# A Spectrum Correction Method Based on Optimizing Turbulence Intensity
## 3 Results and Discussion
### Determine the Analog Frequency Range \([F_{max},F_{min}]\)  
By setting the initial frequency \(F_{min}\in\left[10^{-5},10^{-1}\right]\), setting the cut-off frequency \(F_{max}\in\left[1,10\right]\) (step by \(10^{1}\) units and \(1\) unit respectively to form an \(5\times 10\) analog frequency interval matrix), and performing integral operations one by one according to Equation (13), the ratio of the standard deviation of the truncated interval integral to the standard deviation of the fully defined interval integral is \(\eta=\sigma_{theory}/\sigma\), the result of which is shown in Figure 3a. By combining this with the \(3\sigma\) principle, it can be found that when the initial frequency \(F_{min}<10^{-4}\) and cut-off frequency is given any value, the standard deviation ratio \(\eta\) will always fall within a confidence interval of \(+2\sigma\), as the cutoff frequency increases, \(\eta\) infinity goes to \(+3\sigma\), therefore, it achieves a value of \(F_{min}=10^{-4}\). Under the same simulation conditions, the simulation frequency interval element from the matrix described above was substituted, one by one, into the uniform turbulent wind field simulation algorithm, for simulating fluctuating wind speed of time history in one-dimensional space (at hub height), accompanied with the standard deviation's deviation and truncation deviation. The comparison between the two is shown in Figure 4b. The figure shows that when \(F_{min}=10^{-4}\), the simulation bias \(\hat{\epsilon}\) does not decrease significantly with the increase of the simulation frequency range, the truncated bias step rises when \(F_{max}>6\), instead. To weaken the distortion of the analog signal, the cutoff frequency \(F_{max}=5\) was selected to simulate the uniform turbulent wind field in one-dimensional space (at the height of the hub) and two-dimensional space (at the swept surface of the wind wheel), respectively, (the distribution of simulated points of these two types of spatial wind field simulation are shown in Figure 4a,b, as above) considering the truncation error and simulation error comprehensively, and the compensation coefficient was determined. Figure 3: (**a**) The ratio of cut-off standard deviation and defined standard deviation along with the cut-off frequency change curve; (**b**) Standard deviation scatter distribution under different cut-off frequency intervals. After compensation and correction, the main parameters of one-dimensional space are shown in Table 6.  
A comparison of the power spectral density at simulated points and power spectral density related to simulated wind speed before and after modification is shown in Figure 7. Among which, the line of the 'Corrected Spectrum Simulated' represents the power density spectrum of the fluctuating wind speeds that are generated by the corrected Kaimal spectrum; the line of the 'Corrected Spectrum' represents the corrected Kaimal power density spectrum itself; the line of the 'Original Simulated Spectrum' represents the power density spectrum of the fluctuating wind speed that are generated by the original Kaimal spectrum; the line of the 'Original Spectrum' represents the original Kaimal power density spectrum itself.