The most intermittent behaviour of atmospheric turbulence is found for very short timescales. Based on a concatenation of conditional probability density functions (cpdf's) of nested wind speed increments, inspired by a Markov process in scale, we derive a short-time predictor for wind speed fluctuations around a non-stationary mean value and with a corresponding non-stationary variance. As a new quality this short-time predictor enables a multipoint reconstruction of wind data. The used cpdf's are (1) directly estimated from historical data from the offshore research platform FINO1 and (2) obtained from numerical solutions of a family of Fokker–Planck equations in the scale domain. The explicit forms of the Fokker–Planck equations are estimated from the given wind data. A good agreement between the statistics of the generated and measured synthetic wind speed fluctuations is found even on timescales below 1 s. This shows that our approach captures the short-time dynamics of real wind speed fluctuations very well. Our method is extended by taking the non-stationarity of the mean wind speed and its non-stationary variance into account.

The transition of our energy system, formerly strongly relying on gas and coal, to a decarbonised one, mainly based on wind, solar and hydropower, is still ongoing work, but great progress has been made. From 2005 to 2017 the share in installed capacity of wind (solar) has been increased from 6 % (0.3 %) to 18 % (11.5 %) in the European Union

To aid the design of our future energy systems, for example to size the needed energy storage or the power generation capacity of conventional power plants, much work has been done in the field of long-term wind speed and power modelling, utilising Markov chain models. Whereas simple first-order Markov chain models cannot grasp the characteristics of long-term correlations of wind speeds

Despite their dramatic effect of long-range correlation and fluctuations of wind speeds on the power generation (and thus the grid stability), wind speed fluctuations are known to be most intermittent on short timescales

The knowledge of full three-dimensional wind fields for all three velocity components and the pressure in all details would be desirable. The lack of basic understanding, the impracticability of handling such huge data sets and the complexity of the wind energy conversion process often lead to the demand for simplified models for wind speed. Common approaches for the design of wind turbines are the so-called Mann uniform shear and the Kaimal spectral and exponential coherence model

Within this work we propose a novel stochastic generator of one-dimensional wind speed fluctuations with a sampling interval of 0.1 s. One main novelty is that we show how to grasp by this model multipoint statistics of wind structures in time. While commonly applied methods, like spectral analysis and two-point correlations, limit themselves to two-point statistics, here we extend the methodology to more than two points in time. We obtain generalised correlations between multiple points in time, in terms of probability density functions (pdf's), for the occurrence of a whole sequence of wind speeds. Those pdf's we denote multipoint pdf's, and they constitute the basic concept of our approach. Such a stochastic multipoint approach should in principle be able to grasp wind structures like gusts as well as clustering of wind fluctuations. The method was initially developed by

We will continue as follows. In the first part we discuss the method for a subset of wind data characterised by its mean wind speed and its standard deviation. For such data it is shown how, arising from a Langevin process in scale, a predictor for the upcoming wind speed fluctuation around a mean value can be derived by a nesting of conditional probability density functions. Afterwards we check for Markovian properties of the wind speed fluctuations in scale and set up a Fokker–Planck equation, corresponding to the Langevin process in scale, and we show how it contributes to the improvement of our stochastic prediction method. Finally, in the second part, the non-stationary mean wind speed and its non-stationary variance are incorporated into our approach to achieve more realistic wind speed time series.

In this section we present the stochastic framework used for our multipoint reconstruction scheme. As a simplification we start this discussion for blocks of wind data

Since we assume wind speeds to emerge from a turbulent cascade, increments will play a key role in our method. Having a time series of wind speeds

As a further remark we note that although we consider in this work time series of wind speed, we often talk of multipoint statistics.

Our idea is to predict a wind speed

Now we link Eq. (

It is known that the evolution of cpdf's of a Markov process can be described by the famous Kramers–Moyal expansion

As can be seen, the FPE can be used to determine the factorised pdf's from Eq. (

Next we check if wind speed data are suitable for the reconstruction method just described. According to the right-hand side (rhs) of Eq. (

Comparison of single cpdf's

Comparison of the double cpdf's

As was mentioned in Sect.

This way we obtain estimations of the drift and diffusion functions

To match the functional shape of the estimated

Exemplary estimations of the drift and diffusion functions

We set for

Furthermore we check the validity of the Pawula theorem, requiring

Exemplary estimations of the second and fourth Kramers–Moyal coefficients

Next, we check if the parameterisation of the FPE, given by Eqs. (

Comparing cpdf's estimated directly from the data and from the numerical solution, we see (Fig.

Isoline plots of a cpdf

Alternatively to the presented approach to obtain the cpdf's from numerical solutions of the FPE, it is of course possible and much less cumbersome to estimate them directly from observational data. (Note that due to the use of the FPE, the obtained pdf's are less noisy and extend to large values as seen in Fig.

Comparison of original wind speed time series

To start the reconstruction scheme, we provided a short piece of the original time series of

Comparison of the marginal increment pdf's computed from the empirical data (black), the simulated data using the directly estimated pdf's (red) and the simulated data using the cpdf's obtained from numerical solution of the FPE (blue). The scales range from

To confirm this visual impression in a quantitative way, we compare the increment pdf's

As shown in Fig.

A striking difference between the increment pdf's from the stochastic simulation can be noted. Whereas the tails of the original pdf's are systematically underestimated by the reconstruction using the directly estimated pdf's, the reconstruction from the numerically obtained cpdf's is able to keep track of the tails of the original pdf. This stems from the fact, mentioned above, that by estimating a pdf from observational data one in general underestimates the outer tails, since there are only a few measured points available. Considering the estimation of cpdf's, the estimation error of course worsens, which is even more severe when additional conditioning is applied, like in our case in terms of the additional condition on

The tails of cpdf's

Certainly this approach indirectly suffers from the limited number of observations as well, as the estimation of the functions

But we see from Fig.

From Fig.

Evolution of

In the preceding part for the reconstruction scheme, block-wise normalised wind speeds with a window length of 1 min were used. These blocks were defined by common mean wind speed

There are different methods to generate more general non-stationary wind data. Knowing the slow variation in

A third possibility is a self-adaptive procedure which we show here. Instead of using given values of

We presented a stochastic approach based on multipoint statistics to generate surrogate short-time wind speed fluctuations with stochastic processes. Note these stochastic processes can be estimated self-consistently from given data. By using the normalised wind speeds

It was shown that our method works well in describing the dynamics of block-wise normalised wind speeds

For timescales larger than the response times of wind turbines, the turbines operate with fully adapted control systems in a stationary state. To estimate effects, like loads, of such stationary states, the temporal order of the states becomes unimportant. It is sufficient to know how often which wind situation emerges. Thus the knowledge of the valid Weibull distribution

Finally we emphasise that the presented stochastic multipoint approach to small-scale wind speed fluctuations should encompass automatically extreme short-term wind fluctuations, commonly added to wind investigations in terms of standard or multi-year gusts. These methods can be applied easily to other wind quantities like the temporal behaviour of shears, or wind veers, eventually combined in higher-dimensional stochastic processes

To link Eq. (

Here we present the polynomial coefficients used to parameterise the first and second Kramers–Moyal coefficients

Coefficients for the drift function

And for the diffusion function

Data sets are available upon request by contacting the correspondence author.

CB did the preliminary analysis of the data, implemented the algorithms for the multiscale reconstruction and parametrization of the FPE, conducted all simulations, and wrote the main part of the manuscript. JP had the initial idea, and JP and MW supervised the work and helped in preparing the manuscript.

The authors declare that they have no conflict of interest.

The authors acknowledge helpful discussions with André Fuchs and Hauke Hähne.

This research has been supported by the VolkswagenStiftung (grant no. 8482) and by the Ministry for Science and Culture of the German Federal State of Lower Saxony (grant no. ZN3024, nieders. Vorab to M. W.).

This paper was edited by Horia Hangan and reviewed by two anonymous referees.