These authors contributed equally to this work.

Lidar-assisted control (LAC) of wind turbines is a control concept that takes advantage of a nacelle-mounted lidar (a remote sensing device) to measure upstream wind speeds of a turbine to allow a preview of the incoming turbulence. Because the turbine will not be exposed to the identical turbulence as that measured by the lidar in advance, the simulation of a LAC system will be more realistic if wind evolution can be modeled in the wind field generation. Since the commonly used 3D stochastic wind field generation method does not include wind evolution, the main goal of this research is to extend the 3D method to 4D to enable the modeling of wind evolution along the wind direction. The most novel part of this research is that we propose a two-step Cholesky decomposition approach for the factorization of the coherence matrices in the wind field generation. With this approach, 4D wind fields can be generated by combining multiple statistically independent 3D wind fields. To enable better integration of the 4D method into the common workflow of wind turbine simulations, we implement the 4D method as the open-access tool evoTurb in combination with TurbSim and Mann turbulence generator. Moreover, since 4D wind field generation is supposed to be coupled with lidar simulations, and considering the range weighting effect of lidars and eventually multiple range gates, a 4D wind field will contain many more simulation points than a 3D one. To avoid excessive computational effort, we further investigate the impacts of the spatial discretization in 4D wind fields on lidar simulations to provide some insights to optimize the application of 4D wind field generation.

Wind turbines are highly dynamic systems operating in turbulent wind fields in the atmospheric boundary layer, with interacting effects of aerodynamics, structural dynamics, control systems, soil dynamics, and hydrodynamics (only for offshore locations)

TurbSim was initially developed on the basis of the 3D wind field simulation method proposed by

The Mann turbulence generator (MTG) is based on the Mann uniform

It is worth mentioning that both tools create a 3D wind field: TurbSim creates time series of wind vectors at points in a 2D vertical rectangular grid fixed in space, namely

Wind evolution refers to time-dependent variation of turbulence structures (eddies). In practice, wind evolution is usually quantified with the longitudinal coherence between the wind speeds measured at different locations in the mean wind direction

Some attempts have been made to simulate the effect of wind evolution or integrate wind evolution models into 3D simulations. For example,

In this work, we aim to extend the Veers method of 3D stochastic wind field generation to 4D in a general form so that the simulation of multi-distance lidar measurements can be better integrated into the current framework of the aeroelastic simulation of wind turbines. We first derive the mathematical expression of how to combine the longitudinal coherence into a conventional 3D wind field simulation. Based on this, a two-step Cholesky decomposition approach is proposed to factorize the matrices of the lateral–vertical coherence and the longitudinal coherence, respectively, to make the 4D wind field generation more feasible in practice.

The two-step Cholesky decomposition approach also makes it possible to generate a 4D wind field by combining multiple statistically independent 3D wind fields. To facilitate practical application of our 4D method, we implement it as an open-access tool evoTurb (evolving turbulence) published on GitHub (coded in MATLAB and Python). This tool takes 3D wind fields generated using standard wind field simulation tools – TurbSim or MTG, so that the longitudinal coherence can be introduced in synthetic wind fields without changing any other turbulence properties. Figure

Concept for integrating the 4D wind field simulator evoTurb into the aeroelastic simulation of wind turbines.

Since the 4D wind field simulation is supposed to be applied in combination with lidar simulations, it is expected to generate wind speed time series for a 3D grid, and thus the computational effort is much larger than that of the 3D method. Therefore, we further look into the possibility to reduce the size of the simulation grid. For LAC, the auto-spectrum of line-of-sight (LOS) measurements is an important indicator since it is related to its variance in the time domain, which can be further used to estimate turbulence intensity

This paper is organized as follows: Sect.

This section focuses on the methodology and implementation of the 4D wind field generation method proposed in this research. Section

The Veers method

As explained above, the key to the Veers method is to compute the CFCs for each frequency component. Here, we take the example of generating time series of the

Consider

In principle, Eq. (

To explain this idea, we continue with the example of the

Currently, there is no simple model for the 3D coherence available. Therefore, a general approach to create the 3D coherence is combining the lateral–vertical coherence and the longitudinal coherence

In fact, Eq. (

A problem may occur when directly applying Eq. (

As mentioned above, the 3D coherence

Considering that regular grids are more commonly used in practice, we define a 3D grid with

Similar to the lateral–vertical coherence, the longitudinal coherence only depends on the spatial separation between the two points on the

With the two assumptions, the 3D coherence matrix

With the two-step Cholesky decomposition, Eq. (

Following the idea mentioned above, Eq. (

Based on this, the CFCs of the

In comparison to the direct generation of 4D wind fields, the advantage of this concept is that it can introduce the longitudinal coherence into the stochastic wind field generation without changing any other wind field properties generated by these standard tools. This is conducive to the integration of the 4D wind field generation with the current framework of the aeroelastic simulation of wind turbines. Moreover, this concept makes it possible to use pre-generated 3D wind fields (with different random seeds) for the generation of 4D wind fields. The input 3D wind fields should be randomly selected and non-repetitive. Different combinations of the same 3D wind fields, i.e., assigning the 3D wind fields to the vertical planes in a 4D wind field differently, can form different 4D wind fields, and thus a 3D wind field can be used multiple times. This can significantly reduce the computational effort required to generate 4D wind fields, considering that in the aeroelastic simulations of wind turbines, a design load case requires several simulations using wind fields generated with different random seeds

Based on this concept, we developed an open-access 4D wind field generator – evoTurb (GitHub:

For the Kaimal model in TurbSim, we just need to apply Eq. (

The Mann model

The functionality of evoTurb (referred to the main script of the codes) is briefly introduced as follows.

In fact, the Mann model additionally contains the spatial coherence of

The validation of evoTurb mainly focuses on two aspects: whether the longitudinal coherence is correctly simulated and whether other wind field properties generated by TurbSim and MTG are not affected by evoTurb.

The validation is done by two examples coupling with TurbSim and MTG, respectively. The relevant parameters of the 4D wind field generation are summarized in Table

The wind field parameters for the validation.

To validate the coherence, Fig.

Comparison of the theoretical and simulated coherence between different horizontal separations in 4D wind fields. The simulated coherence is calculated by dividing the averaged cross-spectra by the auto-spectra of 16 samples. Sim.: simulated. Theo.: theoretical.

As presented in Sect.

Illustration of the

Regarding the special issue related to the Mann model raised in Sect.

Comparison of the simulated and theoretical spectra of the Mann model at

As mentioned in the introduction, 4D wind field generation is supposed to be applied to the simulation of lidar-assisted control systems, and thus this section intends to study its integration with lidar simulations. Section

Lidars in this article refer specifically to coherent Doppler wind lidars whose measuring principle is based on the optical Doppler effect. Such lidar systems measure wind speed by transmitting narrow bandwidth laser signals into the atmosphere and detecting the Doppler shift in the backscattered signals from aerosol particles in the atmosphere using coherent detection

Two types of wind lidars are commonly available for the wind energy applications: continuous-wave lidars and pulsed lidars. In this research, we take the example of a pulsed lidar. As its name implies, a pulsed lidar emits regularly spaced short laser pulses. In the data processing of pulsed lidars, the return signals of each pulse are first divided into range gates, and the averaging of power spectra is done for the same range gate from different pulses

Due to the measuring principle of lidars, two effects should be considered in lidar simulations in general: the volume averaging effect and the time averaging effect

The volume averaging effect can be simulated by applying a range weighting function

The typical coordinate system for simulating an upstream-looking nacelle lidar.

For pulsed lidars, the range weighting function can be analytically computed as the convolution between the pulse power profile and the range gate profile

In lidar simulations, Eq. (

In this section, we derive the analytical expressions of the auto-spectrum of LOS measurements and the coherence between the REWS and the lidar-estimated REWS to serve as a theoretical basis for the analysis of the impact of spatial discretization in lidar simulations in Sect.

We formulate the mathematical derivation according to the Kaimal model and perform the derivation mainly based on the linearity of 1D Fourier transform because this procedure is easier to understand. As for the Mann model,

Inspired by

In the Kaimal model, besides the auto-spectra of

It is worth emphasizing that in 4D wind fields,

The REWS is often assumed to be the averaged

As introduced in Sect.

As indicated in Eq. (

For simplicity, the simulated laser beam is assumed aligned to the wind direction. In this case, Eq. (

To visualize this issue, we take the following three-point (3pt), five-point (5pt), and seven-point (7pt) cases as examples:

Based on the above analysis, we suggest to consider spacing rather than the number of discrete points when applying a discrete range weighting function in lidar simulations. The spacing could be chosen according to the maximum relevant wavenumber

In practice, the lidar simulation points are not necessarily located on the grid points of the simulated wind fields. To tackle this issue, one may consider using interpolation to approximate the values of the desired points for lidar simulations. In this section, we discuss the impact of interpolation on the auto-spectrum of the

For simplicity, we consider a 2-by-2 square grid on the

Visualization of 2D linear interpolation. The interpolated value at

Nearest-neighbor interpolation takes the value of the nearest point. In this example,

Two-dimensional linear interpolation calculates

Figure

Comparison of the impact of nearest-neighbor interpolation and linear interpolation on the auto-spectrum of the

As discussed in Sect.

If the position and the measuring trajectory of the lidar to be simulated are fixed, we can directly generate wind speeds at all the points required for lidar simulations, including the discrete points for modeling the volume averaging effect of lidars. We define this type of grid as the direct grid (see Fig.

Illustration of three types of grids for lidar simulations.

To simulate a nacelle-mounted lidar, the grid of wind fields must cover all the range gates of the lidar and the space in their vicinity so that LOS measurements can be properly modeled when the required points constantly change due to the nacelle motion. For this purpose, we define the grid consisting of the vertical planes (perpendicular to

With the full grid method, the overall effects of both discrete ranging weighting functions (taking three-point and five-point cases as an example) and interpolation methods (linear or nearest) are illustrated in Fig.

Parameters of the simulated pulsed lidar system.

Comparison of the simulated lidar spectral properties with different discrete weighting functions and interpolation methods.

However, using full grid for lidar simulations requires relatively high computational effort for the 4D wind field generation because it needs to generate the same number of 3D wind fields as the unique

To evaluate if the semi-frozen grid is a proper approximation, we compare these three grids by considering the auto-spectra of LOS measurements and the coherence between the REWS and the lidar-estimated REWS. Four-dimensional wind fields are generated in each of these three grids with the Kaimal model (for parameters, see Table

Comparison of the simulated lidar spectral properties with different grids.

Lidar-assisted control (LAC) of wind turbines is a control concept which takes advantage of a nacelle-mounted lidar (a remote sensing device) to measure upstream wind of a turbine to enable the turbine to preact to the incoming turbulence

Out of the need for a wind field generator capable of simulating wind evolution, in the first part of this paper, we present a general method for 4D (space–time) stochastic wind field generation based on the extension of the Veers method

Because 4D wind field generation is supposed to be applied to simulations of LAC systems, in the second part of this paper, we study lidar simulations in 4D wind fields with respect to the following three aspects and provide corresponding suggestions.

There is still space to improve evoTurb. For example, the impact of assuming the identical longitudinal coherence for the

In the stochastic wind field generation, the wind velocity fluctuations are assumed to be statistically stationary Gaussian processes or Gaussian fields, which can be completely characterized by their mean, variance, auto-spectrum, and cross-spectra between any two spatial points

Before giving the formulas of both turbulence models, a general definition of the spatial coherence between the wind components

The Kaimal spectral and exponential coherence model

In this model, only the spatial coherence for the

The Mann uniform

The magnitude coherence (no square) for spatial separations perpendicular to the longitudinal direction can be obtained by

Summary of the wind evolution models supported in evoTurb.

Wind evolution refers to the decorrelation of turbulence structures (eddies) dependent on evolution time. Following previous research

Currently, only the longitudinal coherence of the

The Kronecker product

The Cholesky decomposition

With the above-mentioned properties, we can extend the Kronecker product of two real, symmetric, positive-definite matrices

The open-access tool evoTurb has been published on GitHub:

YC and FG conceived the concept, derived the 4D wind field simulation method, programmed the open-access tool evoTurb, and prepared the manuscript. FG generated the figures presented in the manuscript. DS supported to derive the 4D simulation method. DS and PWC provided general guidance and reviewed the paper.

The contact author has declared that neither they nor their co-authors have 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 the Joint Graduate Research Training Group Windy Cities funded by the State Ministry of Baden-Wuerttemberg for Sciences, Research and Arts, as well as the European Union's Horizon 2020 research and innovation program under Marie Skłodowska Curie grant agreement no. 858358 (LIKE – Lidar Knowledge Europe). This open-access publication was funded by the University of Stuttgart.

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