# An Optimization Framework for Wind Farm Design in Complex Terrain
## 3 Optimization Framework
### Constraints  
Wind farm design is subject to various constraints, which may come from technical, logistical, environmental, economical, legal, and/or even social considerations. In the current framework, three types of constraints are considered: (1) inclusive and exclusive boundaries; (2) minimal distance requirements; and (3) bounds on certain site condition variables. Inclusive and exclusive boundaries denote the feasible and infeasible area for placing turbines, which may come from limitations and requirements on leased land, existing roads and properties, soil conditions, and so on. As a general method, polygons can be used to model inclusive and exclusive zones defined by the inclusive and exclusive boundaries. Thus, for a wind farm with \(N_{inc}\) inclusive and \(N_{exc}\) exclusive boundaries, the individual inclusive and exclusive zones and the overall feasible zone can be modeled as:  
\[S_{inc}^{i}=\Big{\{}(x,\ y)\Big{|}\ \ a_{k}^{i}x+b_{k}^{i}y\leq c_{k}^{i},\ \ \ k=1,2,\ \ldots,\ m^{i}\ \ \Big{\}},\ \ \ i=1,2,\ \ldots,\ N_{inc} \tag{10}\]  
\[S_{exc}^{j}=\Big{\{}(x,\ y)\Big{|}\ \ a_{l}^{j}x+b_{l}^{j}y\leq c_{l}^{j},\ \ \ l=1,2,\ \ldots,\ n^{j}\ \Big{\}},\ \ \ j=1,2,\ \ldots,\ N_{exc} \tag{11}\]  
\[S_{\text{feasible}}=\Big{\{}(x,\ y)\Big{|}\ \ (x,\ y)\in\cup_{i=1}^{N_{ \text{inc}}}S_{inc}^{i},\ \ \ (x,\ y)\
otin\cup_{j=1}^{N_{\text{exc}}}S_{exc}^{j}\ \ \Big{\}} \tag{12}\]  
where \(m^{i}\) and \(n^{j}\) are the polygon edge numbers of the \(i\)th inclusive and \(j\)th exclusive boundaries, respectively. Thus, the constraints on inclusive and exclusive boundaries of the wind farm design can be written as:  
\[(x_{i},\ y_{i})\in S_{\text{feasible}},\ \ \ \text{for}\ i=1,2,\ \ldots,\ N_{ \text{wt}}. \tag{13}\]  
The second type of constraint is also typical in the engineering practice, which requires the distance between any two turbines larger than a minimal value. This is because a shorter distance between turbines will give a larger wake loss and a higher turbulence intensity, which results in a higher level of fatigue loads and maintenance cost, and even a shorter lifetime of turbines. Also, a minimal distance is required to make sure two turbines' blades never contact each other and one turbine never falls on the other turbine. In the current framework, a fixed minimal distance requirement, \(Dist_{\text{min}}\), is assumed for any two turbines. Thus, the constraints on the minimal distance requirement are governed by:  
\[\sqrt{\big{(}x_{i}-x_{j}\big{)}^{2}+\big{(}y_{i}-y_{j}\big{)}^{2}}-Dist_{\text {min}}\geq 0\ \ \ \text{for}\ i,j=1,2,\ \ldots,\ N_{\text{wt}}\ \text{and}\ i\
eq j. \tag{14}\]  
The third type of constraints exposes requirements on certain site condition variables to rule out certain unfavorable sites, such as those with too low a mean wind speed, too high a turbulence intensity, or too rugged terrain. When applied properly, this type of constraint can help the optimizer to focus on the more favorable design space and speed up the convergence process, as found in [18]. To measure the degree of the terrain ruggedness, a terrain ruggedness index (TRI) is introduced. This index was first proposed by [PERSON] et al. (2007) as a quantitative measure of topographic heterogeneity. A dimensionless version of this index is developed in the current framework, which can be calculated based on the elevation data of grids as defined in Section 2.1.