends in wind farm performance, with a focus on variability.

Parametric studies first map the effect of various parameters on large aligned wind farms, including wind turbine spacing, wind shear and atmospheric turbulence intensity. The results are then aggregated and compared to engineering models as well as LES results from other investigations to provide an overall picture of how much power can be extracted from large wind farms operating below the rated level. The simple engineering models, although they cannot capture the variability features, capture the general trends well. Response surfaces are constructed based on the large number of aggregated LES data corresponding to a wide range of large wind farm layouts. The response surfaces form a basis for mapping the inherently varying power characteristics inside very large wind farms, including how much the turbines are able to exploit the turbulent fluctuations within the wind farms and estimating the associated uncertainty, which is valuable information useful for risk mitigation.

As renewable energies are expected to take an increasing share of future electricity production, wind energy is progressing where wind farms are being built in ever increasing sizes, especially offshore. Wind turbines operating far downstream in very large farms are subject to complex flow conditions, comprising the combined interaction between the atmosphere and the complex wake dynamics introduced by the wind turbines. Several factors come into play and contribute to the complexity of wind farm flow. These factors can be grouped into atmospheric conditions (e.g., stability, shear, veer and turbulence intensity), wind farm conditions (turbine size, farm layout) and combined effects as the turbines affect the atmospheric flow. A better understanding of how the flow develops in large wind farms is crucial in order to better plan and control the wind farms and to optimize their production.

A decade ago,

This complex wake problem has attracted the interest of numerous researchers for many years, with work being performed using several numerical methods, including both engineering type models, such as those by

The effect of the streamwise and spanwise turbine spacing on power output and turbulence intensity in the case of infinite aligned wind farms was for its part investigated, using LES, by

On the experimental side, work by

The aim of the present article is to present a methodology that can be used in a systematic way to further understand the general flow behavior in large wind farms. As outlined above, a number of research groups are today frequently simulating the flow in large wind farms using high-fidelity methods to further understand basic flow features. However, since there is a large variety of parameters, e.g., flow directions, choice of verification cases with different turbine spacings, atmospheric conditions, it is often very difficult to draw general conclusions through direct comparisons. The aim with the developed methodology is to capture key parameters from different setups to be able to investigate the global trends in wind farm performance. Here, results from high-fidelity simulations are combined and systematically analyzed. As will be shown in this article, the quality of the conclusions that can be drawn depends on the extent of data that can be used. By quantifying the variability for different situations, the uncertainty can be estimated.

In the present work, data derived from LES will be used, as these kind of high-fidelity data have been shown to produce very reliable results as regards the development of the flow within wind farms; see e.g.,

The work is a continuation of previous work that studied the variability in the flow statistics in LES performed on large wind farms by

The paper is arranged as follows: in Sect. 2, the methodology used to perform this work in terms of numerical methods is outlined, followed by the simulation setups considered to run each of these methods in Sect. 3. Results are then presented and discussed in Sect. 4, where works from other researchers are also included, before the main conclusions from the work are summarized and discussed in Sect. 5.

In this section, an overview of the main differences as regards the methodology used by the different participants is provided. Detailed information on the theoretical background associated with each method can be found in the publications that are referred to.

Results from two different CFD codes are used.

EllipSys3D is a 3D flow solver that was developed at DTU

PALM (the Parallelized LES Model) was developed at Leibniz University Hannover and has been applied for several years for the simulation of a variety of atmospheric and oceanic boundary layers. Recently, it has been enhanced by a wind turbine model; see

The wind turbines are modeled by DTU and Uppsala University (UU) by using the actuator line (AL) and actuator disc (AD), respectively. In the former, body forces are distributed along rotating lines, while they are distributed along a rotating disc in the latter. Details about the implementation of the AD and AL in EllipSys3D can be found in

The PALM implementation considers an AD model with rotation (FW-AD-R) in which local body forces are derived from airfoil data. The PALM simulations were performed by ForWind (FW). In contrast to the AL method, the forces are distributed across the rotor plane. This model also includes tower and nacelle effects that are modeled by a drag force approach. See

The three models used in this work include a turbine controller. This causes the applied body forces to be governed by the inflow conditions, meaning that the turbines are not constantly loaded but operate as “real turbines”.

Two different three-bladed horizontal-axis wind turbines have been considered in the simulations, i.e., the NM80 and the NREL 5

In the coordinate system used in this work,

All participants simulated a neutrally stratified atmospheric boundary layer (ABL). Details about the methods used to model the ABL and associated turbulence in, respectively, EllipSys3D and PALM are provided below.

EllipSys3D uses the prescribed boundary layer (PBL) method, in which body forces are used to impose any arbitrary vertical wind shear profile; see

Ambient turbulence is modeled by introducing pregenerated synthetic ambient turbulence using the Mann model; see

PALM uses a no-slip bottom boundary condition and the Monin–Obukhov similarity theory between the surface and the first grid level to model the atmospheric boundary layer. Random perturbations are initially imposed on the velocity fields until atmospheric turbulence has developed in a precursor simulation. The latter is performed on a smaller domain with periodic boundary conditions in the streamwise and lateral directions. The precursor results are used to initialize the full simulations with nonperiodic boundary conditions in the streamwise direction. Turbulence recycling is also applied; see

An overview of the numerical methods described in the previous sections are summarized in Table

Summary of methods. The participating institutes are ForWind (FW), Uppsala University (UU) and the Technical University of Denmark (DTU).

A total of 18 large wind farms have been simulated and analyzed. A majority of the simulations are performed for below rated conditions at approximately

Overview of simulations performed by DTU. The simulations include 16 turbines and

Overview of simulations performed by UU. The simulations include 16 turbines and

Overview of simulations performed by FW. The simulations included two rows of 50 turbines and

The present analysis is an extension of the previous work on the inherent variability in the flow statistics in LES as presented by

First, the inherent variability in LES is described, before the effects of freestream turbulence intensity and of turbulence and shear combined, as well as of turbine spacing, are investigated using the different numerical setups. Finally, the large number of data are aggregated, and a more generalized analysis is performed on mechanical power production and variability within large wind farms.

Simulations DTU3, UU2 and FW5 (cf. Tables

The results from DTU and UU are very comparable in terms of level of mechanical power production, while the FW results are approximately

Box plots of the

The variability is investigated more specifically in terms of the turbulence intensity, shear and turbulence intensity, as well as the turbine spacing, in the following.

Box plots of all 50 turbines for FW5 with

The simulations from DTU and UU utilize body forces to introduce ambient turbulence into the flow. This enables direct investigation of the isolated effects of changing the ambient turbulence by changing the forcing.

Here, the distributions of instantaneous power production of the 16 turbines are compared directly in violin plots in Fig.

Violin plots comparing the influence of turbulence intensity on the instantaneous power production in DTU2 (black) with

Turbulence intensity and shear are inherently linked in the simulations performed by FW, as a change in equivalent roughness yields different shear and turbulence profiles. Figure

Violin plots comparing the influence of turbulence intensity and shear on the instantaneous power production in FW2 (black) with

The initial turbulence intensity and shear discussed in the previous sections develops through wind farms, and the flow development is closely related to the turbine spacing.
Figure

Violin plots comparing the influence of turbine spacing on the instantaneous power production in DTU5 (black) with

Mean power production of all turbines from the sixth to the end of the row for all simulations as function of representative turbine spacing. The mean power productions have been normalized by the mean power production of first wind turbine. Error bars show standard deviation of all the 10 min periods. Simulations with turbulence intensity of 0 %, 3 %, 10 % and 15 % are shown in green, blue, cyan, and red, respectively. Two simulations with

The simulation data are aggregated in terms of

It is clear how the 18 simulations follow the same trends as the data derived from

The effect of atmospheric turbulence is also clear, both when comparing the general trends in the plot and when comparing the DTU and FW results for different turbulent intensities. A higher atmospheric turbulence yields a higher production deep inside the farms, while low or even no atmospheric turbulence results in a lower boundary in terms of production.

Finally, the figure includes the resulting power production based on two asymptotic expressions derived by

The model developed by

Calibrated geostrophic wind speeds (

Both models capture the general trends very well, although the Jensen model underestimates the actual power production for the recommended values. The Frandsen model performs very well and captures both the high and low turbulence intensity levels as well as the gradual change for the lower turbine spacings where the data by

The total number of aggregated data in Fig.

The power per ground area, or power density, compared to the standard deviation of power normalized by the mean power for different relative spacings is shown in Fig.

The data show significant spread in both power per area and standard deviation of the power although all simulation results generally cluster together. The binned values are generally very consistent except at low standard deviations, in particular Fig.

Power per ground area plotted against standard deviation of power normalized by mean power for

Contours based on multiple linear regression fit of bin-averaged power production per area to the standard deviation of power production normalized by mean power production and relative turbine spacing. Points mark the binned data, where blue and red shades indicates whether the fit underestimates or overestimates, respectively, the binned averaged values.

A multiple linear regression is applied to the full set of bin-averaged data with a freestream velocity of

Figure

Contours based on multiple linear regression fit of standard deviation of the bin-averaged power production per area to the standard deviation of power production normalized by mean power production and relative turbine spacing. Points mark the binned data, where blue and red shades indicate whether the fit underestimates or overestimates, respectively, the binned averaged values.

Figure

As shown previously, the simple engineering model by

The variability around the mean is up to 0.4

Contours based on multiple linear regression fit of the standard error in the bin-averaged power production per area to the standard deviation of power production normalized by mean power production and relative turbine spacing. Points mark the binned data, where blue and red shades indicate whether the fit underestimates or overestimates, respectively, the binned averaged values.

The increased variability for smaller spacing also comes with an increased uncertainty as shown in Fig.

The response surfaces are only fitted to second order, because the aim here is merely to provide general insights into the global trends and hence to avoid overfitting. It should be strongly emphasized that this is a rather crude approach. However, the response surfaces yield a first attempt at constructing global response surfaces of the power density including the inherent variability based on significant numbers of LES data for a wide range of wind farm layouts operating at

Figure

Heatmap showing number of

The response surfaces could also be made dependent on more parameters by adding more LES data. Currently, the turbine spacings in the lateral and streamwise direction have been collapsed to a single dimension, but the dependency could be unfolded. Similarly, the dependency on a number of additional parameters could be investigated, for instance,

free wind speed,

turbulence level,

atmospheric stability,

shear,

turbine size.

One way to circumvent the large computational costs would be to utilize supervisory control and data acquisition (SCADA) data in combination with the LES. Similar response surfaces could be constructed based on SCADA data from operating wind farms, which would enable a more global verification of LES and the actuator disc and line methods on a wind farm scale. Such a verification would be valuable as direct comparison of time series of specific events between LES and actual wind farms is extremely difficult, if not impossible, to achieve given the complexity and amount of information required on the atmospheric conditions to enable such a comparison.

A successful verification would facilitate the direct integration of LES data and SCADA data to construct more certain response surfaces covering a larger range of scenarios and parameters. It could act as a lodestar and inform researchers about in which regions of turbine spacing and turbulence intensity to perform the expensive LES in order to fill the gaps and explain physical trends not captured by the simpler models.

Finally, the response surface could be extended to include e.g., fatigue loads for turbines operating in wind farms. Such a surrogate model for fatigue loads on a single wind turbine was developed by

This work aimed at providing a general overview of the global trends in power performance for large wind farms, with a focus on variability. This was done through the analysis of large eddy simulation (LES) performed on large wind farms from the three institutions behind this work. LES results of large wind farms obtained from other researchers as well as simulations performed using simpler engineering models were also included to provide a more complete envelope for the results.

As LES requires large amounts of computational resource, emphasis was placed on extracting as much information as possible from the existing set of simulations performed using different setups and incoming flow conditions. As such, emphasis is not put on comparing the simulations to each other but rather on using as many results as possible to cover a wide range of possible scenarios that can provide a global picture of the power characteristics within large wind farms.

Parametric studies were first performed to inform about the effect of atmospheric conditions as well as turbine spacing on production and its variability. An increase in atmospheric turbulence intensity, by increasing energy entrainment, was shown to raise the mean level of power production. It was also associated with wider distributions of the production values. A larger spacing between the turbines was also associated with greater levels of production, as expected.

The analysis was extended further by aggregating the large number of LES runs performed under various conditions. This was carried out in terms of 10 min statistics for each turbine operating in deep-farm conditions. LES works from other researchers as well as simulations performed with simpler engineering models were also included in a first step when looking at the power produced deep inside the farm as a function of a representative spacing. All results were shown to fall within a clear limit showing how much power can be extracted from a wind farm operating below rated wind speed, as a function of representative turbine spacing. Whereas higher turbulence levels lead to larger production levels deep inside the farms, cases without incoming turbulence were shown to provide lower power production. While LES provides more information in terms of variability, simple engineering models were shown to produce a reasonable envelope for the results obtained using the high-fidelity methods.

As a second step, response surfaces encompassing the total number of aggregated LES data, i.e., 12 016 different albeit overlapping 10 min realizations, were created. They revealed information regarding various aspects of the power production within large wind farms, including the amount of power the turbines are able to extract from the turbulent fluctuations, as well as the variability and uncertainty associated with the mean power densities.

The work presented in this paper serves to provide valuable information regarding power and its variability deep inside large wind farms. Nonetheless, the response surfaces presented here would gain from being complemented with more LES results to provide an even more complete picture. This could be done by considering further turbine spacings to fill existing gaps. The dependency of response surfaces on more parameters could also be investigated, including individually considered spanwise and streamwise spacings and the freestream velocity as well as the atmospheric stability. As LES is known to be very computationally demanding, SCADA data could also be used to provide more complete response surfaces. Future work could also go one step further by investigating the behavior of turbine loads in similar terms to what was performed here regarding power production.

The data can be made available on request.

SJA, SPB and BW performed the LES. SJA performed the preliminary analysis and first draft, while all authors contributed to the further analysis and reporting of the research presented in this paper.

The authors declare that they have no conflict of interest.

Computer time was granted by the Swedish National Infrastructure for Computing (SNIC) and by the North German Supercomputing Alliance (HLRN). The proprietary data for Vestas' NM80 turbine have been used. Finally, the authors wish to thank Juan Pablo Murcia Leon for many fruitful discussions on data analysis.

This research has been supported by the Danish Council for Strategic Research (grant no. 2104-09-067216/DSF) for the project Center for Computational Wind Turbine Aerodynamics and Atmospheric Turbulence (COMWIND:

This paper was edited by Sandrine Aubrun and reviewed by two anonymous referees.