# A Learning-Based Methodology to Optimally Fit Short-Term Wind-Energy Bands
## 3 Optimization of Wind-Energy Bands
### Experimental Evaluation  
The experimental evaluation of this work is based upon a later open data from the Uruguayan Electricity Market. From that past repository, we chose two independent forecast sources: Garrad Hassan and Meteologica. The data were pre-processed using a power assimilation methodology, which fits forecasts along the first 6 h in order to match the starting state (\(w_{0}^{d}\)). The exact process is described in paper [3]. The used forecasts from Garrad Hassan were those issued at 1 AM between 5 April 2016, and 10 March 2017. Within this period there are 302 days where both, forecast and actual data, are complete. Regarding the other provider (Meteologica), the number of complete records is 394, with dates of issue ranging from 1 January 2016, to 10 March 2017. Regarding our own forecast (PSF44), synthesized up from a series of actual power registers, we used the same 730 days of between years 2014 to 2016 that were used upon the first part of Section 2.1. Best performance was found by using \(K=4\) and \(W=4\) (acronym PSF44 refers to those parameters). Throughout this work, we relied upon IBM(R) ILOG(R) CPLEX(R) Interactive Optimizer12.6.3 as the optimization solver. The server was an HP ProLiant DL385 G7, with 24 AMD Opteron(tm) Processor 6172 with 64 GB of RAM. After running model (2) over a training set comprising around 30% of Meteologica's days, we find bands like those sketched in Figure 9.  
The x-axis represents the number of hours ahead for each forecast, while the y-axis corresponds to the PLF. Blue curves are associated with power forecasts while red ones are the actual values. Finally, the grey area represents the wind-power band for \(\theta=0.05\) and \(\lambda=1\). Since \(\lambda\) equals 1, every day within the training set must be effectively included; that is, \(y_{d}=1\) for each \(d\in D\), so all days are treated as regular. Furthermore, when \(\lambda=1\) then (2) turns out to be a pure linear programing problem, and running times are within  
Figure 9: Wind-energy bands for three random days within Meteologica’s training set [\(\lambda=1\), \(\theta=0.05\)]. the second. Fixing \(\lambda\) to 1, it is of interest to explore how \(\theta\) modifies the bands. Figure 10 shows the result over the same training set when \(\theta=\) 0.01. Observe that bands in Figure 10 are wider than in Figure 9, which is expected since we are less tolerant respect to how much energy lies outside those bands. In order to balance reliability and thickness, it is of interest to compute how much area do bands cover as we change \(\theta\) while keeping \(\lambda=1\). The training in all of the previous cases was performed over a set \(D\) of 120 randomly selected days out of a set of 300 days in common for all providers. The common complement, i.e., the set of (180) days shared by these three forecasts and not being in the _training-set_, is used as the _test-set_. The calibration of PSF44 was crafted using the set of 430 contiguous days previous to those of training and test sets. Experimental evaluation (see leftmost of Figure 11) verifies that the average width of the bands, when trained over the entire training set of forecasts (\(\lambda=1\)), falls down rapidly to 0, which is reached upon both companies when \(\theta\) is close to 0.2. Although similar, Meteologica's bands (blue) are always better than Garrad Hassan's (red). PSF44 (green) requires much wider bands to achieve the same grades of reliability. The middle plot shows the relative difference between widths of original bands (those of the training set), and widths computed over the test-set using the corresponding \(x\) vector found for each \(\theta\). It is worth mentioning that widths are always similar (divergence is low), so the objetive function in (2) when evaluated over the training set is representative of what happens outside it. This is sustained even for relatively higher \(\theta\) values right below 0.2, where widths tend to zero and the relative deviation makes no sense to be accounted.