## 1 Introduction
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Category** & **Algorithm** & **Percentage of Articles** & **Domain** \\ \hline \multirow{8}{*}{Heuristic} & Genetic algorithm & 32.82\% & Discrete \\  & Genetic algorithm & 6.56\% & Continuous \\  & Evolutionary algorithm & 4.65\% & Continuous \\  & Particle swarm & 3.28\% & Discrete \\  & Particle swarm & 6.55\% & Continuous \\  & Ant colony optimization & 1.64\% & Continuous \\  & Monte Carlo simulation & 1.64\% & Discrete \\  & Simulated annealing & 4.92\% & Continuous \\  & Harmony search & 1.63\% & Discrete \\  & Pattern search & 4.92\% & Continuous \\  & Random search & 3.28\% & Continuous \\  & Greedy heuristic algorithm & 3.28\% & Discrete \\  & Greedy heuristic algorithm & 1.64\% & Continuous \\ \hline Mathematical & Mixed integer programming & 13.11\% & Discrete \\  & Gradient-based optimization & 13.11\% & Continuous \\ \hline \hline  & Total & 61 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Distribution of wind farm optimization search papers according to the optimization technique. The computation time can increase if, in addition, the Hessian matrix is calculated at each step. [PERSON], in [9], develops differentiable mathematical models to handle the Concession Area (CA) constraints and compares the performance (for an interior point method) using FDAG and the exact gradients, showing, for the former, significantly lower computation times at the expense of a negligible deviation in accuracy. The main problem is that it introduces numerical errors that can affect the convergence of the algorithm, especially for highly constrained problems. This same author, in [10], compares a GD-based multi-objective algorithm to an NSGA-II, claiming the superiority of the former in terms of coverage, spread, and computational cost. Including spatial constraints in the search space is a design feature addressed in many other works related to layout optimization. A solution for irregularly shaped and discontinuous polygons is also proposed in [11] by identifying the nearest edge and polygon, and in [12] by means of dynamic penalty functions. In other works where GD methods are used, they are often restricted to very local searches [13], or focused on fine-tuning a heuristic algorithm [14; 15]. Furthermore, the optimal location problem is not convex and GD methods may be incapable of finding the global optimum. Consequently, these methods are often combined with some other ones to escape from local maxima. In [7], triangulation is used as the starting point of the gradient method, with the additional idea of maximizing turbine dispersion, thus reducing maximum turbine loads. [PERSON] et al., in [16], use an stochastic gradient descent (SGD) and estimate the gradient of the AEP using the Monte Carlo simulation, achieving lower computation times as long as the farm size is above 225 turbines. This article presents a method whose search engine is a GD-based optimization. The traditional method has been improved with the following contributions:  
**Linear aggregation of power deficit (LAPD) as the wake aggregation method. ** When a turbine is immersed in the wakes of several upstream turbines there are different aggregation methods to obtain the resulting deficit. A new power deficit aggregation method is proposed and used that takes advantages of the linear properties of conservative fields. Based on this linearity, it drastically simplifies and speeds up the computation of the objective function gradient. Two consequences are derived:  
* It enables the use of look-up table (LUT). * The linearization reduces the complexity of the optimization problem, transforming it from an original problem involving 2 \(nt\) variables, into \(nt\) optimization problems, each involving two variables. **Greedy reposition of local maxima. ** Turbines with higher power deficits will be reallocated to avoid local maxima. The linearization in the aggregation method allows the calculation of a map of power deficits in a few seconds (typically 2.5 s for 200,000 locations).