2.1. Turbulent Wind Field Model
In a turbulent wind field, the wind speed can be decomposed into average wind speed and fluctuating wind speed. Thus, the simulated three-dimensional wind speed at any simulation point is the linear superposition of the average wind speed and fluctuating wind speed in longitudinal, transverse, and vertical directions [
2], which can be calculated by:
where,
denotes the average wind speed,
denotes the fluctuating wind speed of a simulated point,
denotes simulated height, and
denotes time.
When setting the average wind speed, the influence of wind shear must be considered for large wind turbines whose installation height is usually higher than 100 m. Commonly used wind shear models include exponential model and logarithmic model, and the stable modified logarithmic model was used in this paper [
8], which can be calculated by:
where,
denotes friction velocity, it can be represented as
,
, which means the reference mean wind speed;
denotes Von Karman constant, under neutral atmospheric conditions,
;
denotes terrain roughness parameter, which can be calculated as
, where
when the terrain type is an offshore area,
denotes the acceleration of gravity; and
denotes the stability function when in the neutral condition, its value is 0.
Pulsating wind speed is regarded as a stationary Gaussian random process. According to the stochastic process theory, the power spectral density function combines with the coherence function, which is set to describe the wind speed correlation between two different points that may generate different wind speeds during a period in a stochastic wind field while a smaller separation distance of any two points will bring a greater correlation, is used to simulate the pulsating wind speed time history [
15].
2.2. Basic Algorithm for Simulating Fluctuating Wind Speed Time History
Considering the simulation process at a certain point in the turbulent wind field in space, the basic algorithm for simulating and solving the one-dimensional M-variable turbulent wind field is described, and its algorithm flow is shown as follows:
- (1)
According to the sampling theorem [
16], the simulation parameters are set in the frequency domain, including:
Sampling frequency , in order to ensure that the analog signal can be reconstructed accurately without aliasing, is required to be greater than two times the value of , indicates the upper limit of the analog frequency range, which denotes also the cut-off frequency in this paper;
The initial frequency
and cutoff frequency
, which are selected by considering the dimensionless power spectral density function image and the influence of truncation error in the simulation frequency range on the simulation variance [
17], the frequency step size is denoted by
, where
represents the frequency sampling number.
- (2)
Set simulation parameters in the time domain, including:
Sampling interval , ;
Analog time points , generally .
- (3)
Set the basic parameters of the simulated wind field, including simulation points , simulation height , and spacing between simulation points , etc.
where,
represent the abscissa and ordinate values of simulation point
and simulation point
, respectively.
- (4)
Calculate the cross-spectral density matrix of each frequency sampling point , :
where,
denotes the power spectral density function,
denotes the autocorrelation spectrum of corresponding points, and
denotes the cross-correlation spectrum between two simulated points,
, which can be calculated as:
Since the cross-correlation function of the fluctuating wind speed in one-dimensional space does not involve the change of height, the cross-correlation spectrum between the simulated point
and other simulated points is consistent, and the cross-correlation spectrum of the fluctuating wind speed in one-dimensional space can be obtained by the following formula:
- (5)
Combined with the double-indexed frequency method, the Cholesky decomposition method is performed on the cross-spectral density matrix of sampling points at each frequency, and the decomposed lower triangular matrix is obtained, that is:
where,
denotes the double-indexed frequency;
denotes the decomposed lower triangular matrix.
- (6)
Random phase obedience is introduced and subjected to independent random distribution among intervals , carrying out the Fast Fourier transformation, namely:
- (7)
Employing harmonic superposition method to generate the wind speed time history of the simulated point samples, and can be calculated by Equation (11):
where,
.
2.3. Power Spectrum Correction Method
In the practical application of simulating a uniform pulsating wind field, since the dimensionless power density spectrum of one-dimensional space accords with Gaussian distribution (
Figure 1 shows the longitudinal dimensionless wind speed power density spectrum curve of several classical spectra), the relationship between the simulating pulsating wind speed variances
and the power spectrum density function
is defined as below [
18,
19]:
Therefore, when the analog frequency range
is a segment interval and is cut out from the spectral density function originally defined interval
, theoretically, the variance of the simulated fluctuating wind speed
is defined as:
The defined standard deviation of the fluctuating wind speed
can be determined by an empirical formula in the simulation algorithm (e.g., Equation (20) in
Section 3.1), according to the calculation formula of turbulence intensity
, choosing the segment interval
will cause the simulated results to produce truncated standard deviation, which will inevitably affect the numerical error of the simulated turbulence intensity due to the existence of the deviation of
and
. The deviation between the two can be calculated as:
The standard deviation’s deviation affects the simulated turbulence intensity to a certain extent, and the smaller the deviation is, the smaller the influence is.
It can be seen from Equations (12)–(14) that when increasing the proportion of the interval length in the simulated frequency interval, the theoretically calculated standard deviation’s deviation will decrease. However, it can be seen from the simulation algorithm introduced in
Section 2.2 that the frequency step has a significant impact on the simulation results. On the other hand, according to the formula
, after determining the basic range of the simulation frequency interval, a slight change caused by the minor increase in cutoff frequency relative to the magnitude of
is almost negligible. When the ratio of the turbulence integral length to the average wind speed
is constant, the upper limit of the interval expands to the right indefinitely, and the deviation will not significantly decrease [
10].
This is also consistent with the simulation conclusion in this paper: under the same simulation conditions, when the initial frequency changes by one step unit, the impact on the standard deviation’s deviation
is significantly greater than that when the cutoff frequency changes by one step unit (see
Section 3.2 for details). In addition, the simulation results also show that the theoretical truncation bias
caused by the truncation of the simulation frequency range will affect the numerical value of the simulated object in the simulation process, which leads to the simulation bias
. Simulation deviation is not only related to truncation deviation, but also to the setting of other simulation parameters and the application of interpolation methods. The simulated deviation is a manifestation of the error generated by simulating the turbulence intensity, which can be expressed by the following calculation formula:
where,
denotes the standard deviation of actual simulated wind speed.
Considering
, if the magnitude of the frequency sample
is increased by an order (usually
is an exponential form with base 2), the simulation bias can indeed be reduced, but at the same time the computational memory will be doubled and the simulation speed will be seriously slowed down. Take the analog frequency range
, and the other simulation conditions are consistent with the case of one-dimensional simulation in
Section 3. The simulation experiment results are shown in
Table 1.
Table 1 and
Figure 2 show that increasing the scale
causes the whole standard deviation bias of the turbulent wind field simulation to almost linearly reduce, or even when increased to
, the overall standard deviation’s bias of the turbulent wind field has reached the truncation error, but with a significant disadvantage of longer running time, and a great computing cost if more simulated points are set in the turbulent wind field space. On the other hand, this result is not very ideal. When increasing the magnitude of
, the simulated turbulence intensity of more simulation points falls outside the range of reference turbulence intensity and the defined turbulence intensity and needs to be removed, resulting in an unnecessary waste of computing resources.
To sum up, the correction method should realize the minimum truncation deviation in theory and the minimum simulation deviation generated in the actual simulation process, without adding more frequency sampling points to reduce the simulation efficiency. In addition, the method should generate as many effective simulation points as possible (effective simulation points are defined as the simulation points whose simulated turbulence intensity falls within the range of reference turbulence intensity and defined turbulence intensity ).
Consider the error values of truncation bias
and simulation bias
:
Introduce compensation coefficient
:
If the variance in the PSD function is substituted into the compensation coefficient , then the original factor will be corrected .