## 4 Application
### Simulation of Alpha-Stable Random Variables  
To generate new cases by the methodology proposed, it worked with three standardized test systems. * System I: Based on the IEEE-104 bus electric energy system of Mainland Spain with 104 nodes, 62 thermal units, and 160 transmission lines [31]. * System II: Based on the IEEE-24 bus test system with 24 nodes, 24 thermal units, and 38 transmission lines [29]. * System III: Based on the IEEE-118 bus test system with 118 nodes, 54 thermal units, and 186 transmission lines [30]. All instances consider a 24-h planning horizon with one period per hour. We use alpha stable distribution to model the demand from the original systems analyzed. The parameterization one of alpha stable distribution is used, which is usually applied for modeling heavy tail and high-frequency data. This parameterization is represented in Equation (25) [46; 47]. \[\phi(\mathrm{k})=\mathrm{Infc}(k)=\left\{ \begin{array}{cc}\mathrm{i}\delta \mathrm{k}-\gamma^{\alpha}|k|^{\alpha}\Big{\{}1-\mathrm{i}\beta\mathrm{sgn}(k) \tan\frac{\mathrm{I}\alpha}{2}\Big{\}},&(\alpha\
eq 1)\\ \mathrm{i}\delta\mathrm{k}-\gamma|k|\Big{\{}1+\mathrm{i}\beta\mathrm{sgn}(k) \frac{\mathrm{I}}{\mathrm{I}}\Big{\}},&(\alpha=1)\end{array} \right\}, \tag{25}\]  
The parameters that characterize the stable distribution were obtained through the three methods generally accepted: (1) maximum likelihood, (2) quantile, and (3) regression. For the simulation of the alpha stable random variables in this work, the parameters generated by the regression method will be used. Better fit has been documented by the regression method for heavy-tailed data [47; 48]. Table 1 shows the estimated parameters for the fitted alpha stable distribution for each system, with the three estimation methods (maximum likelihood, quantile, and regression). The parameter alpha in three systems shows the presence of impulsivity in the series; all series are positive asymmetric and have high dispersion denoting the presence of heavy tails. That shows that demand in the 24 h of the day and the electricity consumption habits. The Chambers-Mallows-Stuck method [42] was used to simulate random variables that follow an alpha stable distribution. With the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**IEE-104 bus system** & \(\alpha\) & \(\beta\) & \(\gamma\) & \(\delta\) \\ \hline Maximum likelihood & 1.2000 & 1.0000 & 84.3954 & 423.414 \\ Quantile & 1.6132 & 1.0000 & 121.3540 & 291.3820 \\ Regression & 1.6938 & 1.0000 & 113.085 & 227.7650 \\ \hline
**IEEE-24 bus system** & \(\alpha\) & \(\beta\) & \(\gamma\) & \(\delta\) \\ \hline Maximum likelihood & 1.4000 & 1.0000 & 35.4172 & 160.6370 \\ Quantile & 1.6045 & 1.0000 & 41.7026 & 151.6880 \\ Regression & 1.8430 & 1.0000 & 42.8166 & 121.925 \\ \hline
**IEE-118 bus system** & \(\alpha\) & \(\beta\) & \(\gamma\) & \(\delta\) \\ \hline Maximum likelihood & 1.2000 & 1.0000 & 10.7543 & 54.0993 \\ Quantile & 1.7539 & 1.0000 & 15.1875 & 32.7828 \\ Regression & 1.5092 & 1.0000 & 13.3914 & 35.9108 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Alpha stable parameters. estimated parameters, 1000 alpha stable random numbers were simulated for each original test system according to the original energy demand ranges. To obtain the values in demand units of energy, the inverse transform method was applied. Figure 2 shows the simulation of random instances based on the IEEE-104 bus system with the alpha stable parameters \(S(\alpha=1.6,\beta=1.0,\gamma=113.3,\delta=227.7)\) and the histogram of these instances. The presence of extreme values is observed as well as heavy tails in the distribution of the generated data. This system is the one with the greatest impulsiveness and dispersion in the data. Figure 2: IEEE-104 bus system simulations. Figure 3 shows the simulation of instances based on the IEEE-24 bus system, with the alpha stable parameters \(S(\alpha=1.8,\beta=1.0,\gamma=42.8,\delta=121.9)\) and the histogram of these instances. Given the characteristics of the system, the demand peaks are observed.