## 6 Conclusions and Discussion
### Discussion  
The power deficit aggregation method can be applied beyond the gradient method. In fact, it has also been used for the EA as a reference for comparison with the proposed gradient method but it can also be used with other heuristic algorithms. However, it is in the gradient method where, taking advantage of its linear superposition character, a problem of \(2 nt\) variables can be decomposed into \(nt\) problems of two variables, making the improvement more evident. From a purely mathematical point of view, this method does not guarantee the achievement of the global optimum. Given the uncertainties in the modeling of the system, mainly due to inaccurate wind condition forecasting and imprecise wake modeling, deviations in the assumed conditions or models can lead to unrealistic objective function values. Therefore, an exhaustive search for the global optimum is not worthwhile if its neighborhood of monotonicity is very small. In this sense, flatter local optima are preferable, even though they may be of a lower value. Another problem cited in the literature, which appears for large farms, is the high computing time. Throughout this work, three tools (PS, LAPD and LUT) have been provided to simplify and accelerate the calculation of the objective function and its gradient. Thanks to these tools, the search for the optimum is carried out in a much shorter time compared to previous works. This last tool (LUT) is only possible due to the linearization achieved with LAPD. Compared to the EA, it is observed that the GD actively seeks the contours as positions for the turbines, noting that these are locations where the potential (i.e., the power deficit) is minimized. On the contrary, for a search engine like an EA, based on crossovers and mutations, it will be more difficult to occupy such a contour, especially when it is irregular. In summary, this method is by far the fastest of all layout optimization methods found, it does not present limitations regarding the number of turbines or the area of the CA, and it naturally manages land-use constraints for irregular CAs. The solutions offered by the algorithm are consistent, similar to those of other authors who have thoroughly studied layout optimization, and also similar to those of real parks. As a limitation, it is imposed that the turbines must be of the same model and height, and there should not be a nearby farm or obstacle that could cast a shadow on the study area causing heterogeneous wind distribution patterns. It is also restricted to OWFs. For onshore wind farms, several factors generally cause the disturbance exerted by turbine \(i\) on turbine \(j\) to differ from that exerted by turbine \(j\) on turbine \(i\). This symmetry is the basis of the mathematical formulation, and as a result, the proposed reasoning would not be valid. These factors include differences in the rotor height causing different ground interactions; variations in free-stream wind conditions due to a rough orography; different surface roughness and the presence of nearby obstacles. As a future line of research, disconnected CAs with exclusion zones within them will be studied, as well as the effect of existing parks in the vicinity of the CA. Conceptualization, A.G.G.-R; Data curation, J.S.-G. and J.V.M.-D.; Formal analysis, A.G.G.-R. and J.M.R.-F; Funding acquisition, J.M.R.-F. and [PERSON]; Investigation, [PERSON] and [PERSON]; Methodology, [PERSON]-R. and [PERSON]-[PERSON]; Project administration, [PERSON]-[PERSON] and [PERSON]; Resources, [PERSON]; Software, [PERSON]-R. and [PERSON]-[PERSON]; Supervision, [PERSON]-R.; Validation, [PERSON]-R. and [PERSON]-[PERSON]; Visualization, [PERSON] and [PERSON]-[PERSON]; Writing--original draft, [PERSON]-R.; Writing--review and editing, [PERSON] and [PERSON]-[PERSON]. All authors have read and agreed to the published version of the manuscript. This paper is part of the research project PID2021-127550 OA-I00 funded by MICIU/AEI/10.13039/501100011033. Not applicable.