## 3 Risk Assessment of Technical Constraints Violation
\begin{array}{cccc}C_{F}\leq B_{cap}&-C_{F}\leq 0\\ &-\sum\limits_{g=1}^{N_{RES}}\underline{a}_{d,h,h,h}^{V,ps}\Delta p_{gh}^{RES}+ \sum\limits_{d=1}^{N_{DR}}\underline{a}_{d,h,h}^{V,ps}\Delta p_{dh}^{DR}- \sum\limits_{k=1}^{N_{N}}\left(\underline{d}\right)_{i}\Delta p_{kh}^{B}-s_{ i,h}^{Vmax}\quad\forall i=1,\ldots,N_{N}\;;\;\forall h=1,\ldots,\;T\\ &\sum\limits_{g=1}^{N_{RES}}\underline{a}_{d,h,h}^{V,ps}\Delta p_{gh}^{RES}+ \sum\limits_{h=1}^{N_{DR}}\underline{a}_{d,h,h}^{V,ps}\Delta p_{dh}^{DR}+\sum \limits_{k=1}^{N_{N}}\left(\underline{d}\right)_{i}\Delta p_{kh}^{B}-s_{i,h}^{ Vmax}\quad\forall i=1,\ldots,N_{N}\;;\;\forall h=1,\ldots,\;T\\ &\sum\limits_{g=1}^{N_{RES}}\underline{a}_{d,h,h}^{V,ps}\Delta p_{gh}^{RES}- \sum\limits_{h=1}^{N_{DR}}\underline{a}_{d,h,h}^{V,ps}\Delta p_{dh}^{DR}+\sum \limits_{k=1}^{N_{N}}\left(\underline{d}\right)_{i}\Delta p_{kh}^{B}-s_{i,h}^{ Vmax}\quad\forall i=1,\ldots,N_{N}\;;\;\forall h=1,\ldots,\;T\\ &-\sum\limits_{g=1}^{N_{RES}}\underline{a}_{d,h,h}^{IP,ps}\Delta p_{gh}^{RES}+ \sum\limits_{d=1}^{N_{DR}}\underline{a}_{d,h,h}^{IP,ps}\Delta p_{dh}^{DR}- \sum\limits_{k=1}^{N_{DR}}\underline{a}_{d,h}^{IP,ps}-\sum\limits_{j=1}^{N_{DR }}\left(\underline{d}\right)_{i}\Delta p_{kh}^{B}-s_{j,h}^{Vmax}\leq\sum \limits_{l=1}^{Vmax}\quad\forall i=1,\ldots,\;T\\ &-\sum\limits_{g=1}^{N_{RES}}\underline{a}_{d,h,h}^{IP,ps}\Delta p_{gh}^{RES}+ \sum\limits_{d=1}^{N_{DR}}\underline{a}_{d,h,h}^{IP,ps}\Delta p_{dh}^{DR}- \sum\limits_{k=1}^{N_{DR}}\underline{a}_{d,h}^{DP}-\sum\limits_{k=1}^{N_{DR}} \underline{a}_{d,h}^{DB}-s_{j,h}^{Vmax}\leq\sum\limits_{l=1}^{Vmax}\quad \forall i=1,\ldots,\;T\\ \end{array}  \tag{21}\]  
The group of constraints (16) remains valid and unaltered, so it has not been repeated just for simplicity. The uncertainty set size of constraint (19) is \(|J_{1}|=(1+T)N_{RES}+(1+T)N_{DR}\), while the number of uncertain parameters for the generic \(n\)th inequality of group (21) is \(|J_{n}|=N_{RES}+N_{DR}\). Finally, the robust counterpart of the LP optimization used to find the robust exploitation of flexibility from DER has been obtained by applying model (4). To avoid a tedious and repetitive mathematical description, only the application to constraint (18) has been shown:  
\[\begin{array}{cccc}\sum\limits_{g=1}^{N_{RES}}\left[\underline{a}_{g}^{ RES}c_{F}^{RES}+\sum\limits_{h=1}^{T}R_{h,h}\left(\underline{a}_{gh}^{RES} \Delta p_{gh}^{RES}\Delta h\right)\right]+\sum\limits_{d=1}^{N_{NR}}\left[ \underline{a}_{d}^{DRc}c_{F}^{DR}+\sum\limits_{h=1}^{T}R_{h,h}\left(\underline{ a}_{d,h}^{DRc}\Delta p_{dhtertiary (2) and industrial (2). Their typical load profiles have been taken from the daily curves of the ATLANTIDE project [25], as depicted in Figure 6. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Node**} & \multicolumn{2}{c}{**Load**} & \multirow{2}{*}{**Generator**} & \multirow{2}{*}{**Node**} & \multicolumn{2}{c}{**Load**} & \multirow{2}{*}{**Generator**} \\ \cline{2-2} \cline{5-6}  & **P [kW]** & **Q [kvar]** & **P [kW]** & & **P [kW]** & **Q [kvar]** & **P [kW]** \\ \hline
2 & 216 & 105 & - & 13 & 637 & 308 & - \\
3 & 368 & 178 & - & 14 & 456 & 220 & - \\
4 & 483 & 89 & - & 15 & 230 & 11 & - \\
5 & 364 & 176 & - & 16 & 370 & 18 & - \\
6 & 608 & 295 & 1000 & 17 & 240 & 6 & - \\
7 & 770 & 37 & 1000 & 18 & 100 & 3 & - \\
8 & 244 & 118 & - & 19 & 100 & 3 & - \\
9 & 456 & 75 & 9000 & 20 & 100 & 3 & - \\
10 & 293 & 93 & - & 21 & 210 & 10 & - \\
11 & 441 & 213 & - & 22 & 240 & 6 & - \\
12 & 422 & 204 & - & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Load and Generator data.