# A Learning-Based Methodology to Optimally Fit Short-Term Wind-Energy Bands
## 3 Optimization of Wind-Energy Bands  
Optimization of wind-energy bands is in the core of this framework. We provide an expression to compute a band around any forecast, and, for that concrete formula, we seek for the narrowest band that satisfies a set of constraints, which imposes limits to the actual process in its deviations in accordance with the historical behavior. A traditional approach would go the way of setting constraints to keep the power deviation under certain boundaries. Conversely, this work aims on minimizing the expected off-band energy. The previous is explained by the particulars of the Uruguayan electricity installed plant, but it is also justified by trends of new technologies. Around 98% of the electricity annually consumed in this country comes from renewable sources (see [17]). In average, 50% is from hydroelectric sources, while 35% is from wind-power. All of the hydroelectric dams in Uruguayan have water reservoirs; two of them (Bonete and Salto Grande, see Figure 1) are particularly huge. Almost 40% of the hydroelectric capacity is located after the greater lake (Bonete's), which would take 5 months to empty at full-power. Therefore, in fact, the hydroelectric plant also constitutes an accumulator, i.e., a kind of battery that can plenty compensate short-term fluctuations of the power coming from wind turbines. Hence, regarding Uruguayan short-term planning concerns, an accurate prediction of energy boundaries is more convenient than a power forecast of limited punctual quality. In a complementary manner, smart-grids capabilities are rapidly advancing towards active applications, capable of dynamically adjusting portions of the demand to adapt them to fit system needs (see [18]), while electricity storage units based on batteries are just around the corner (read [19] and also see the \"Neoen & Tesla Motors\" project in Australia). Therefore, in the near future, this work could be a useful experience for other countries. The information required to determine an instance of our optimization problem comprises the following data sets. At first place, we need a historical of wind-power forecasts. We consider a collection \(\mathcal{P}\) of deterministc registers that involves short-term point forecasts over a horizon of a few days ahead; i.e., a family of vectors \(p^{d}\in[0,1]^{T}\), with fixed \(T\), which is set by the number of samples along the time horizon. Here, \(d\) is the index for each day on which the construction of a band begins; \(d\in D\), being \(D\) the set of indices for days with historical observations. Wind-power forecasts usually span from one up to three days, i.e., from 24 to 72 h, and time is discretized at a rate of one sample per hour. Let \(T-1\) be the limit of hours ahead available for each forecast. We assume that all forecasts share the same time horizon, and that in \(t=0\) the current power is the only data known for sure. As we mentioned earlier, for simplicity the wind-power is expressed as the PLF, which corresponds to the actual power generated divided by the sum of the installed power capacity of wind turbines in the system at each moment. Thus \(p^{d}_{t}\in[0,1]\) is the normalized point forecast of the wind-power \(t\) hours ahead, within the vector associated to the forecast issued on the day \(d\). The second part of the input data set comprises the actual historical wind-power time series samples, grouped into a collection: \(\mathcal{W}\), whose elements \(w^{d}\in[0,1]^{T}\) are also assumed normalized. Hence, \(w^{d}_{t}\in[0,1]\) is the actual PLF measured \(t\) hours after the beginning of the day \(d\). For consistency, since the current state can be measured rather than forecasted, \(p^{d}_{0}=w^{d}_{0}\) for each day \(d\). Observe that the set \(\mathcal{W}\) usually has duplicated records, for instance:  
Figure 7: Example of matching subsequences within a historical of symbols. \(w_{24}^{d}=w_{0}^{d+1}\). Despite that, we have chosen this format to simplify those expressions that link with forecast information.