## 3 Relevant Research in Wind Farm Design and Optimization
### Objective Functions
#### 3.1.1 Power Conversion and Annual Energy Production (AEP)  
The WT mechanical/electrical power conversion and the WT Annual Energy Production (AEP) are the basic metrics used to evaluate the performance of wind farms. More than one third of the reviewed works used one of these metrics as an objective function [11; 12; 13; 14; 15; 27; 156; 157; 159; 163; 172; 173; 174; 175; 176; 177; 178; 179; 180; 181; 182; 183]. Most of the non-linear behavior of the wind farm planning process comes from the quantification of both metrics. The mechanical power of uniform, isotropic and steady wind (\(U_{\infty}\) [m/s]) crossing the swept rotor area (\(A\) [m\({}^{2}\)]) of a WT, derived from the first law of thermodynamics, is the product of the wind mass flow rate (\(\rho AU_{\infty}\) [kg/s]) times the kinetic energy of the wind (\(U_{\infty}^{2}/2\) [m\({}^{2}\)/s\({}^{2}\)]), where \(\rho\) [kg/m\({}^{3}\)] is the air density. The instantaneous mechanical power conversion of a WT, as shown in Equation (2), is the product of the power conversion potential times the WT power coefficient (\(C_{p}\)), a variable which summarizes both the aerodynamic and the electromechanical properties of the WT under consideration and is often considered as the WT efficiency metric. The power coefficient is a complex function of many variables including the free-stream wind speed, Turbulence Intensity (TI [%]), wind shear, yawing angle misalignment (\(\Delta\theta\) [\({}^{\circ}\)]), the blade aero-structural condition, blade pitch setting, and the  
Figure 2: Timeline of Objective Functions used in the WFDO Problem. Tip Speed Ratio (TSR, \(\lambda_{\text{W}}=\Omega R/U_{\infty}\), where \(R\) [m] is the rotor radius and \(\Omega\) [rad/s] is the rotational speed), among other variables [29, 214, 215]. Its mathematical representation is strongly influenced by the adopted control strategy (e.g., stall regulation or pitch regulation) and is commonly reported by WT manufacturers as an empirical function of the wind speed at hub height (\(U_{\text{Hub}}\) [m/s]), while assuming standardized inflow conditions. Due to the complex requirements of the empirical characterization procedures, it is hard to express the power coefficient as an analytical expression of the previously mentioned variables. Therefore, the power coefficient is often considered to be a function of the assumed homogeneous, isotropic, free-stream wind speed at the rotor (\(U_{\text{Hub}}=U_{\infty}\)), while accounting for the selected control operation strategy, which in turn accounts for the TSR and other electromechanical variables:  
\[P_{\text{Mech}}(U_{\infty})=\frac{1}{2}\rho AU_{\infty}^{3}C_{P}(U_{\infty}) \tag{2}\]  
Since wind speed and density (which is a function of other thermodynamic variables such as pressure, temperature and humidity) are time- and space-dependent, as mentioned in Section 2.1, they are often considered stochastic variables that in most cases show well-defined statistical distributions; often these distributions are assumed to be Weibullian [216] and Normal [217] distributions, respectively [43, 218]. Each statistical distribution is defined in terms of statistical parameters, namely the Weibull scale (\(c\)) and shape (\(k\)) parameters, and the normal mean (\(\mu\)) and standard deviation (\(\sigma\)) parameters, respectively. The wind direction is also considered a stochastic variable with an unknown general statistical distribution. However, it has been shown in [43] that the von Mises distribution, depending on the concentration (\(k_{c}\)) and mean (\(\mu_{M}\)) modulation parameters, might fit the statistical wind direction behavior well. In order to properly estimate the mechanical power conversion from the wind, considering the stochastic nature of the atmospheric variables involved, expected values of the power conversion are calculated.