# A New Affinely Adjustable Robust Model for Security Constrained Unit Commitment under Uncertainty
## 5 Tests and Results
### IEEE RTS 24-Bus Power System  
To further explore the performance of the proposed AARO-SCUC model, several experiments were performed with the IEEE RTS 24-bus power system. This system comprises 24 buses of which 17 are demand buses, 26 conventional generators, nine wind generators and five solar generators that are connected to 31 transmission lines. The data for this test system are also available in GitHub [70]. The maximum levels of uncertainty considered for the model were set at \(\pm 5\%\) for demand, and \(\pm 10\%\) for wind and solar generation. Figure 6 shows the behavior of the optimality gap (blue bars) and the computation time of Algorithm 1 (dashed orange line) based on the amount of uncertain information disclosed from different periods in the past (\(n\)). For the sake of conciseness and based on the results presented in Section 5.1, this figure only includes the results from \(t-1\leq\tau\) to \(t-6\leq\tau\). In this case, both the computation times and the optimality gap tend to increase as more uncertain information revealed from past periods (\(n\)) increases. Figure 4: Number of violated constraints vs. materialization of forecast errors, 6-bus system. Figure 5: Zoom of Figure 4.  
For all the experiments, the dual solution (i.e., the lower bound of the cost) was 0.49430 MUSD. Initially, the differences in the optimality gap and computation times when comparing the results for \(t-1\leq\tau\) and \(t-2\leq\tau\) are very small. By contrast, in \(t-1\leq\tau\), the optimality gap is 0.0149%, obtained with an MILP solution of 0.494373 MUSD of cost; meanwhile, for \(t-6\leq\tau\), the gap is 0.5200%, obtained with a solution of 0.496880 MUSD of cost. The solution with \(t-6\leq\tau\) produces a solution with a 0.508% cost increase when compared to the cost of the system for \(t-1\leq\tau\). Furthermore, the computation times of Algorithm 1 vary considerably as \(n\) increases. It takes 1417 s for \(t-1\leq\tau\), whereas it takes 81,000 s for \(t-6\leq\tau\), the computation time at \(n=6\) being 57 times the computation time it takes to solve the model for \(n=1\). This confirms the results of Section 5.1, that is, high quality solutions can be obtained in short running times with uncertain information revealed from very few past periods. Long computing times are obtained in this system for \(t-6\leq\tau\) because we assume that all buses have uncertain loads, something that considerably increases the computational complexity of the AARO-SCUC model. For instance, this feature renders the solution of the AARO-SCUC model impossible for this system without the dynamic generation of \(N-1\) security constraints of Algorithm 1. Note, however, that the solution of the AARO-SCUC model using Algorithm 1 may work well in large-scale systems by exploiting one of the advantages of affine policies. In that case, linear decision rules precisely allow the tractability of these kind of systems by taking into account some practical considerations. For instance, one should define, within the affine policies, a subset of only those hydrothermal generators that have sufficient technical capacities to respond to uncertainty deviations. Likewise, one can define in the affine policies, only the subset of buses with uncertain loads. These considerations decrease the computational burden of the model and allow for obtaining proper solutions. The next step is to verify that such solutions are robust using again the post-hoc analysis with MCS described in Section 5.1. Figure 7, as well as the results presented in Section 5.1, show that realizations of the uncertain variables within the uncertainty box do not generate violated constraints (infeasible unfeasible solutions). On the contrary, for realizations of the uncertain variables greater than 100% (values outside the uncertain box), the number of violated constraints is greater than zero. This means that at least one constraint is violated when evaluating the solution of the AARO-SCUC model.