That is, it generates time series of wind speeds at a set of points on a surface, say the plane of the rotor of a wind turbine.
Then, the time series are obtained by inverse fast Fourier transforms.
In its original form, the Sandia method only simulates the u-component of the wind; that is, the wind was modelled as propagating in a direction perpendicular to the plane of the rotor.
Work carried out by NREL, specifically Kelley, suggested that a considerable amount of turbulent energy existed in the v-component (the v-component is parallel to both the plane of the rotor and the Earth).
As such, the Sandia method was upgraded such that it included the v-component and w-component.
Further upgrades have been performed such that the wind profile exhibits cross-axis correlation (turbulent fluctuations in one component being somehow connected to turbulent fluctuations in another).
Several models of frequency domain representations of point wind speeds have been developed: the von Kármán wind turbulence model and Dryden Wind Turbulence Model are examples of such.
A spectrum in its original form is a continuous function.
However, computer programmes operate on discrete functions.
is the size of the step between consecutive frequencies being considered.
When generating a time series of wind speeds for a set of points across a surface, coherence needs to be taken into account.
In addition, one would expect low frequency components of the wind speeds at points A and B to show more correlation than high frequency components.
As such, many coherence functions have been proposed: Davenport, Solari, etc.
The Solari coherence spectrum is provided as an example: where
To cover all relationships between all points, the coherence function must be an
As such, in the following section, a power spectrum refers to the value of the power spectrum at a given frequency and not the full set of values across the frequency range being used.
With the power spectra, the spectral matrix can be formed.
The main diagonal of the spectral matrix contains the previously defined spectra for all
The off-main diagonal elements contain all the cross spectra between the points.
The cross spectra are determined by the following function: Due to the symmetry of the coherence matrix, only
This property can be exploited to lighten memory requirements when writing a programme to simulate the Sandia method.
is ultimately needed to obtain the complex Fourier co-efficients of the Fourier transforms of the time series of the wind speeds at all the points on the surface.
Note - if the Fourier transform of a time domain function,
; for multiple time domain functions, the Fourier co-efficients can be stored in a matrix, which then means that the above equation is applicable.
Obviously, there are an infinite number of solutions to the above expression; consequently, the assumption that
is a lower triangular matrix is made such that only one solution exists.
matrix can be thought of as the weighting factors for the linear combination of N independent, unit-magnitude, white-noise inputs that will yield N correlated outputs with the correct spectral matrix.
, as shown below: The column vector gives the Fourier co-efficients for all points on the grid at a given frequency.
This is then built up into a two dimensional matrix which covers the complex Fourier co-efficients for all points across all frequencies.
Then, an inverse fast Fourier transform is performed to get the time series.