## 2 Wind Data Modeling
\begin{bmatrix}a_{11}&\cdots&a_{1r}\\ \vdots&\ddots&\vdots\\ a_{r1}&\cdots&a_{rr}\end{bmatrix}  \tag{6}\]  
where, \(a_{ij}\) is the desired correlation between \(i^{\text{th}}\) and \(j^{\text{th}}\) column of matrix \(X\). Upper triangular matrix \(G\) can then be calculated using Cholesky Decomposition. Matrix \(X_{c}\) which contains p correlated random number series as defined in matrix A can be deduced using Equation (7). \[X_{c}\ =\ X\cdot G \tag{7}\]  
This method can be simplified when only two random number series are present as shown in Equation (8). \[X_{c}=X_{1}\cdot\zeta+X_{2}\cdot\sqrt{1-\zeta^{2}} \tag{8}\]where, \(X_{1}\) and \(X_{2}\) are series of uncorrelated random numbers. \(\zeta\) is the desired correlation coefficient. Series \(X_{c}\) calculated using Equation (8) has a correlation of \(\zeta\) with series \(X_{1}\). A particular case was considered taking \(\zeta\) = 0.5. The scatter plots of 1000 pairs of uncorrelated and correlated random numbers thus obtained are compared in Figure 1.  
Two separate wind speed series can be synthesized using correlated random numbers \(X_{1}\) and \(X_{c}\) in Equations (2) and (3). The hourly mean and hourly standard deviation values of Swift Current site were obtained from Environment Canada. Two pairs of wind speed data series were simulated using \(\zeta\) = 0 and \(\zeta\) = 0.5. The scatter plots for 1000 data pairs for two cases are shown in Figure 2.  
It can be seen from Figures 1 and 2 that the pairs of wind speed series simulated using Equations (2) and (3) have a correlation close to the correlation of the random numbers used. The actual correlation between the two pairs of simulated wind speed series using random numbers with \(\zeta\)= 0 and \(\zeta\)= 0.5 were 0.11 and 0.56 respectively. The wind speed profile for the 21 st and 22 nd day are shown in Figures 3 and 4 respectively for the two cases to illustrate how the wind speeds follow each other at the two correlation levels. Figure 1: Generation of correlated random numbers using Cholesky decomposition, (**A**) Uncorrelated random numbers; (**B**) Correlated random numbers with \(\zeta\)= 0.5. Figure 2: Simulated wind speeds using uncorrelated and correlated random numbers. (**A**) Using uncorrelated random numbers; (**B**) Correlated random numbers with \(\zeta\)= 0.5. Wind speeds in Figure 4 follow each other more closely than the ones in Figure 3 due to the induced correlations. The actual correlations obtained in simulated wind speeds of two sites were very close to the correlations of random numbers used. Pairs of correlated wind speed data were simulated with correlations 0, 0.25, 0.5, 0.75 and 1 using the same ARMA model of Swift Current for 8000 sample years. The probability distribution of wind speed for all of the generated series with varying correlation is given in Figure 5.  
Figure 4: 2-day sample simulation of wind speeds for two sites with correlation of 0.56. Figure 5: Probability distribution of wind speeds simulated at different correlations. Figure 3: 2-day sample simulation of wind speeds for two sites with correlation of 0.11. It can be seen in Figure 5 that the simulated wind speeds for two sites have almost identical probability distributions even if they have different correlations because of the use of same wind characteristic for both sites. Some of the simulated speeds obtained from above method are negative in magnitude, which cannot be defined. Different methods to deal with negative values produced during simulation are presented in [10]. A straight forward method proposed by [10] is to set all the negative values to zero after the completion of simulation process that has been used in this paper.