## 3 Relevant Research in Wind Farm Design and Optimization
### Objective Functions
#### 3.1.1 Power Conversion and Annual Energy Production (AEP)
Thus, it is difficult to perform accurate long-term extrapolations of the local wind resource. Therefore, it is common in practice to assume that the \(p_{1}\) modulation parameters will not vary within the period of time \(T\) and are well characterized based on the experimental information obtained in the characterization campaign. The standard AEP (or expected annual energy production [kWh/y]) of an isolated WT can be calculated using Equation (6), where 8766 is the number of hours in a year:  
\[\mathrm{AEP}_{\mathrm{WT}}=\int_{0}^{8766}\overline{P_{\mathrm{Mech}}}\ \mathrm{d}T=8766\int_{0}^{U_{\mathrm{Max}}}\frac{1}{2}\rho AU^{3}C_{P}(U, \rho)\ p_{1}(U)\ \mathrm{d}U \tag{6}\]  
For a wind farm composed of \(N\) equal WTs, wind direction cannot be neglected during the AEP calculation [136], since WT wakes will reduce the AEP expectations of the wake-affected WTs. Therefore, the AEP of the wind farm (\(\mathrm{AEP}_{\mathrm{Farm}}\)) is calculated by multiplying the expected mechanical power conversion of the wind farm with 8766 h/y, as proposed by [55, 159, 166, 199, 203, 219]:  
\[\mathrm{AEP}_{\mathrm{Farm}}=8766\sum\
olimits_{m=1}^{N}\int_{0}^{366}\int_{0 }^{U_{\mathrm{Max}}}\frac{1}{2}\rho AU^{3}C_{P}(U,\rho)\ p_{m,H}(U,\theta)\ \mathrm{d}U\ \mathrm{d}\theta \tag{7}\]  
where \(p_{m,H}(U,\theta)\) is a bivariate probability density function that represents the probability of occurrence of a wind condition defined by a wind speed \(0\leq U\leq U_{\mathrm{Max}}\) from a direction \(0\leq\theta\leq 360\) at the position of the \(m\) WT and having a hub height \(H\). To the wind farm designer, the mathematical form of \(p_{m,H}\) is unknown _a priori_. However, two different approaches that help to approximate the form of \(p_{m,H}\) were identified in the literature. The first approach [55, 159, 219] presumes: (1) that the statistical distributions of wind speed are Weibullian at all the possible WT locations and WT heights within the wind farm; and (2) that the Weibull modulation parameters for each of the WT locations  
Between Failure (MTBF), and the Mean Downtime (MDT), which in addition provide information for WT maintenance scheduling [221; 222; 223; 224; 225]. In wind energy, the operational availability measure and the concept of reliability are extended by including external elements to the wind farm (e.g., the main electrical system), which are controlled by logisticians (e.g., main electrical system operators). Therefore, issues related to electrical instabilities and energy curtailments, among other external factors, are often included in the availability and reliability definitions. It is generally acknowledged that energy curtailments produce important impacts on wind economics [226; 227]. Curtailments occur for a number of reasons including electrical grid congestion, energy oversupply, and operational issues. Each type of curtailment occurs with different frequency and depends on the regional and the local system's energy generation and its electrical characteristics. Electrical system operators typically curtail wind-based energy to maintain the system's balance and reliability. Curtailments can be regarded as electrical system-dependent and country-dependent issues of operating wind farms. The modeling and prediction of curtailments requires specific knowledge of the wind resource, wind farm design, wind farm control system, main electrical system condition, PPAs, and the energy regulatory framework (which, for example, states if any excess in energy conversion may be purchased by the energy supplier/operator or not), among other relevant factors. The WT and the wind farm total availability are often included as constant factors multiplying Equations (6) and (7), respectively. It is common to assume conservative values for the availability factor ranging from 95% to 98% [55; 56; 135].