## 3 Wind Power Modeling  
The output power of a WTG at any time depends on wind speed of the site at the time, forced outage rate (FOR) and the characteristics of the WTG influenced by the cut in, rated and cut out speed. Cut-in speed (\(V_{ci}\)) is the minimum wind speed that is required for a WTG to generate any power. Rated speed (\(V_{r}\)) is the wind speed required for a WTG to generate maximum or rated power. Cut-out speed (\(V_{co}\)) is the maximum wind speed that the WTG can safely handle, _i.e._, the WTG is shut down for safety reason at the cut-out speed. The relationship between wind speed (\(v\)) and the corresponding output power of a WTG is presented in [3] and can be expressed as Equation (9). \[p(v)= \begin{cases}\begin{array}{c}0,0\leq v\leq V_{ci}\\ (A+Bv+Cv^{2})P_{r}\,V_{ci}\leq v\leq V_{r}\\ P_{r}\,V_{r}\leq v\leq V_{co}\\ 0,v\geq V_{co}\end{array}\end{cases}  \tag{9}\]  
where \(P_{r}\) is the rated capacity of the WTG, and constants A, B and C depend on \(V_{ci}\), \(V_{r}\) and \(V_{co}\)[3]. The \(V_{ci}\), \(V_{r}\) and \(V_{co}\) of 14.4 km/h, 46.8 km/h, and 90 km/h respectively were used in this study. The simulated hourly wind speeds were converted to hourly power output values using (9). The effect of the FOR of the WTGs in a wind farm can be considered in the evaluation of the power output model of the wind farm. It has however been shown in [13] that the FOR of a WTG can be neglected during reliability evaluation of WECS without losing reasonable accuracy. The effect of FOR has therefore been neglected in this study to simplify the wind model. Another assumption that has been made in this study is that all the WTGs in a wind farm experience same amount of wind speed in any particular hour. Five pairs of hourly power output series were developed using the above mentioned procedure for correlations of 0, 0.25, 0.5, 0.75 and 1. Both the wind farms were assumed to have equal installed capacities for all the cases in this study. The total hourly power generated by two wind farms denoted by \(P_{\mu i}\) can be calculated using Equation (10). \[P_{\rho i}=\sum\
olimits_{j=1}^{NWF}p_{\rho i} \tag{10}\]  
where \(p_{\rho ji}\)= power output of wind farm \(j\) at hour \(I\); \(i=1\) to \(8760\times N\); \(N\) = number of simulated years. \(NWF\) = number of wind farms considered. \(\rho\) = correlation between wind farms which can be in the form a correlation matrix if more than two wind farms are considered. The wind power series for multiple sites can be obtained using this process in which the time chronology and the cross-correlation between wind power outputs are maintained. Hourly power generation obtained by above method can then be grouped into suitable number of class intervals in ascending or descending order to obtain the wind capacity model. This model is a probability distribution of different capacity output states which is generally expressed as a capacity outage probability table (COPT) [2]. Sturges' Rule [14] can be used to determine the appropriate number of class intervals, NoCI while grouping these data calculated using Equation (11). \[NoCI=1+log_{10}(8760\times N) \tag{11}\]  
where, \((8760\times N)\) is the number of data points considered. The number of class intervals for 8000 sample years calculated from Equation (11) is therefore 27. The class size CS is calculated using Equation (12):  
\[CS=TIC/NoCI \tag{12}\]  
where, \(TIC=\left(\sum_{j=1}^{NWF}\sum_{k=1}^{NWTGj}P_{rjk}\right)\) is the sum of the installed capacities of all the wind farms connected to a power system; \(P_{rjk}=\) installed capacity of \(WTG\)\(k\) in wind farm \(j\); \(NWTG_{j}=\) number of \(WTG\) in wind farm \(j\). The wind capacity model is represented by a COPT with NoCI capacity states. The capacity outage, \(CO_{i}\) for a state \(i\) is given by Equation (13).