# Simulation of Wind Speeds with Spatio-Temporal Correlation
## 5 Conclusions  
The growing trend of incorporating wind energy into electric power networks makes its characterization a mathematical problem. The inherent uncertainty due to errors made during the data collection process, together with the unpredictability nature of wind speed, makes it difficult to accurately obtain the power to be injected into the network at each moment. The simulation of wind speed series with a certain degree of accuracy can be carried out after obtaining the models of the behavior of each location and the relationships among them through the calculation of the spatial and temporal correlations. The method presented in this paper is intended to deal with the errors that occur in the data collection period in which a wind speed value is either unknown or known, but its reliability cannot be assessed. By calculating the spatial correlations, it is possible to identify the trends of the wind speeds for the set of locations. If temporal correlations are also considered, it is possible to characterize the wind speeds in different time intervals. By simulating random values, it is possible to characterize the wind speed in different periods or stages. The simulation makes it possible to cover the gaps that arise due to the lack of knowledge of the wind speeds and also to estimate the values that can occur in subsequent periods of time. This facilitates knowing the power injected into the electrical network with a certain degree of reliability. Additionally, the accuracy of the method is as close to reality as desired when a level of tolerance is established. In this work, a value of 0.05 is considered reduced enough. As a result, errors are lower than 5% in all simulated cases. The data used in this paper correspond to 2019 in the meteorological stations of Fragavella, Labrada and Lanzos, all of them selected from the MeteoGalicia website. The wind speeds are considered to fit Weibull distributions. It was verified that the studied locations meet this assumption. There seems to be an inverse relationship between distance and correlation. When the distance between locations is high, the calculation time is shorter because the Cholesky decomposition initializes the iterative algorithm closer to the solution, so the number of iterations is low. The consideration of a higher number of locations directly affects the calculation time. On one hand, a large number of operations per iteration is needed, having to perform  
Figure 10: Evolution of the calculation time required to carry out each iteration of the simulation stage considering a different number of locations, where the dotted line corresponds to the first iteration and the solid line to the rest of iterations. \(\frac{3m^{2}+m}{2}\) calculations, where m is the number of locations, and on the other, the solution of the correlated series obtained before Cholesky decomposition is increasingly far from the objective because values corresponding to the temporal correlations increases more rapid, \(m^{2}\), then the spatial correlations, \(\frac{m^{2}-m}{2}\), which can influence the number of iterations required, making the problem with a large number of locations become unapproachable. The modeling stage, where the relationships between locations are reproduced, is practically instantaneous, with a duration of a few seconds. Nevertheless, the simulation stage, where the randomness of the wind speed is reproduced, is more time-computing, reaching several hours in some cases. Moreover, on one hand, the variation of the collection data period does not involve a significant time. On the other, the number of simulated values affects the calculation time, and this can make the problem unapproachable in the case of the simulation of a large one. Conceptualization, M.C.-C. and D.V. ; methodology, M.C.-C. and D.V. ; software, M.C.-C.; validation, M.C.-C. and J.M.-T.; formal analysis, M.C.-C. and A.E.F.-L; investigation, M.C.-C. and D.V. ; resources, D.V.