## 3 Relevant Research in Wind Farm Design and Optimization
### Objective Functions
#### 3.1.1 Power Conversion and Annual Energy Production (AEP)
The general definition of the expected mechanical power conversion of an isolated WT, assuming anisotropic wind at the rotor, is given in Equation (3):  
\[\text{E}[P_{\text{Mech}}]=\overline{P_{\text{Mech}}}=\int\limits_{A}\int\limits _{0}^{U_{\text{Max}}}\int\limits_{0}^{\rho_{\text{Max}}}\int\limits_{0}^{ \tau_{\text{Max}}}\int\limits_{0}^{10}\frac{1}{2}\rho U^{3}C_{P}(U,\rho,\text{ Tl},\Delta\theta)\;p_{0}(U,\rho,\text{Tl},\Delta\theta)\;\text{d}\Delta\theta\; \text{d}\text{Tl}\;\text{d}\rho\;\text{d}U\;\text{d}A \tag{3}\]  
where \(U_{\text{Max}}\), \(\rho_{\text{Max}}\) and \(\text{Tl}_{\text{Max}}\) are the maximum wind speed, the maximum density and the maximum turbulence intensity observed at the WT location in a differential segment of area within the WT swept rotor area. \(p_{0}(U,\rho,\text{Tl},\Delta\theta)\) is a multivariate probability density function used to describe, at least approximately, the probability of occurrence of the correlated, real-valued random variables \(U\), \(\rho\), \(\text{Tl}\) and \(\Delta\theta\). Therefore, \(p_{0}\) represents the probability of occurrence of a wind condition defined by a wind speed \(0\leq U\leq U_{\text{Max}}\), density \(0<\rho\leq\rho_{\text{Max}}\), turbulence intensity \(0\leq\text{Tl}\leq\text{Tl}_{\text{Max}}\) at yaw angle misalignment \(0\leq\Delta\theta\leq 180\) in a differential segment area within the WT swept rotor area. Unfortunately, it is almost impossible to define \(p_{0}\) and \(C_{P}\) as a function of such stochastic variables in practice. Therefore, simplifications in the expected mechanical power conversion quantification are performed; it is commonly assumed that: (1) WTs operate in steady state conditions; (2) the atmospheric variables are uniform within the rotor swept area (e.g., \(U=U_{\infty}=U_{\text{Hub}}\)); (3) the air density is constant at the rotor swept area and has no effect on \(p_{0}\); (4) TI has no direct effects both on \(C_{P}\) and \(p_{0}\); and (5) yaw misalignment effects are completely neglected as it is assumed that WTs can instantaneously realign themselves with wind direction. The simplified relation for the quantification of expected mechanical power conversion is given in Equation (4):  
\[\mathrm{E}[P_{\mathrm{Mech}}]=\overline{P_{\mathrm{Mech}}}=\int_{0}^{U_{\mathrm{ Max}}}\frac{1}{2}\rho AU^{3}C_{P}(U,\rho)\ p_{1}(U)\ \mathrm{d}U \tag{4}\]  
where \(p_{1}(U)\) is a univariate probability density function that represents the probability of occurrence of a wind condition with wind speed \(0\leq U\leq U_{\mathrm{Max}}\) at hub height and \(C_{P}(U,\rho)\) is the WT power coefficient corrected for the given air density \(\rho\). Even after simplifications, closed-form analytical solutions for Equation (4) cannot be obtained [159], so its value is approximated numerically using Riemann discrete summations, Monte Carlo integration or Gaussian quadrature integration, among other numerical integration methods. The expected mechanical energy conversion of an isolated WT over a time period \(T\left[\mathrm{h}\right]\) (ignoring down time or technical availability) is given in Equation (5):  
\[\mathrm{E}[E_{\mathrm{Mech}}]=\int_{0}^{T}\overline{P_{\mathrm{Mech}}}\ \mathrm{d}T=\int_{0}^{T}\int_{0}^{U_{\mathrm{Max}}}\frac{1}{2}\rho AU^{3}C_{P} (U,\rho,T)\ p_{1}(U,T)\ \mathrm{d}U\ \mathrm{d}T \tag{5}\]  
Equation (5) presumes that the WT power coefficient might change in time, as a consequence of WT degradation. It also implies that variations in wind resource might occur during the period time \(T\), which in turn affect the probability density function modulation parameters. It is common in practice to neglect time variations of the power coefficient, since appropriate maintenance is assumed to be performed during the period of time \(T\). In addition, wind resource characterization campaigns are commonly performed in short periods of time (e.g., one or two years), and the measured data is not always reliable due to unexpected issues (sensor calibration, sensor malfunction, _etc._).