## 2 Uncertainty Modeling
### Modeling the Wind Power Generation Uncertainty  
Considering the intermittent nature of the wind speed, the Weibull distribution is often considered as the probability density function that can approximate the behavior of the wind with a reasonable error. Therefore, by defining the Weibull distribution for wind speed, the probability of wind speed at different intervals (scenarios) can easily be calculated. Equation (3) is a general expression for Weibull distribution [20]. Equation (4) can be used to calculate the probability of a wind speed interval (scenario). The corresponding value of each wind speed interval can be achieved using Equation (5). \[PDF(v)=\frac{\beta}{\alpha}\bigg{(}\frac{v}{\alpha}\bigg{)}^{\beta-1}\exp{(- \bigg{(}\frac{v}{\alpha}\bigg{)}^{\beta})} \tag{3}\]  
\[\pi_{w}=\int_{v_{in,w}}^{v_{in}}\frac{\beta}{\alpha}\bigg{(}\frac{v}{\alpha} \bigg{)}^{\beta-1}\exp{(-\bigg{(}\frac{v}{\alpha}\bigg{)}^{\beta})}dv \tag{4}\]  
\[v_{w}=\frac{1}{\pi_{w}}\int_{v_{in,w}}^{v_{in}}v\frac{\beta}{\alpha}\bigg{(} \frac{v}{\alpha}\bigg{)}^{\beta-1}\exp{(-\bigg{(}\frac{v}{\alpha}\bigg{)}^{ \beta})}dv \tag{5}\]  
where \(v\) denotes the wind speed, and \(\alpha\) and \(\beta\) are the wind speed parameters that vary depending on the region in which the wind blows. Considering \(v_{i,w}\) and \(v_{f,w}\) as the initial speed and final speed of the hypothetical scenarios for wind speed, the probability (\(\pi_{w}\)) of occurrence of any wind speed scenario can simply be obtained. Thereafter, the wind speed (\(v_{w}\)) associated with each scenario is gained using the calculated probabilities. The output power of a wind turbine is highly dependent on wind speed. Therefore, any wind turbine has a characteristic named power curve that exactly shows the capability of a wind turbine in power generation versus existing wind speed. Knowing a specific wind speed (\(v_{w}\)), one can estimate the output power of a wind turbine (\(P_{w}^{est}\)) through its power curve. The power curve is generally defined by a set of equations as it is stated in Equation (6) [21], which in terms, (\(v_{in}^{c}\)), (\(v_{ratel}\)), and (\(v_{out}^{c}\)) denote the cut-in wind speed, rated wind speed, and cut-out wind speed for a wind turbine,respectively. The rated power (\(P_{w}^{r}\)) and the estimated output power of the wind turbine are also evident from Equation (6). \[P_{w}^{est}=\left\{ \begin{array}{ll}0,&v_{w}\leq v_{in}^{c}\\ \frac{v_{w}-v_{in}^{c}}{v_{mot}-v_{in}^{c}},&v_{in}^{c}<v_{w}<v_{rated}\\ P_{w},&v_{rated}<v_{w}<v_{out}^{c}\\ 0,&v_{w}\geq v_{out}^{c}\end{array} \right. \tag{6}\]  
In most research studies, the concept of the power curve is extended to a wind farm. Hence, instead of studying a single wind turbine, it is preferable to focus on a group of wind turbines that are in a special area and usually known as wind farms. Considering several scenarios for a probabilistic problem is generally not an easy procedure. Depending on the problem type, various methods exist for scenario generation [22; 23]. However, in this paper, a technique based on [19; 24; 25] is applied to generate a desirable number of scenarios with reasonable accuracy. In order to have a combination of load and wind scenarios, the following steps are taken:  
1. Several scenarios for the load level are considered. 2. The probability of each system loading scenario (level of the load) and its corresponding value using Equations (1) and (2) are calculated. 3. Several scenarios for wind speed are considered. 4. The probability of each wind speed scenario and its corresponding value using Equations (4) and (5) are calculated. 5. The output power of the wind farm using the estimated wind speed in each scenario and Equation (6) is generated. 6. The final number of combined load-wind scenarios is obtained by multiplying the number of load scenarios by the number of wind scenarios.