Lidar systems have the potential of alleviating structural loads on wind turbines by providing a preview of the incoming wind field to the control system. For a collective pitch controller, the important quantity of interest is the rotor-effective wind speed (REWS). In this study, we present a model of the coherence between the REWS and its estimate from continuous-wave nacelle-mounted lidar systems. The model uses the spectral tensor definition of the Mann model. Model results were compared to field data gathered from a two- and four-beam nacelle lidar mounted on a wind turbine. The comparison shows close agreement for the coherence, and the data fit better to the proposed model than to a model based on the Kaimal turbulence model, which underestimates the coherence. Inflow conditions with larger length scales led to a higher coherence between REWS and lidar estimates than inflow turbulence of smaller length scale. When comparing the two lidar systems, it was shown that the four-beam lidar is able to resolve small turbulent structures with a higher degree of coherence. Further, the advection speed by which the turbulent structures are transported from measurement to rotor plane can be estimated by 10 min averages of the lidar estimation of REWS. The presented model can be used as a computationally efficient tool to optimize the position of the lidar focus points in order to maximize the coherence.

The control system is an integral part of a wind turbine and has substantial influence on its behaviour. Its aim is to maximize the power production while keeping the turbine structural loading within the design limits. In order to decrease the levelized cost of energy, several novel sensors and control strategies have been proposed. One of them is a lidar-assisted pitch controller and one of the first was introduced by

Block diagram of a lidar feedforward collective pitch controller to assist a traditional feedback controller.

For such a controller, the important information about the wind is the rotor-effective wind speed (REWS) (

the contamination from lateral and vertical wind speed components,

the spatial averaging due to the lidar's probe volume,

the scarcity of measurement points in the rotor plane

and the uncertain estimation of the time delay between lidar measurement and disturbance arrival at the rotor.

Previously,

Comparisons between data gathered during field experiments and models were conducted in several studies. In

The integration of lidar measurements into turbine control by suitable controllers and their associated benefits have been the topic of various analyses. FF additions to FB controllers have been studied in, e.g.

To verify simulated performance, field tests have been pursued. In

In this paper, we present a model of the coherence between REWS estimated from turbine and lidar measurements. The model uses the description of a turbulence field according to the model by

In this section, we present a coherence model between nacelle lidar systems and a wind turbine. The theoretical expressions to calculate the variances of turbine and lidar measurements have already been derived in

The fluctuating part of a three-dimensional (3-D) wind field can be represented by the vector field

A lookup table of one-dimensional spectra obtained from the Mann model can be obtained from the second author at jmsq@dtu.dk

. This is where the Kaimal model has an advantage as the one-dimensional spectra are given as simple analytic expressionsIn this study, we have used an estimation of the coherence to evaluate the correlation between models and measurements. Specifically, we were interested in the magnitude-squared coherence between the REWS measured at the turbine and estimated from lidar measurements:

The REWS

To calculate the auto-spectrum of

To estimate

The entire turbine is modelled by a simple drive train model:

Lidar systems are able to measure the frequency shift of light backscattered at aerosols moving with the wind in the atmosphere. This frequency shift is proportional to the velocity of the aerosols, and thus the wind speed can be determined. The measurement of a continuous-wave lidar system can be expressed mathematically as the convolution of the line-of-sight (LOS) component of the wind vector and a weighting function given by the laser light intensity along the laser beam:

The typical setup of a nacelle lidar looking forward is shown in Fig.

In wavenumber domain, the auto-spectrum of the REWS estimate from lidar measurement using Eqs. (

When evaluating Eq. (

Integrating the induction corrections into Eq. (

This represents measuring the wind speed component perpendicular to the turbine rotor even when the turbine is misaligned with the free-stream wind direction. A turbulent structure travelling along the mean wind direction, which is not aligned with the rotor axis if yaw misalignment is present, can arrive at the different position at the rotor than predicted by the model. The model assumes that the turbulent structures travel along the mean wind direction and that the turbine is aligned with that wind direction. However, in the case of small yaw misalignment, the shortcoming of the model can be assumed to be insignificant.

Similarly, the cross-spectrum between the REWS and its estimate from lidar measurements

For the implementation of the model, a C++ code has been created to numerically solve Eqs. (

The adaptive cubature integration scheme was written by Steven G. Johnson and is available on GitHub:

The software can be downloaded free of charge at

The results of the coherence analysis can be found in Fig.

Coherence between the estimation of REWS from the turbine and the lidar. The comparison between the numerical simulations (Simu) and the implementation of the model (Theo) shows very good agreement.

Field measurements have been conducted at DTU's test site at Risø, located at the Roskilde Fjord in Denmark. The site consists of one row of wind turbines intended for testing, and several meteorological masts are installed around the turbines; see Fig.

Digital terrain model (DTM) of the DTU's test site at Risø, where the Vestas V52, its meteorological mast and the Nordtank turbine are indicated. Zone 32 UTM coordinates centred at the Vestas V52 turbine were used. The DTM data were obtained from the Danish Map Supply

For this experiment, two continuous-wave coherent Doppler lidars manufactured by Windar Photonics A/S have been mounted on a Vestas V52 turbine. The lidar systems, a two-beam and a four-beam lidar, are mounted on the nacelle of the turbine and have been staring forward to measure the inflow of the turbine. An illustration and a photo of the four-beam lidar can be seen in Fig.

The specifications for both lidars can be found in Table

Information of lidar setup parameters and measurement periods. The azimuth angle refers to the position on the scanning cone surface with

The Vestas V52 turbine has a diameter of 52 m and a hub height of 44 m with a rated power of 850 kW. It is heavily instrumented with several mechanical strain gauges, in particular a strain gauge setup to measure the torque on the low-speed shaft. Also a meteorological mast is located approximately

The wind rose derived from wind direction and horizontal wind speed of the sonic anemometer measurements during the periods of the experiment is presented in Fig.

To get a clearer picture of the inflow conditions, the data set was grouped into sectors of 30

We separated the following analysis into two regions. The information on the two regions can be found in Table

Measurement sectors and fitted Mann model

The first step in the analysis of the results was to apply appropriate data filters, which reject measurements based on certain criteria. It was necessary to identify periods where the turbine was in a normal power production state. Thus, a lower threshold on the minimum power production (i.e.

The filter applied to the lidar data consisted of a minimum number (

Additionally, inflow from all yaw position except from the wake sector (

Next, the 10 min average REWS estimates of lidar and turbine are compared to the sonic anemometer mounted on the meteorological mast. The comparisons for the two lidar systems and the turbine can be found in Fig.

Comparison of 10 min REWS estimates from of the lidars and the turbine to the meteorological mast's sonic anemometer. The data were taken from periods when the turbine was operational and facing the mast.

Corresponding comparisons are performed between the REWS estimated from the lidar systems and the turbine, respectively. The correlation plots can be found in Fig.

Comparison of 10 min REWS estimates between the lidars and the turbine. Data from the wake sector and non-operational periods of the turbine were removed. All units are m s

For illustrative purposes, the next plots in Fig.

The effect of probing two versus four focus locations is now studied in wavenumber domain by comparing the squared coherences. The experimental data are also compared to two models: the model based on the Mann turbulence model introduced in Sect.

Time series example of lidar and turbine estimates of REWS. A high degree of similarity between the signals can be seen. Also, the preview ability of the lidar system is evident.

The coherence analysis for region 1 can be found in Fig.

Squared coherence between the REWS estimation of the lidar and the turbine for region 1. The two models are also included in the plot.

The results for region 2 are presented in Fig.

Equivalently to region 1, the Mann turbulence model fits very well to the measured data. There are however some slight deviations for both lidars in the region of 0.01 to 0.1 rad m

Squared coherence between the REWS estimation of the lidar and the turbine for region 2. The two models are also included in the plot.

To quantify the error between measured and model-derived coherence, we use the root-mean-squared error (RMSE) calculated from the data presented in Figs.

Comparison of the RMSE between measured and model-derived coherence for the two- and four-beam lidars in regions 1 and 2.

It can be seen that the errors between measured data and the model based on the Mann turbulence model are consistently lower than those of the model based on the Kaimal turbulence model. For the two-beam lidar, the errors are approximately twice as high, while the difference is slightly less for the four-beam lidar. Also, in region 2, the model based on the Kaimal turbulence model performs worse, which also can be seen when comparing Figs.

Next, to better compare the measured coherence of the two- and four-beam lidar systems for regions 1 and 2, all measured coherences have been plotted in one figure. It can be seen that the coherence of the four-beam system drops at larger wavenumbers than the two-beam lidar due to probing the incoming wind at two additional positions. Differences between regions 1 and 2 can also be observed. For both lidars, the coherence measured in region 2 is higher than that in region 1. This can be explained by the larger turbulence length scale, which implies that there are more large-scale fluctuations in the flow. These large-scale fluctuations can be better resolved by both lidar systems.

Comparison of the measured coherence of the two- and four-beam lidars for regions 1 and 2.

Further, it is observed that even without the inclusion of the evolution of turbulence the model is able to predict the coherence very accurately. This implies that effect of disregarding turbulence evolution can be neglected. For bigger turbines, where larger focus distances are required, turbulence evolution might become more significant.

Next, the delay between lidar and turbine estimations of REWS is analysed. The delay stems from the perpendicular distance between the rotor plane and the measurement plane

Since the experiment was performed with very good time synchronization (with a maximum time delay of a few

The result can be seen in Fig.

In general, towards high wind speeds, the available preview time provided by the lidar becomes smaller. The required preview time for the filtering is also shown for the two lidar setups. Low-pass filtering the lidar system is crucial to reject high-frequency fluctuations that are sensed by the lidar but not experienced by the rotor and if not filtered would cause detrimental pitch actuation. In this study, a first-order Butterworth filter is used following the approach presented in

Delay time analysis results for the two-beam and four-beam lidars. Delays and advection speeds are calculated based on 10 min averages. The required filter time of a first-order Butterworth filter is also shown.

In this study, we presented a model of the coherence between REWS estimated from turbine and lidar measurements. The underlying model of the 3-D turbulent field is the Mann spectral tensor and allows the direct calculation of auto- and cross-spectra of REWS estimations for lidar and turbine. It is compared to field data obtained from two continuous-wave lidar systems mounted on top of the nacelle of a wind turbine. To retrieve the turbulence model parameters, measured spectra from a sonic anemometer have been fitted to the spectral tensor. The comparisons of squared coherence show that the presented model fits the field data better than previously used models, which are based on the Kaimal model defined in IEC standard. Thus, this study gives confidence that the proposed model can accurately represent the important lidar properties and it can be used to optimize the lidar focus point positions to maximize the coherence between lidar and turbine. A common parameter used in the lidar optimization is the wavenumber where the coherence drops to a value of 0.5

We have found that larger turbulence length scales led to higher coherences between REWS estimated of turbine and lidar compared to inflow turbulence of smaller length scale. It was also shown that the smallest detectable eddy size can by reduced by almost 50 % when using the four-beam compared to the two-beam system. Further, the advection speed by which the turbulent structures are transported from measurement to rotor plane can be estimated from 10 min averages of REWS from lidar measurements. This is important information for the correct timing of the measured fluctuations of the lidar systems. In the case of the four-beam lidar, there is enough preview provided by the lidar to perform the necessary low-pass filtering, while the two-beam lidar lacks preview time for filtering for high wind speeds.

Since some of the physical mechanisms have not been modelled, future work includes additions to both the lidar and turbulence modelling. First of all, the evolution of turbulence as it travels from measurement to rotor plane has been neglected. An amendment of turbulence evolution to the Mann model has been proposed in

The presented model in the current form can be applied to nacelle-mounted continuous-wave lidars and by modifying the spatial averaging of the lidar it can be extended to nacelle-mounted pulsed lidars as well. To cover spinner-mounted lidar systems, the rotational sampling effect of the lidar as it rotates with the rotor needs to be modelled.

The data are not publicly available since they contain confidential information owned by Windar Photonics A/S.

For the calculation of the

Calculated

In this appendix, an example of the flow field around a rotor operating at the aerodynamic optimum according to the model of

Example of the flow speed reduction and diversion around the rotor of a wind turbine. The red lines indicate the laser beam and the red dots show the focus points.

In Sect.

Average spectra and the corresponding Mann model fits from Table

DPH performed the research work and prepared the manuscript. JM conceived the research plan, supervised the research work and the manuscript preparation.

The authors declare that they have no conflict of interest.

This study was supported by Innovationsfonden Danmark in the form of an industrial PhD stipend. The authors also want to acknowledge the research project “Cost-efficient lidar for pitch control”, funded by Energiteknologiske Udviklings- og Demonstrationsprogram, and specifically Hector Villanueva for providing the turbine data in the form of a database, Qi Hu for designing and constructing the four-beam lidar, Ebba Dellwik for the coordination within the project and the technicians at DTU Wind Energy for conducting the experiments on the Vestas V52 turbine at the Risø test site. The authors also want to thank Pedro Santos and Ebba Dellwik for their comments and suggestions on the manuscript.

The work of Dominique P. Held was partly funded by Windar Photonics A/S in the form of an industrial PhD stipend (project no. 5016-00182).

This paper was edited by Carlo L. Bottasso and reviewed by three anonymous referees.