## 4 Method: Optimization Using the GD + GR Method
### Formulation of the Gradient Descent Method  
Among the variants of the GD method, one of the most used for solving large-scale nonlinear optimization problems is the conjugate gradient method, introduced for nonlinear systems by [PERSON] and [PERSON] in the 1960s. For linear systems, it is faster than the basic GD method. However, in the case under study, obtaining an analytical expression for the  
Figure 5: Proposed Algorithm. Evolution of the overall power deficit along the optimization process. In magenta, the \(\mathbb{E}[\delta P]=\overline{\delta P}\) estimation using LAPD, which is the aggregation method used during the optimization. In green, the overall power deficit \(\overline{\delta P}\) estimation using LAVD. The pulse in red represents the GR of turbines. function to be minimized (AEP), as in [8; 9; 10], increases the computation time. This time increases even more if the inverse of (33) or its Hessian is required to obtain a correct value of the learning rate to meet the inequalities of the strong Wolfe conditions [24]. Traditionally, the GD method is based on the calculation of the gradient of an objective function, in this case \(\mathrm{U}=\delta P\), which depends on the positions of the \(nt\) turbines, giving rise to a gradient vector of 2 \(nt\) components. \[\
abla U=\left[\frac{\partial U}{\partial q_{1}}\cdots\frac{\partial U}{ \partial q_{2 nt}}\right]^{T} \tag{34}\]  
\[U(\mathbf{q})\quad\text{defined and differentiable in }\ S\subset\mathbb{R}^{2\,nt}\]  
where \(\mathbf{q}\) is the vector of components (x and y) for all of the \(nt\) turbines  
\[ \begin{array}{l}q_{2i-1}=x_{i}\\ q_{2i}=y_{i}\end{array} \quad\forall\ 0<i\leq nt \tag{35}\]  
and the set \(S\) defined as  
\[\mathbf{q}\in S\subset\mathbb{R}^{2\,nt}\iff\mathbf{t}_{i}\equiv(x_{i},y_{i} )=(q_{2i-1},q_{2i})\in CA\quad\forall\ 0<i\leq nt \tag{36}\]  
In this work, a method based on the basic GD will be used, but with the proposed modifications, which proves to be faster than any other optimization method based on GD or not. The common expression dictating the evolution of the solution \(\mathbf{q}\) at step \(k\) is given by the following:  
\[\mathbf{q}^{k+1}=\mathbf{q}^{k}-\gamma^{k}\
abla U(\mathbf{q}^{k}) \tag{37}\]  
where \(\gamma^{k}\in\mathbb{R}^{+}\) is the so-called learning rate and establishes the size of the steps taken in the direction of the negative gradient during the optimization process. 2.1 Decoupling the Optimization Problem in \(\mathbb{R}^{2\,nt}\) into \(nt\) Problems in \(\mathbb{R}^{2}\)  
As discussed in Section 2, when multiple wakes interact with the same downwind turbine, several aggregation methods can be used to estimate the speed deficit experienced by the air stream passing through the downwind turbine. With adequate modeling of wind conditions, the quadratic and linear methods (QAVD and LAVD) yielded results comparable to those obtained directly from SCADA systems. Another method with similar results to LAVD is LAPD; the power deficit generated by a downwind turbine affected by several upstream ones is the sum of the power deficits calculated considering each interaction separately. Despite a higher estimation deviation than QAVD or LAVD, the proposed method, LAPD, produces a similar power deficit map and, therefore, a similar distribution of gradients, as shown by comparing maps (a) and (c) of Figure 2.  
Additionally, it will be shown that by using LAPD, the optimization problem involving \(2\,nt\) variables shown in (33) is reduced to optimizing \(nt\) problems of two variables. Indeed, taking partial derivatives with respect to, for example, component \(x_{i}\), yields  
\[\frac{\partial U}{\partial x_{i}}=\frac{\partial\sum_{k}\overline{\delta P_{ k}}}{\partial x_{i}}=\sum_{k}\frac{\partial\sum_{j\
eq k}\overline{\delta P _{j\to k}}}{\partial x_{i}}\overset{\mathrm{LAPD}}{=}\sum_{k}\sum_{j\
eq k} \frac{\partial\overline{\delta P_{j\to k}}}{\partial x_{i}} \tag{38}\]  
and since the power deficits \(\overline{\delta P_{j\to k}}\) consider their interaction separately  
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