## 3 Relevant Research in Wind Farm Design and Optimization
### Sub-Modeling
#### 3.2.1 Wind Turbine Wakes and Turbulence Modeling
The wake expansion is commonly assumed to be proportional to the downwind distance (\(x\) [m]) from the WT hub such that \(R_{\mathrm{W}}=\alpha x+R\), where \(R_{\mathrm{W}}\) [m] is the radius of the wake and \(R\) [m] is the rotor radius. \[\hat{d}=d_{0}\left(1+7.12\left(\dfrac{\sigma_{\theta}x}{b_{\text{W}}}\right)^ {2}\right)^{-\frac{1}{2}} \tag{25}\]3 Timeline of Wake Models used in the WFDO Problem. Like [PERSON]'s wake model, the Jensen-Katic wake model disregards pressure gradients, the gravity force, viscous forces (e.g., shear forces in the edge of the wake were neglected), rotational effects, ground effects, and tower effects. The free-stream wind speed was considered constant, homogeneous, stationary, and incompressible. Moreover, the wake was considered axisymmetric and its transversal profile constant (_i.e._, a top-hat profile was considered). The Jensen-Katic wake model relies on momentum conservation in a fully developed wake, thus its validity is limited to the far wake region. However, conflicting views about which is the axial length at which the wake can be considered to be in the far region arise in the literature. It is often considered that the far wake starts at an axial distance of about three to four rotor diameters downstream from the WT. A formal definition for such a distance, however, should be based on the necessary distance required for the reestablishment of the axial and radial pressure field, turbulent kinetic energy and angular momentum values, which in turn are a function of the WT operation mode, as previously described. It has been observed that the total velocity deficit, composed of several merged wakes, depends mostly on the closest WT [59, 299, 300]. However, the interaction of multiple wakes is not fully understood and is subject of many studies in the fluid mechanics field. Different methods for calculating the total effect of merged WT wakes have been proposed in the literature [23, 269, 300]. The wake merging process is based on the concept of constructive superposition, where several wakes can be combined to produce a stronger wake. One effective and recurring way to calculate the effective value of the velocity deficit (\(U_{\text{Def}}^{\text{eff}}\)) in merged wakes is to sum the kinetic energy deficits of all WTs contributing to the wake (_i.e._, the sum of the square of the velocity deficits) as shown in Equation (41):  
\[\hat{d}=d_{0}\left(1+7.12\left(\dfrac{\sigma_{\theta}x}{b_{\text{W}}}\right)^ {2}\right)^{-\frac{1}{2}} \tag{25}\]4  
where \(U_{\text{W}}^{i}(x,r)\) is the velocity deficit contribution of the upstream WT \(i\), and \(N_{u}\) is the number of upstream WTs. The effect on the energy conversion of a WT operating under partial wake conditions is commonly modeled through the use of a geometric factor [156, 178, 301] that affects the value of the effective wind speed deficit (_i.e._, Equations (39) and (41)). The geometric factor is commonly defined as the ratio of the wake affected rotor area to the WT rotor area (\(A_{\text{Overlap}}/A\) ). Therefore, the modeled partial wake operation is less harmful, in terms of power conversion, than full wake operation. The assumption, however, does not account for the increased fatigue loading the WT will undergo while operating in partial wake conditions, which is important as denoted by the experimental findings of [PERSON] _et al._[69]. Moreover, the tower and nacelle contributions to the wake are usually not considered explicitly by EWMs. Nevertheless, its effects can be included indirectly through the use of empirical values of \(c_{T}\), which may contain contributions from all passive and active parts of the WT.