The present study introduces a new method to characterize ramp-like wind speed fluctuations, including coherent gusts. This method combines two well-known methods: the continuous wavelet transform and the fitting of an analytical form based on the error function. The method provides estimation of ramp amplitude and rise time, and is herein used to statistically characterize ramp-like fluctuations at three different measurement sites. Together with the corresponding amplitude of wind direction change, the ramp amplitude and rise time variables are compared to the extreme coherent gust with direction change from the IEC wind turbine safety standard. From the comparison we find that the observed amplitudes of the estimated fluctuations do not exceed the one prescribed in the standard, but the rise time is generally much longer, on average around 200 s. The direction change does however exceed the one prescribed in the standard several times, but for those events the rise time is a minute or more. We also demonstrate a general pattern in the statistical behaviour of the characteristic ramp variables, noting their wind speed dependence, or lack thereof, at the different sites.

The IEC wind turbine safety standard prescribes various models of extreme wind conditions that a wind turbine must withstand during its operational lifetime

With the increasing rotor size of modern wind turbines, resent research has focused on how the gust models in the IEC standard are unrealistically represented by a uniform wave

There are however many studies in the field of atmospheric science that investigate large coherent structures in turbulent flow (e.g.

In this study we focus on large-scale, high-amplitude (extreme) fluctuations, which are coherent across the rotor of any multi-megawatt wind turbine. We examine data from three sites with different terrain types and characterize the fluctuations. In order to characterize the amplitude and rise time of the investigated fluctuations, we provide a new combination of two well-known methods: the continuous wavelet transform and the fitting of an idealized ramp function (based on the error function), which is inspired by detection of atmospheric boundary-layer depth

The measurements used for the characterization of the ramp-like events come from three different sites. The locations of the measurement sites may be seen in Fig.

A map of Denmark and southern Sweden showing the locations of the measurement sites.

The Høvsøre National Test Centre for Wind Turbines is located on the western coast of Jutland, approximately 1.7 km east of the coastline. The site is in a coastal agricultural area where the terrain is nearly flat.

Several masts with measurement instruments are located at the site, which has been in operation since 2004. In the current analysis we use measurements from a light mast with cup anemometers and wind vanes installed at 60, 100, and 160 m heights.
The light mast is located between two of the test wind turbines which are separated by approximately 300 m in the north–south direction.
The dominating wind direction is from north-west, the annual average 10 min wind speed at the light mast is

The Østerild National Test Centre for Large Wind Turbines is located in a forested area in northern Jutland. The distance to the coast is approximately 4 km to the north and 20 km to the west.
The site has two 250 m tall light masts equipped with sonic anemometers at 37, 103, 175, and 241 m. In this analysis we use measurements from the southern mast, where the terrain around the mast is flat and the surrounding forest has a canopy height between 10 and 20 m. To the west of the mast there is a narrow clearing of the forest with a grass field. The clearing is approximately 1 km long in the east–west direction and 200 m wide in the north–south direction.
The mast is located approximately 300 m south-west of a row of seven wind turbines aligned in the north–south direction.
At the southern light mast, the annual average 10 min wind speed is

The Ryningsnäs measurement site is located approximately 30 km inland from the south-eastern coast of Sweden. The terrain is forested and generally flat. The forest has a 200 km fetch in the westerly direction and the tree height around the site is between 20 and 25 m. There is a 138 m tall meteorological mast equipped with sonic anemometers at 40, 59, 80, 98, 120, and 138 m measuring at 20 Hz sampling frequency. In this analysis we use the measurements at 59, 98, and 138 m heights from a period between November 2010 and December 2011. There are two wind turbines approximately 200 m from the mast, one in the southerly direction and the other in the north-easterly direction. The annual average 10 min wind speed is

In this section we go through the steps of selecting and characterizing the ramp-like coherent structures. There are three steps in the procedure.

Identify events of extreme variance, indicating large-scale fluctuations, and acquire 30 min wind speed measurements for each event.

Estimate the timescale and position in time (timing) of the dominating fluctuation using wavelet transform.

Characterize the amplitude and rise time of the dominating fluctuation by fitting an idealized ramp function to a subset of the wind speed signal, whose timing and scale are found by the wavelet transform.

Here we select the ramp events by comparing two different data sets: one where the 10 min standard deviation is calculated from the raw measurements (

The filtering is performed with a second-order Butterworth filter where the cut-off frequency is chosen as

In order to identify where the 10 min standard deviation is reduced the most by filtering, we calculate the ratios of

Here

An example of an extreme-variance event may be seen in Fig.

This example is taken from the light mast in Høvsøre at 100 m. The 10 min standard deviation of the raw measurements is 2.66 m s

Wind speed measurements from 100 m at Høvsøre, raw measurements (blue line), and high-pass filtered measurements (dashed black line).

The continuous wavelet transform (CWT) unfolds a signal in both frequency and time and provides an efficient way to identify and localize abrupt changes or transients in non-stationary time series.
The CWT is often used to identify and characterize coherent structures in turbulent flow (e.g.

The CWT is formally defined as the inner product of a function

The choice of analysing wavelet influences the results of the wavelet transform, since it reflects characteristics of the wavelet. We have therefore chosen a wavelet that includes features similar to those we look for in the signal, i.e. one dominating increase at the centre of the wavelet function.
The analysing wavelet chosen here is the first derivative of a Gaussian (DOG1) wavelet

The wavelet transform is performed using Python package PyWavelets

Figure

The continuous wavelet transform of a ramp-like coherent structure.

The definition of the idealized ramp function is borrowed from

For the optimization fitting procedure we employed the SciPy curve_fit function.

.Three examples of the idealized ramp function fit to the wind speed measurements. Measurements from

Figure

A brief summation of the detection: a subset of extreme-variance events is found. The CWT is performed on each event and the timing and scale of the ramp-like wind speed increase are estimated. The scale (

As we want to compare the wind speed ramps with the ECD load case of the IEC standard, we investigate the direction change during the ramps.
Here we use the directional data at

The direction change during the ramp-like wind speed increase is determined as the difference between the maximum value and the minimum value of the moving average.

An example of a ramp event at Østerild is shown in Fig.

The amplitudes and rise times are characterized for each measurement height. Afterwards the values are averaged over the three different heights to give the characteristic rise time and amplitude for each event.

The idealized ramp function fit to Østerild measurements at three different heights and the corresponding direction change.

The extreme coherent gust with direction change (ECD) is modelled with an amplitude of

The extreme coherent gust with direction change from the IEC wind turbine safety standard.

In this section we look at the amplitudes, rise times, and direction change in the detected events and how these variables are distributed.
Selecting the 0.1 % highest ratios of

The estimated

The distributions of detected amplitudes (

The number of analysed events and average estimated variables from each site.

The sample means and the corresponding standard deviations of

We see that the average (

The average

Figure

The detected amplitudes (

The CWT is ideal for finding abrupt changes in a wind speed signal and can provide useful information on different scales of the flow. Here we use the wavelet transform to provide an objective estimate of the timescale of the ramp-like wind speed increase as well as the precise timing in the signal. To obtain characteristics of the amplitude and rise time of these fluctuations, we need an additional step, which is inspired by mixed-layer height detection performed by fitting an idealized profile to backscatter measurements.

The main difference between backscatter profiles and wind speed time series is that the wind speed continuously fluctuates through time and the period of the coherent structure we investigate is finite. This difference is why the wavelet analysis is important prior to the fitting of the idealized function, where the limited period of the ramp and the timing is identified. We found the optimal period for the fitting to be 3 times the scale dilation (

The first step in the selection, choosing high-variance events, is used for two purposes: first, to ensure that the selected ramp-like fluctuations are associated with scales that are large enough to cover any rotor of a multi-megawatt wind turbine. We have seen in a previous study that these fluctuations occur approximately simultaneously at two different measurement masts in Høvsøre that are separated by 400 m transverse to the mean wind direction

Second, by choosing a subset of events, we avoid performing a CWT on the whole data set of high-frequency measurements, which is computationally demanding on a 10-year data set like the one from Høvsøre.
If a CWT is performed on the whole data set, an extra step would be needed in the analysis to decide whether a structure is coherent or not, e.g. to apply a threshold on the scale-averaged wavelet coefficients or wavelet spectrum (e.g.

The main difference between the observed fluctuations analysed in the current study and the classic ECD (investigated in

We observe that the average amplitudes of ramp-like fluctuations

The rise time of the ramp-like fluctuations is generally much higher than that of the ECD. But the range is large: e.g. at Høvsøre the rise time ranges over 2 orders of magnitude (from 9 to 952 s). The rise time of the extreme direction events is of the order of a minute or more. Although these extreme direction events generally have a longer rise time than the defined ECD, they could readily be considered for load simulation purposes. The reason is that a wind turbine reacts much more slowly to changes in wind direction than to changes in wind speed. The yaw speed of a wind turbine is typically less than 0.5

We observe ramp events that either have an amplitude, or rise time, or direction change of the same order of magnitude as the ECD. However, no single event is comparable to the ECD on all three variables at once. In order to predict an extreme event considering all three variables simultaneously, one would need a multivariate distribution model including the parameter distributions. That way it would be possible to model the probability of different positions in the three-parameter space and extrapolate to desired return periods.

The combination of the wavelet transform and the fitting of an idealized ramp function is a new and efficient way to characterize extreme wind speed ramps. The characterization provides variables that are relevant for wind energy, particularly for wind turbine load simulations, probabilistic design, and wind turbine safety standards.

We use measurements from three measurement sites in different terrain to calculate statistics of the amplitudes, direction change, and rise time of extreme ramp-like fluctuations, and also compare the estimated variables with the ECD load case of the IEC standard.
Here we find the following.

The amplitudes of these coherent structures do not exceed the amplitude of the ECD (using 10, 3, and 1 year of data respectively).

The amplitudes show no clear wind speed dependence at Høvsøre and Østerild, but at Ryningsnäs the amplitudes increase with increasing wind speed.

The direction change may exceed that of the ECD, but for those events the rise time is a minute or more.

Future related work includes further analysis of ramp events, in particular, using a multivariate distribution model based on the marginal distributions of the ramp variables to estimate ramp events with a 50-year return period.

The high-frequency measurements used in this study are stored in an SQL database at DTU that is not publicly accessible. However, we provide a subset of the data with six ramp events of 30 min duration. A Python script may be applied to these data to perform the ramp characterization described in the paper. This Python script and data are available at

ÁH provided the detection method, performed the data analysis, and made the figures. MK provided guidance and comments. ÁH prepared the manuscript with contributions from MK.

The authors declare that they have no conflict of interest.

This work was part of Ásta Hannesdóttir's PhD at DTU Wind Energy under the supervision of Mark Kelly.

This paper was edited by Ola Carlson and reviewed by Anders Wickström and Matti Koivisto.