## 2 Wind Data Modeling  
It is very important to have a wind speed model that adequately represents the wind characteristics of a wind farm for wind energy studies. It has been shown in [12] that the long term wind characteristics of a particular site can be represented by an ARMA model that can be written in the form of Equation (1):  
\[y_{t}=\ \Phi_{1}\cdot y_{t-1}+\Phi_{2}\cdot y_{t-2}+\cdots+\Phi_{s}\cdot y_{t- s}+\alpha_{t}-\theta_{1}\cdot\alpha_{t-1}-\theta_{2}\cdot\alpha_{t-2}-\cdots- \theta_{m}\cdot\alpha_{t-m} \tag{1}\]  
where \(\Phi_{i}\) (\(i=1\), \(2\), \(3\)...s) and \(\theta_{j}\) (\(j=1\), \(2\), \(3\)...m) are autoregressive and moving average coefficients of the wind model respectively. These coefficients can be calculated using the historical data of a site [12]. \(\alpha_{t}\) is a Normally and Independently Distributed(NID) white noise process with zero mean and \(\sigma^{2}\) variance generally expressed in the form \(\alpha_{t}\in\) NID (\(0\), \(\sigma^{2}\)). Equation (1) represents a time series for \(y\), which can be generated using value of \(\alpha_{t}\) randomly generated each time interval and previous values of \(y\) and \(\alpha\). An hourly time interval is used in this study. The time series of y obtained using Equation (1) can then be used to calculate the simulated hourly wind speeds using Equation(2). \[SW_{t}=\mu_{t}+\sigma_{t}\cdot y_{t} \tag{2}\]  
where \(SW_{t}=\) Simulated wind speed for hour \(t\). \(\mu_{t}=\) Hourly mean wind speed for hour \(t\). \(\sigma_{t}=\) Hourly standard deviation of wind speed for hour \(t\). It should be noted that there are each of 8760 hourly mean and standard deviation values for a wind site obtained from its historical data. An ARMA model for Swift Current site, which is located in Saskatchewan, Canada is presented in [12] and expressed in Equation (3). \[ \begin{split} y_{t}=1.1772y_{t-1}+0.1001y_{t-2}-0.3572y_{t-3}+0.0379 y_{t-4}+\alpha_{t}-0.5030\alpha_{t-1}-0.2924\alpha_{t-2}+0.1317\alpha_{t-3}\\ \alpha_{t}\in NID(0,0.5247602)\end{split}  \tag{3}\]  
\(\alpha_{t}\) can be generated using a suitable normally distributed random number generator. For the initial four calculations, _i.e._, \(y_{1}\) to \(y_{4}\), the preceding values of y and \(\alpha\) that at the right hand side of Equation (3) were assumed to be zero if they were not yet obtained. Two sites each having the wind characteristics of Swift Current as represented by Equation (3) are considered in this work. Independently simulating the two sets of wind speed data using Equation (3) resulted in correlations being close to zero. The two sets of wind speed data are therefore simulated with random numbers in such a way to obtain a desired correlation between the two sites. The impact of wind correlation on the system reliability is then investigated. Correlation between two sets of simulated wind speed data can be established by using correlated random numbers during the hourly simulation [6]. The correlation between the two sets of simulated wind speed data thus obtained is close to the correlation between the random numbers used. Various methods are available that can be used for the generation of correlated random numbers. The Cholesky Decomposition [11] method is used in this paper. Cholesky Decomposition is the process of decomposing any symmetric and positive definite matrix into the product of two triangular matrices as represented by Equation (4). \[A=\ G^{T}\cdot G \tag{4}\]  
where, A = A symmetric positive definite matrix, G = An upper triangular matrix with positive diagonal entries. G\({}^{T}\) = Transpose of matrix \(G\). A set of \(r\) uncorrelated random number series may be represented by matrix \(X\) shown in Equation (5). \[X= \begin{bmatrix}x_{1}^{1}&\cdots&x_{1}^{r}\\ \vdots&\ddots&\vdots\\ x_{q}^{1}&\cdots&x_{q}^{r}\end{bmatrix}  \tag{5}\]  
where, each column of matrix \(X\) is an uncorrelated random number series with \(q\) members. The desired correlation between any two sets of series may then be represented by Equation (6). \[A=