## 6 Results and Discussion
### Single Hour Analysis  
The 19 th hour of the typical day in ordinary operating conditions (Figure 7b) has been used to analyse the performances of the optimization tool. Preliminary, no uncertainties for the flexibility prices and no budget cap have been considered. Therefore, the only uncertainty source is the actual response of the resources to the DSO request. In the robust optimizations, the a priori risk has been changed from 1% to 30%, while the deterministic optimization is characterized by a 50% risk, being obtained by using the expected values. Because the technical issue is an excessive voltage drop, the only flexibility service required is a demand curtailment. Without storage, the deterministic solution involves 9 secondary substations (SS), from 13 th to 22 nd node (except the node 14), for an overall curtailed demand of 801 kW. All the loads are curtailed to their maximum availability, apart from node 13 (Figure 9a and Table 5). This result is quite obvious because further the load from the primary substation, more effective its curtailment for raising the nodal voltage profile. All the results of robust optimization require a higher amount of curtailed demand and a greater number of resources involved, due to the uncertainty in the actual curtailment that can be the 20% lower than requested. By decreasing the acceptable residual risk, the amount of flexibility to purchase increases, with the higher values registered for the 1% a priori risk (11 secondary substations involved for a total curtailed demand of 1145 kW). This observation confirms the correctness of the robust optimization methodology implemented. Indeed, if a higher protection against adverse changes of uncertain coefficients is desired (low residual risk), constraints must be checked against all cases where up to \(\Gamma\) of these coefficients can change. When the pre-fixed a priori risk grows from 1% to 30%, the uncertainty budget \(\Gamma\), calculated for the minimum nodal voltage constraints, lowers from 9 to 3, allowing less resources to assume their worst response. In Table 5 the secondary substations that change their expected response in the most critical constraint  
Figure 9: Comparison of flexibility exploitation (differentiated by resources with different colours) by varying the a priori risk: (**a**) deterministic and robust optimization without storage; (**b**) deterministic and robust optimization with storage. (minimum voltage limit for node 22) have been highlighted (greyed cells). It is interesting to notice that the adverse behaviour has not been assigned to the DR resources strictly in order of distance from the primary substation, because the effectiveness of their flexibility depends also on the amount of demand that can be curtailed. For instance, SS 17 can reduce its electricity demand more than twice the curtailment of nodes 18, 19 and 20; so, the improvement of the nodal voltage profile along feeder _F2_ is greater by acting on node 17, even if it is closer to the primary substation. The repercussion of the robustness against problem uncertainties is a deterioration of the objective function: Figure 10a shows the higher flexibility costs for any robust solution versus the deterministic result, and the increment of this cost with the reduction of the accepted residual risk. The relative high value of this cost (61.3 kf for the deterministic solution and 87.6 kf for the robust solution with 1% of a priori risk) derives from the extreme hypotheses adopted for this case study (minimum nodal voltage limit of 0.96 p.u., maximum existing risk of technical constraint violation with \(R_{b,h}=365\) h per annum) that bring to a huge exploitation of demand curtailment. Due to the choice of \(T=1\) h, the optimal amount of flexibility accepted from each DR resource in the local ancillary services market (\(F_{d}^{DR}\)) always equates the provision of flexibility service (\(\Delta P_{d,h}^{DR}\)). Figure 10: Comparison of the solutions in terms of flexibility cost (**a**) and residual risk (**b**).