## 4 Testing the Problem Formulation
### Case I: Low Wind Speed with Fully Charged Battery
BESS management is shown in Figure 15, where the battery is charged during the morning in most situations. However, discharging actions are also advised sometimes in the morning, and this is observed when wind generation reduces as a consequence of wind speed profile oscillations. Initially, the battery is empty, so that discharging the battery in this situation does not result in any load satisfaction, because the battery is not able to provide any power. In other words, battery discharge when \(SOC_{B(t)}=SOC_{\text{B}}^{\text{t}min}\) is equivalent to the battery disconnection obtained by setting \(g_{(i,t,k)}\gets 0\). Figure 16 shows the results obtained from the simulation of the management signal previously shown in Figure 15. Battery power (Figure 16 left) and SOC (Figure 16 right) are presented, and the battery is charged during the morning and then discharged during the afternoon. The situations of high wind speed oscillations result in a reduction of energy surplus, and consequently less power is available to be used in the peak-shaving process during the afternoon. Figure 14: GA evolution (Case II). Figure 15: Battery management (Case II). Figure 17 shows the NL profile depending on if the effects of BESS are considered (left) or not (right), calculated by using (22) and (21), respectively. By comparing these results, it is possible to observe how NL-peak is reduced. As NL is so low when BESS is incorporated, energy surplus is produced because the diesel generator is forced to operate at its minimum capacity in order to satisfy a very low load. This operating mode produces a fixed amount of GHG emissions. This reasoning can be verified in Figure 18 for THC (left) and CO (right), in Figure 19 for NO\({}_{\chi}\) (left) and CO\({}_{2}\) (right), and in Figure 20 for PM. Figure 16: Battery power and SOC (Case II). Figure 17: Net load with and without battery (Case II). Figure 5:0 THC and CO emissions (Case II). Figure 21 shows the daily profiles of energy surplus (left) and ENS (right), respectively. As wind generation is abundant from the high wind speed, energy excess is observed at almost any time. Conversely, there is no ENS. Tables 5 and 6 report the cumulative daily values of GHG emissions for different wind speed profiles. Then, the reduction on the emitted pollutants as a consequence of BESS integration is reported in Figure 5:1 Because the diesel generator is operating at its minimum allowed power (\(P_{D(t)}=P_{D}^{min}\)), the reduction of GHG emissions is the same for all the factors considered, up to 12.5%. To improve understanding of the HES operation, Table 8 presents the output power of the diesel generator with and without considering the BESS operation. In bold are shown those situations where the diesel unit is disconnected, all of them during the first peak between \(t=12\) h and \(t=13\) h.  
Figure 5:2 Excess and ENS (Case II). Figure 5:3 NOx and CO\({}_{2}\) emissions (Case II). Figure 5:4 PM emissions (Case II). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{S_{W}^{b}}\) & **THC (kg)** & **CO (kg)** & **NO\({}_{\mathbf{\chi}}\) (kg)** & **CO\({}_{\mathbf{2}}\) (kg)** & **PM (kg)** \\ \hline
0 & 1.17 & 0.62 & 8.48 & 629.60 & 0.14 \\
0.1 & 1.17 & 0.62 & 8.48 & 629.60 & 0.14 \\
0.2 & 1.17 & 0.62 & 8.48 & 629.60 & 0.14 \\
0.3 & 1.53 & 0.82 & 11.13 & 826.35 & 0.19 \\
0.4 & 1.68 & 0.90 & 12.19 & 905.05 & 0.21 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Total GHG emissions without battery (Case II). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\mathbf{S_{W}^{b}}\) & **THC (\%)** & **CO (\%)** & **NO\({}_{\mathbf{\chi}}\) (\%)** & **CO\({}_{\mathbf{2}}\) (\%)** & **PM (\%)** \\ \hline
0 & 12.5 & 12.5 & 12.5 & 12.5 & 12.5 \\
0.1 & 12.5 & 12.5 & 12.5 & 12.5 & 12.5 \\
0.2 & 6.25 & 6.25 & 6.25 & 6.25 & 6.25 \\
0.3 & 0 & 0 & 0 & 0 & 0 \\
0.4 & 0 & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Reduction of GHG emissions (Case II).