## 6 Results and Discussion
### Single Hour Analysis
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Core Setting**} & \multicolumn{4}{c}{**Residual Risk**} & \multirow{2}{*}{\(\Gamma\)} & \multicolumn{4}{c}{**DR Resources Involved–Curtailment**} & \multicolumn{4}{c}{**Requested [kW]**} & \multicolumn{4}{c}{**BESS**} \\ \cline{2-19}  & \multirow{2}{*}{**Ante**} & \multirow{2}{*}{**Post**} & & \multirow{2}{*}{\(\Gamma\)} & & **SS  
With storage, a mix of the two kind of resources has been obtained. The deterministic optimization identifies as optimal the installation of two BESS in the furthest nodes 22 and 21. Due to \(T=1\) h and the hypothesis of \(SoC_{0}=50\%\) of the battery capacity, the optimal BESS sizing and its operation are quite intuitive. Indeed, once the optimization algorithm identifies the allocation node and the amount of power that has to be discharged (because the technical issue is an excessive voltage drop), the power rate is fixed to this value (\(Pn_{k}^{B}=\left|\Delta B_{k,h}^{B}\right|\)) and the capacity rate is chosen so that at the end of the single hour considered the state of charge achieves the minimum admissible capacity (\(SoC_{k,1}=\sigma_{m}En_{k}^{B}\)). For instance, the two BESS of the deterministic solution discharge at their nominal power during the 19 th hour of the typical day. This power discharged (157 kW + 93 kW = 250 kW) is added to the demand curtailed (263 kW). Thanks to the increment of flexibility available in the furthest nodes, the total amount of flexibility service required is drastically reduced in comparison to the previous cases without storage (Figure 9), because the actions are more effective. This general result occurs also for the robust solutions. However, by reducing the acceptable residual risk, the weight of BESS increases to the detriment of DR actions. For instance, with an a priori risk of 1%, three storage devices are installed on nodes 22, 21 and 17, for a total power discharged at the 19 th hour of 253 kW + 96 kW + 48 kW = 397 kW. Simultaneously, only three secondary substations (21, 20 and 19) are involved in DR actions, for a total curtailed demand of 135 kW. This behaviour is justified mainly by the need of limiting the resort to uncertain resources in favour of the more reliable BESS, and by the flexibility costs adopted. Indeed, higher unitary costs of storage or lower remuneration prices for DR actions may change the exploitation ratio between BESS and DR, fostering this latter. In terms of objective function, the total cost of the robust solutions with BESS reduces slightly with the growth of the residual risk, passing from 49.4 k\(\xi\) with 1% risk to 48.8 k\(\xi\) with 30% risk. Also, the gap with the deterministic solution is lower if compared to the solutions without BESS. Obviously, this conclusion is son of the limited resort to uncertain resources that, consequently, contains the extra cost for guaranteeing the desired solution's robustness. The a posteriori risk of each robust solution (with and without BESS) is confirmed always lower than the corresponding a priori value (Figure 10b and second and third columns of Table 5). However, the most valuable result is that the installation of BESS in parallel to the exploitation of flexibility services from available DER leads to less risky and cheaper solutions. Revealing are the solutions with lowest a priori risk (1% and 5%), for which the optimal siting and sizing of BESS nullify the residual risk, i.e., the presence of storage allows covering all the uncertainties associated to the use of DR actions. Indeed, looking deeply at these robust solutions, all the auxiliary variables \(z_{i}\) of constraint (4) are set to zero, so excluding the uncertainty budget addendum \(\Gamma_{i}z_{i}\) and making evident that the few DR resources involved are all considered with their most adverse behaviour (worst-case scenario), neutralizing any risk of constraint violation.