## 2 Characterization of Wind-Solar Power Generation for Earth Regions
### The Renewable Power System
However, not only can this be estimated deterministically, but it is also possible to estimate generation ranges by analyzing the PDFs followed by the different variables appearing in Equation (4). In other words, it is possible to use one of the various statistical uncertainty propagation techniques to estimate confidence intervals of electrical generation over the desired time period. In relation to wind generation, in the field, the power output of wind turbines usually deviates to some extent from the rated power curve presented in the equation below [48]:  
\[P_{T}=\frac{1}{2}\cdot\rho\cdot A\cdot W^{3}\cdot c_{p} \tag{6}\]  
where \(\rho\) is air density, \(A\) the area swept by a turbine's rotor, \(W\) the wind speed and \(c_{p}\) the power coefficient. Wind speed is obtained in some databases at specific heights, such as 10 or 50 m [30]. The wind speed at different heights, such as at a specific hub height that differs from the provided height, can be estimated using the Hellman exponent \(g\), which ranges from 0.1 to 0.4. This range depends on surface roughness and atmospheric conditions, and the wind speed estimation can be made through exponential extrapolation:  
\[W_{h}=W_{ref}\cdot\left(\frac{H}{H_{ref}}\right)^{g} \tag{7}\]  
The power output of the wind turbine is determined using its turbine characteristic power curve, which depends mainly on the wind speed at hub height. These characteristic curves relate the wind speed with the power produced. However, three main speed ranges need to be taken into account:  
\[ \begin{array}{ll}P_{T}=0&\text{if}\;\;W\leq W_{LC}\\ P_{T}=\frac{1}{2}\rho\cdot A\cdot W^{3}\cdot c_{p}&\text{if}\;\;W_{LC}<W<W_{UC} \\ P_{T}=0&\text{if}\;\;W\geq W_{UC}\end{array}  \tag{8}\]  
When the wind speed is less than or equal to the lower cutoff wind speed \(W_{LC}\), there is no generated power. The same occurs when the wind speed is above the upper cutoff wind speed \(W_{UC}\). This occurs up to the upper cutoff, after which the turbine stops or reduces its power output to protect itself. Other issues, such as the air density variations \(\rho\), for example, play a significant role. There are widely adopted international regulations to analyze these influences, which describe simple, physics-informed methods to account for these effects. Furthermore, there are technical reasons affecting the performance of the wind turbine, namely intended or unintended deviations from the optimal control scheme and, therefore, the optimal value of \(c_{p}\) or power coefficient, which can lead to deviations from the standard power curve [48]. In any case, \(c_{p}\) can be estimated from the wind generator's power curve versus wind speed [48]. Its value is estimated from models, calculated using an exponential relationship that accounts for rotor dynamics and control settings, by the tip-speed ratio \(\lambda\) and \(\beta\), which is the pitch angle of the blades [48]:  
\[c_{p}(\lambda,\beta)=c_{1}\left(\frac{c_{2}}{\lambda}-c_{3}\beta-c_{4}\beta^{ \epsilon_{5}}-c_{6}\right)\exp\left(-\frac{c_{7}}{\lambda}\right)+c_{8}\lambda \tag{9}\]  
where \(\omega\) is the rotor's angular speed and \(r\) is the blade's length. The value of \(\beta\) is set to zero to get the maximum power extraction and the optimum \(\lambda\) is considered [49]. The wind generation is estimated with the previous equations and the uncertainty in its production is forwarded by the wind speed, as mentioned before with the crafting of synthetic wind speed curves and by the different constants \(c_{i}\) (Table 3). Typical values for these constants are taken from [PERSON] et al. [50; 51], and an uncertain range of \(\pm 10\%\) is employed, allowing for deviations, atmospheric and working conditions, and other possible factors. \begin{table}
\begin{tabular}{c c} \hline \hline c & 0.44 \\ \hline c & 124.99 \\ \hline c & 0 \\ \hline c & 2.2 \\ \hline c & 21 \\ \hline c & 18.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of the main coefficient’s values used for windmill calculations.