2.2.2. Wind Turbine DGs

The WT-DGs are renewable sources, as well, where their produced powers are intermittent and varied in each hour with high uncertainty levels due to the dependency on wind speed. Generally, the power output from WT is calculated as Equation (12) [51]:

$$P\_W(v) = \begin{cases} 0 & 0 \le v \le v\_{cl} \\ P\_r \times \left(\frac{v - v\_{cl}}{v\_r - v\_{cl}}\right) & v\_{cl} \le v \le v\_r \\ P\_r & v\_r \le v \le v\_{c0} \\ 0 & v \ge v\_{c0} \end{cases} \tag{12}$$

where *vco* and *vci* are the cut-out and cut-in wind speeds of Wind Turbines (WT), respectively; *v* is the average wind speed of the hour; *v<sup>r</sup>* is nominal operating wind speed of WT; and *P<sup>r</sup>* is the maximum power generated by the WT.

For the WT-DGs, the uncertainty of wind speed is modeled by the Weibull-PDF. The Weibull-PDF *f*(*v*) is formulated as in Equations (13)–(15) [52,53]:

$$f(v) = \frac{K}{\overline{C}} \left(\frac{v}{\overline{C}}\right)^{K-1} \exp\left(\frac{-v}{\overline{C}}\right)^K \tag{13}$$

$$K = \left(\frac{\sigma}{v\_m}\right)^{-1.086} \tag{14}$$

$$C = \frac{v\_m}{\Gamma\left(1 + \frac{1}{k}\right)}\tag{15}$$

where *K* and *C* are the shape and scale indexes of the Weibull-PDF; *v<sup>m</sup>* is the mean wind speed; and *σ* is the standard deviation.

Based on that, in each hour, different states of the wind speeds are generated. Accordingly, the output power of the WT-DG is obtained for each state using Equation (12). Besides, the probability of the wind speed for each state is estimated using Equation (16).

$$\rho(v) = \int\_{vw\_1}^{vw\_2} f(v)dv\tag{16}$$

where *ρ*(*v*) is the probability of wind speed in each state whereas *vw*<sup>1</sup> and *vw*<sup>2</sup> are the regarding limits of the wind speed.

Thus, the average output power of the WT-DG for each hour is calculated using Equation (17) [48].

$$P\_w(t) = \int\_1^{25} P\_w(v)\rho(v)dv\tag{17}$$

Figure 2 describes in detail the calculation of the total average power at each hour including the uncertainties of wind speed, where each hour has 20 states for wind speed with a step of 5% of the maximum wind speed.
