## 4 Testing the Problem Formulation
### Case III: Very High Wind Speed with Empty Battery  
In this case, conditions of extreme wind speed are analyzed by setting the average speed to 24 m/s (\(S_{W}^{a}\) = 24 m/s), while the battery remains empty (\(SOC_{B(0)}\) = 0.15).  
Figure 22 shows the wind speed (left) and wind power (right) for this case. As wind speed becomes higher than the cut-out speed (25 m/s) for those cases with wind speed oscillation, the wind turbine is taken out of service in order to preserve it. Thus, NL suddenly increases during the afternoon.  
GA evolution is shown in Figure 23, where a fast convergence is observed due to the fact that the battery has to be charged during the first hours of the day, reducing the number of possible combinations of the optimization problem.  
Management signal is shown in Figure 24. According to these results, battery should be charged during the morning. Then, a short resting period is advised so that enough energy is available to be discharged during the peak-load hours.  
Figure 23: GA evolution (Case III).  
Figure 22: Wind speed and wind power (Case III).  
Charging and discharging cycles are represented by means of the battery power and SOC shown in Figure 25 at the left and right sides, respectively.  
The reduction of NL as a result of BESS integration can be observed at the left side of Figure 26. NL without considering the influence of BESS is presented at the right side of Figure 26, where the increment of load demand during the afternoon can be clearly observed.  
GHG emissions are fully described in Figures 27-29, where the lack of wind generation during the afternoon, as a result of the extremely high wind speed, directly influences the behavior of all emission factors.  
Figure 26: Net load with and without battery (Case III).  
Figure 24: Battery management (Case III).  
Figure 25: Battery power and SOC (Case III).  
Energy surplus and ENS are reported in the left and right sides of Figure 30, respectively, where the energy excess is directly related to the operation of the diesel generator during the afternoon. As expected, there is no ENS because the diesel generator is able to supply any value of electricity demand.  
Figure 28: NO\({}_{\rm X}\) and CO\({}_{2}\) emissions (Case III).  
Figure 29: PM emissions (Case III).  
Tables 9-12 summarize the GHG emissions and diesel generation for this case. By comparing Tables 9 and 10, it is possible to observe that, in some situations, THC emissions can increase when BESS reduces NL in a considerable manner. Regarding the wind speed oscillation, as \(S_{W}^{b}\) increases, wind speed in the hours close to 15 h (\(S_{W}^{c}=15\) h) increases to a value higher than cut-out speed (25 m/s), removing the wind power generation. Thus, BESS supplies a part of the required power, but essentially most of it is provided by the diesel generator, increasing even more the emissions of CO, NO\({}_{X}\), CO\({}_{2}\), and PM to the atmosphere.  
In Table 12, it can be observed how the diesel generator reduces its power production or is disconnected. This happens in those hours between \(t=12\) h and \(t=20\) h, when BESS has an active role in mitigating NL.  
From the sensitivity analysis of Cases I-III, it is possible to conclude that the objective function defined in (30) offers reasonable results in terms of BESS management for peak-shaving. The next section studies the performance of TVMS-BPSO implemented to minimize this already tested objective function.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{S_{W}^{b}}\) & **THC (kg)** & **CO (kg)** & **NO\({}_{\mathbf{X}}\) (kg)** & **CO\({}_{2}\) (kg)** & **PM (kg)** \\ \hline
0 & 1.02 & 0.55 & 7.42 & 550.90 & 0.13 \\
0.1 & 1.02 & 2.89 & 16.05 & 833.17 & 0.24 \\
0.2 & 0.98 & 3.59 & 18.01 & 886.27 & 0.27 \\
0