Simulating Temporally and Spatially Correlated Wind Speed Time Series by Spectral Representation Method

Abstract

In this paper, it aims to model wind speed time series at multiple sites. The five-parameter Johnson distribution is deployed to relate the wind speed at each site to a Gaussian time series, and the resultant m- dimensional Gaussian stochastic vector process <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{Z}(t)$</tex> is employed to model the temporal-spatial correlation of wind speeds at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m$</tex> different sites. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ln$</tex> general, it is computationally tedious to obtain the autocorrelation functions (ACFs) and cross-correlation functions (CCFs) of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{Z}(t)$</tex> , which are different to those of wind speed times series. In order to circumvent this correlation distortion problem, the rank ACF and rank CCF are introduced to characterize the temporal-spatial correlation of wind speeds, whereby the ACFs and CCFs of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{Z}(t)$</tex> can be analytically obtained. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\text{Then}$</tex> , Fourier transformation is implemented to establish the cross-spectral density matrix of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{Z}(t)$</tex> , and an analytical approach is proposed to generate samples of wind speeds at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m$</tex> different sites. Finally, simulation experiments are performed to check the proposed methods, and the results verify that the five-parameter Johnson distribution can accurately match distribution functions of wind speeds, and the spectral representation method can well reproduce the temporal-spatial correlation of wind speeds.