Emerging stochastic analysis methods are of potentially great benefit for wind turbine power output and loads analysis. Instead of requiring multiple (e.g. 10 min) deterministic simulations, a stochastic approach can enable a quick assessment of a turbine's long-term performance (e.g. 20-year fatigue and extreme loads) from a single stochastic simulation. However, even though the wind inflow is often described as a stochastic process, the common spectral formulation requires a large number of random variables to be considered. This is a major issue for stochastic methods, which suffer from the “curse of dimensionality” leading to a steep performance drop with an increasing number of random variables contained in the governing equations. In this paper a novel engineering wind model is developed which reduces the number of random variables by 4–5 orders of magnitude compared to typical models while retaining proper spatial correlation of wind speed sample points across a wind turbine rotor. The new model can then be used as input to direct stochastic simulations models under development. A comparison of the new method to results from the commercial code TurbSim and a custom implementation of the standard spectral model shows that for a 3-D wind field, the most important properties (cross-correlation, covariance, auto- and cross-spectrum) are conserved adequately by the proposed reduced-order method.

Comparison of the solution processes in a pure deterministic (A), a deterministic–statistic (B), and a stochastic framework (C).

Engineering design tasks frequently face uncertain or random model parameters
(e.g. imprecise component geometries), system properties (e.g. tolerances on
manufacturing quality), and/or boundary conditions (e.g. varying wind
conditions). In a deterministic modelling framework, the analysis of such
uncertain systems produces one specific solution for each realization of the
random quantity. A “realization” (also referred to as one “sample”) is one
specific observation of the random quantity, for example a specific solution
for one specific geometry or one specific set of inflow conditions. In a
numerical experiment, a realization is usually obtained from one specific
random seed. However, through this process the stochastic dimension of the
problem at hand is either ignored entirely, by analyzing the most likely case
only (the purely deterministic approach), or it requires multiple parallel
solutions to asses the statistics of the results a posteriori, for example
via extreme value, sensitivity analysis, or Monte Carlo simulation. Often the
first two options are insufficient, and the latter is computationally too
expensive. To solve this dilemma the focus of recent research has lately
moved towards stochastic analyses and uncertainty quantification

In wind turbine engineering, the driving force is the turbulent atmospheric
wind which is commonly described as a stochastic field derived from turbulent
wind models developed around stochastic 10 min mean wind speed
distributions. This naturally invites the use of stochastic methods to asses
extreme and fatigue loads, annual power production, power fluctuations, etc.,
in a stochastic sense and thus exploit the advantages of stochastic methods.
However, wind turbine design and analysis are usually carried out in a
deterministic fashion or at best as a Monte Carlo-like set of several
subsequent deterministic solutions (path A and B in Fig.

A direct stochastic treatment of the wind loading (path C in Fig.

Recently, progress has been made towards stochastic analysis of wind
turbines. For example, results have been shown for an aeroelastic analysis
with one uncertain system parameter – stiffness or damping

As turbulent wind is already represented as a stochastic field in many common
wind models, a transition from a deterministic aerodynamic model for specific
wind realizations to a stochastic model yielding the whole stochastic load
ensemble at once seems an obvious step. However, this step comes with a
simple, yet fundamental challenge: current wind models, even simple spectral
models, rely on a large number of random variables to set the wind sample's
phase angles. Since realizations of large sets of random variables can be
generated very quickly, this is not a problem for deterministic load
analyses. However, the computational cost of stochastic analysis methods
increases dramatically with the number of random variables included, a fact
commonly known as the “curse of dimensionality”

To address this problem we reformulate an industry standard wind model into a
reduced-order engineering model. The aim of our work is to develop a wind
model that can generate a realistic wind field with appropriate (long-term)
dynamic properties from considerably less random variables than the current
models. In the last 3 decades numerous turbulent wind models have been
proposed.

In the present study, we focus on a formulation for the IEC standard
spectral wind description

The following sections will first briefly review Veers' model to set the
stage for the proposed modifications. Subsequently, the new reduced-order
wind model is introduced, and finally results are presented, which confirm
that key statistical properties (cross-correlation, covariance, auto- and
cross-spectrum) are conserved by the new model. The paper concludes by giving
direction for continued work on integrating the wind model into a turbine
simulation and on refinements with other turbulent wind descriptions. To not
overload this paper, the focus is solely on the details and validation of the
stochastic wind inflow model itself. Interested readers should refer to

Veers' method represents the established method for synthesizing turbulent
wind

In Veers' spectral method, the wind speed time series

Following Veers' method

This method works well to generate multiple (deterministic) wind speed data
sets at many points. However, as already noted by Veers in his original
publication

Raw wind spectra from a single wind speed sample; no averaging.
Kaimal: the analytic spectrum; Veers: sample of the spectrum resulting from
Eq. (

To arrive at a reduced-order model, we follow a two-step process. First is a
reduction in the number of frequencies necessary for the spectral composition
of the wind speed time series at a single point in space, which yields a
reduction in the number of random phase angles associated with each
frequency. This frequency reduction has been done before. For example,

In Veers' model the phase angle matrix

While the phase angles at each point (the columns

The temporal variability (in the columns of

The spatial variability (in the rows of

For each point, all elements in each column vector

Schematic of random phase angle vectors and deterministic phase increments.

To obtain a reduced-order model which requires fewer random variables, we
propose splitting the complex Fourier coefficients

Assuming

It is important to note that focusing on the temporal part does not mean that
each realization of the reduced-order wind field will exhibit the same
spatial structure of gusts and lulls, i.e. that a gust at point

Inserting Eqs. (

Substituting

In contrast to Veers' original model, where

What remains is to obtain the phase angle increments

In the following, we will take a closer look at statistical metrics of the
synthetic reduced-order wind field. As mentioned earlier our goal is not to
develop a more physically faithful wind model but rather to reduce the
number of random variables required while retaining a similar fidelity to the
methods currently in use. TurbSim

The first is our implementation of Veers' model, which allowed us to freely
choose the number of frequencies at each data point and the frequency
binning. Following Veers this implementation relies on an inverse discrete
Fourier transform with random phase angles at each frequency bin. This model
was validated directly against TurbSim. If many frequencies are used and
identical phase angles are enforced perfect agreement of the resulting data
set was found as expected. As shown by

Table

Comparison of random numbers used in different wind models for a common grid size.

Schematic of grid points of wind speed data (minimal test case). We
arbitrarily chose the right-hand top point (

Three 50

Three realizations of wind speed time series at three points
generated from the new reduced-order model with fixed phase increments
(Veers

Since the goal here is not to calculate wind turbine loads, but to merely
asses the quality of the reduced-order wind model, we used a dummy wind field
generated on

Figure

Figure

Beyond this qualitative visual comparison of the wind speed time series, the remainder of this section will show that the phase increment model produces the same statistics as Veers' original model (with only 20 frequencies) as well as the full TurbSim model (with the full set of frequencies) for the most important metrics.

Figure

As can be seen from Fig.

Wind speed cross-correlation for two point pairs generated from
different models: a close pair (

Further investigation with the pair

Now we look at the covariance as a function of the distance between two
points and compare data from TurbSim to Veers' model
(Veers

From Fig.

Wind speed power spectra are again obtained as average from 100 realizations
(from 100 different random seeds). However, this time 6000

Wind speed covariance for points different distances apart.

Wind speed cross power spectral density for three point pairs from
different models, together with the analytic results
(Eq.

The wind speed auto-spectrum is included in Fig.

Again, the phase increment model (Veers

To further assess the validity of the reduced-order wind model, loads were
calculated for one single blade on a three-bladed

Blade thrust load probability distribution from a blade element momentum (BEM) model based on wind fields generated with either TurbSim or from the reduced-order Veers' model with constant phase increments.

As shown in this section the phase increments wind model presented in
Sect.

The results from Sect.

Stochastic analysis and uncertainty quantification are very active fields of research in engineering with the developed methods increasingly adopted by industry. To enable practitioners to apply these methods to wind turbine aerodynamics and more generally wind loading analysis on various structures, we presented a new method, which significantly reduces the number of random variables used in the wind model. This reduction is critical because the computational effort of the common stochastic solutions is very sensitive to the number of random variables involved.

The model introduced here employs a separation of the temporal (correlation in time) and spatial (coherence in space) part of the random dimension of turbulent wind. While the temporal part is still determined from random variables, the spatial part is collapsed into deterministic phase increments. Thus, the number of random variables is reduced by several orders of magnitude compared to the commonly used model developed by Veers and implemented in TurbSim, currently the (de facto) standard tool for synthetic wind generation. A comparison of the most important stochastic metrics (cross-correlation, covariance, auto- and cross-spectrum) showed that the reduced-order model based on phase increments still reproduces these metrics as accurately as Veers' equations or TurbSim. Moreover, preliminary results were presented, which indicate that the reduced-order wind model based on phase increments also preserves wind turbine blade loads well. A detailed study quantifying the impact of using deterministic phase increments on the overall statistics of wind turbine loads is yet to be carried out. Subsequent to the implementation of this reduced-order wind model in a full wind turbine simulator, which is the focus of ongoing work, these ultimate questions can be addressed.

Underlying data are obtained from TurbSim and custom implementations of the spectral wind models. TurbSim input files as well as copies of the implementation of the spectral models may be obtained from the authors upon request.

The authors declare that they have no conflict of interest.

We gratefully acknowledge the funding provided for this study by the Pacific Institute for Climate Solutions (PICS), the German Academic Exchange Service (DAAD), and the Natural Sciences and Engineering Research Council of Canada (NSERC). Edited by: Horia Hangan Reviewed by: two anonymous referees