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 $\mathit{Z}(t)$ is employed to model the temporal-spatial correlation of wind speeds at $m$ different sites. In general, it is computationally tedious to obtain the autocorrelation functions (ACFs) and cross-correlation functions (CCFs) of $\mathit{Z}(t)$, 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 $\mathit{Z}(t)$ can be analytically obtained. Then, Fourier transformation is implemented to establish the cross-spectral density matrix of $\mathit{Z}(t)$, and an analytical approach is proposed to generate samples of wind speeds at $m$ 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.