## 3 Methodology
### Simulation Software
#### 3.1.2 Implemented Mathematical Equations and Correlations  
The power of the movements of winds can be described by a defined mass of air \(m\), flowing through the swept area \(A\) of the rotor with the velocity \(U\) given by the Equations (1)-(3) [29]:  
\[m=\rho AU \tag{1}\]  
The kinetic power of the mass of air is given by the expression  
\[P_{kinetic}=\frac{1}{2}\cdot(\rho A\cdot U)\cdot U^{2}=\frac{1}{2}\cdot\rho \cdot A\cdot U^{3} \tag{2}\]  
where;  
\(P_{kinetic}\) is the power produced by the Pelton hydraulic turbines, KW  
\(\rho\) the air density, kg/m\({}^{3}\)  
\(m\) is the air mass flow rate, kg/sec  
\(U\) is the wind speed, m/sec  
\(A\) is the swept area, m\({}^{2}\)  
The conversion of the kinetic power in the wind is subject to aerodynamic losses as the air passes through the rotor. Those losses are valued by a coefficient \(C_{p}\) to account for the conversion losses of energy. The power coefficient can be obtained from manufacturer data where, apart from the aerodynamic behavior of the blades, the mechanical and electrical losses are considered. The power coefficient can be obtained from manufacturer data where, apart from the aerodynamic behavior of the blades, the mechanical and electrical losses are considered. Therefore, the kinetic power becomes:  
\[P_{kinetic}=\frac{1}{2}\cdot C_{p}\cdot\rho A\cdot U^{3} \tag{3}\]  
where;  
\(C_{p}\) is the aerodynamic capacity factor (wind power fraction captured by the wind turbine). The wind turbines implemented in this study were of the type Vestas V90-2.0 MW. Vestas V90-2.0 MW wind turbine is a pitch regulated upwind turbine with active yaw and a three-blade rotor. The Vestas V90-2.0 MW turbine has a rotor diameter of 90 m with a generator rated at 2.0 MW, depending on wind conditions. The turbine uses a microprocessor pitch control system called OptiTip and the OptiSpeed (variable speed) feature. With these features, the wind turbine can operate the rotor at variable speed (rpm), helping to maintain the output at or near rated power. Wind turbine manufacturers are the source of the power and thrust information for any selected wind turbine. Thrust data are given in kilo Newtons (kN) or as thrust coefficient \(CT\), which is the normalized force, \(CT\) is related to the thrust force by the relationship:  
\[CT=\frac{Thrust}{0.05\rho U^{2}A} \tag{4}\]  
WindFarm Simulation software can calculate the energy yield of a wind farm, including topographic and wake effects, optimizing the turbine layout for maximum energy yield. We then perform noise calculations (showing the noise contours), analyze wind turbine data, create zone-of-visual-influence maps, display wire frame views of wind farms, calculate shadow flicker, and create 3D visualizations. Also, the model performs measure-correlate-predict analysis of wind speed data and deals with the correlation between the measured data at two different sites. Based on the common data, calculations are performed to estimate correlation coefficients (\(r\)), linear regression coefficients (a, b of the best-line fit) and the goodness of fit (\(R^{2}\)). All the above are calculated [30]:* per wind speed range and per direction sector
* per wind speed range but for all directions
* per direction, but for all the wind speeds (above calms)
* for all wind speeds, no matter the direction  
It is reminded here that correlation coefficient values close to 1.0 show similar (in-phase) variations, while values close to 0.0 show irrelevant variations (uncorrelated phenomena). Furthermore, goodness of fit values close to 1.0 show that the best-fit line represents very well the data \"cloud\" (using the linear regression coefficients).