# A New Affinely Adjustable Robust Model for Security Constrained Unit Commitment under Uncertainty
## 5 Tests and Results
### 6-Bus System  
The system shown in Figure 2 has six buses, seven lines and three generation units (two thermal units and one hydro unit). Renewable Energies Sources (RES) are located at buses 1 (PV, 20 MW) and 3 (wind, 200 MW), respectively. Meanwhile, the total load is distributed at buses 3, 4, and 5. The maximum uncertainty levels of load and RES error forecasting are set at \(\pm\)5% and \(\pm\)10%, respectively. The input data and the model can be found on GitHub [70]. Figure 3 shows the value of the operating cost of the power system on the left axis (blue bars), run time using Algorithm 1 to solve the AARO-SCUC (dashed orange line), and run time of the AARO-SCUC solution without using Algorithm 1 (dashed magenta line) on the right axis. This figure shows the behavior of the OF and running time when different numbers of past time periods are considered in the AARO model for the known uncertainty parameters. This analysis was performed by adding the logical condition \(t-n\leq\tau\) into the model, with \(n\) being the number of past time periods containing known uncertainty data (with \(1\leq n\leq 24\)). Figure 3 reveals that using Algorithm 1 brings computational advantages to the proposed AARO-SCUC model, when more known uncertain information from past periods  
Figure 1: Numerical experiments for the uncertainty sets. Figure 2: 6 bus test system. is desired. That is, the execution times of the model using such algorithm (dashed orange line) are much shorter compared to the times it takes for the model without using it (dashed magenta line). For example, if it is desirable to take into account the uncertain information disclosed from 23 past periods including the current period (\(t-24\leq\tau\)), the solution of the AARO-SCUC model takes 30 s when Algorithm 1 is used, whereas it takes 105 s without using it. That is, Algorithm 1 provides a 2.5 speed-up factor when solving the proposed AARO-SCUC model. These results are in line with [60], which have proven the effectiveness of this computational approach for reducing the solution times of SCUC as the complexity and size of the power system increases. Furthermore, Figure 3 shows that, when considering the revealed uncertainty of only a past period (\(t-1\leq\tau\)), the operating cost of the system is 0.12654 MUSD, while the cost of the system for \(t-24\leq\tau\) decreases to 0.12646 MUSD. This value represents a negligible decrease of 0.0567% in the OF that comes at the price of multiplying the run time by a factor of six. Furthermore, from \(t-2\leq\tau\) to \(t-24\leq\tau\), the cost does not change, but the run time has an increasing behavior. This results shows that it is possible to obtain good quality solution in short running times by considering only the information from a few past periods. It is noteworthy that, in this case, the optimality gap was zero for all runs. Once the computational effectiveness of the proposed approach is illustrated, it is also important to verify the robustness and stability of the solutions provided by the AARO-SCUC model. For this purpose, a post-hoc analysis using MCS was executed. Figure 4 shows the feasibility behavior of the solutions of the AARO-SCUC model for different values of \(n\) in expression \(t-n\leq\tau\) (i.e., more (\(n=24\)) or less (\(n=1\)) information about past realizations of the uncertain variables are considered) after evaluating a large number of realizations (scenarios) of the uncertain variables inside and outside the uncertainty sets generated through MCS (represented in the horizontal axis of this figure). This figure shows that the number of violated constraints is zero for those realizations of the uncertain variables that are within the range 0% to 100% of the maximum uncertainty level. As expected, the solutions of the model are robust, generating feasible solutions (valid operating points) for all realizations of uncertain variables within the box. For values outside the box, (i.e., greater than 100%), the behavior of the obtained solution may vary depending on the value of \(n\).