# A Learning-Based Methodology to Optimally Fit Short-Term Wind-Energy Bands
## 1 Introduction
### Main Goals of This Work  
Regardless of the technique used to narrow uncertainty with bands, those works mentioned in Section 1.1 share a common characteristic: the electric-power is the magnitude to be captured. For some applications and/or contexts, the _energy_ coming from wind sources--which is a derivative of the wind's power anyway--is an important magnitude itself, and since it is the outcome of integrating the previous, results less noisy and easy to capture. This work aims on crafting bands representative of the energy to be produced along some periods, rather than focusing on accurate measures of the instant power. It is worth mentioning that since both magnitudes are strictly dependent, a fitted power-process cannot be outside energy-bands for too long. Thus, though this kind of bands could resemble the other in their shape, strictly speaking, they are different.  
Regarding the process to craft such wind-energy bands, the idea goes by designing a learning model that is fed from: wind-power forecasts as a mainstream of what to expect for the days to come, and actual wind-power data to incorporate information related to the historical deviations of the process. The area inside the band is a measure of the quality of such calibration. The smaller the area, the better the quality of the fitting. So, crafting energy-bands of minimal area is our objective. Assuming persistent behavior, an optimal fitting over a training set is expected to replicate reasonably well over other independent instance, so a historical calibration could be used to estimate energy-bands in the near future.  
Summarizing, our interest is not focused on particular power trajectories and their probabilities but aims on crafting optimal area energy-bands around wind-power forecasts, in such a way that the energy outside those bands be bounded. At this respect, our article presents a novel approach. The method is purely nonparametric, since it makes no assumptions on the physical phenomena, nor any hypothesis about the involved random processes (conditions on homogeneity in time, seasonal behavior of the temporal series or any kind of markovian hypotheses, are not necessary for this work). Our proposal is based  
Figure 1: Geographical distribution of renewable sources. Leftmost: wind-power (blue), solar-photovoltaic (yellow) and biomass thermal units (red). Rightmost: hydroelectric dams. [source UTE: 2019 and [12]].  
upon a mixed integer optimization problem, which aims on getting the narrowest average band around a set of forecasts, or a combination of forecasts, that keeps the off-band aggregated energy below a given threshold. As an innovation, to allow the optimization to go as farther as possible, the model enables to discard up to a given percentage of training samples that are treated as _atypical profiles_.  
At first glance this last feature might look risky, in the sense that, in advance, one cannot tell whether the day to come will match or not an atypical profile. However, as we see later, when the method is trained with more than one forecast, whenever they are conditionally independent or weakly dependent, a combination of those forecasts and their bands allows to regain the lost confidence; the result is a more accurate band than those of constructions computed by separate. This is another remarkable point of the present work, since it helps to improve band's quality by taking the best of more than one forecasts provider.