## 5 Conclusions
\begin{tabular}{l l} \(N_{y,d}\) & number of \(d\) typical days in year \(y\) \\ \(P^{g^{\prime}}_{DESS_{\langle y,d\rangle}}\) & active power of the DESS installed at the bus \(i\) phase \(\xi_{i}\) during the time interval \(k\) of typical day \(d\), \\ \(p^{g^{\prime},sp}_{DC_{\langle y,d\rangle}}\) & forecasted active power produced by the DG unit installed a at the phase \(\xi_{i}\) of the bus \(i\) during the time interval \(k\) of typical day \(d\) in year \(y\) \\ \(P^{g}_{L_{\langle y,d\rangle}}\) & net injected active power f the bus \(i\) phase \(p\) during the time interval \(k\) of typical day \(d\) in year \(y\) \\ \(p^{g,sp}_{L_{\langle y,d\rangle}}\) & forecasted active power demand of the load at the phase \(\xi_{i}\) of the bus \(i\) during the time interval \(k\) of typical day \(d\) in year \(y\) \\ \(Pr_{k_{\langle y,d\rangle}}\) & price of electrical energy applied to the time interval \(k\) of typical day \(d\) in year \(y\) \\ \(Q^{g^{\prime}}_{DESS_{\langle y,d\rangle}}\) & reactive power of the DESS installed at the phase \(\xi_{i}\) of the bus \(i\) during the time interval \(k\) of the typical day \(d\) in year \(y\) \\ \(Q^{g^{\prime}}_{DC_{\langle y,d\rangle}}\) & reactive power of the DG unit installed at the phase \(\xi_{i}\) of the bus \(i\) during the time interval \(k\) of the typical day \(d\) in year \(y\) \\ \(Q^{g^{\prime}}_{L_{\langle y,d\rangle}}\) & net injected reactive power of the bus \(i\) phase \(p\) during the time interval \(k\) of typical day \(d\) in year \(y\) \\ \(Q^{g^{\prime},sp}_{L_{\langle y,d\rangle}}\) & forecasted reactive power demand of the load at the phase \(\xi_{i}\) of the bus \(i\) during the time interval \(k\) of typical day \(d\) in year \(y\) \\ \(RC_{DESS}\) & replacement cost of a single element of DESS \\ \(S^{g^{\prime}}_{DESS_{i}}\) & converter size of the DESS installed at the phase \(\xi_{i}\) of the bus \(i\) \\ \(S^{g^{\prime}}_{DC_{i}}\) & converter size of the DG installed at the phase \(\xi_{i}\) of the bus \(i\) \\ \(S_{MV}\) & size of the MV/LV transformer \\ \(V^{\prime}_{L_{\langle y,d\rangle}}\) & voltage magnitude at bus \(i\), phase \(p\) during the time interval \(k\) of the typical day \(d\), year \(y\) \\ \(V_{\max}\) & upper bound of the admissible range for the bus voltages \\ \(V_{min}\) & lower bound of the admissible range for the bus voltages \\ \(a_{L}\) & effective rate of change for the cost of the electrical energy \\ \(\eta^{g^{\prime}}_{Q^{g^{\prime}}_{L_{i}}}\) & charging efficiencies of the DESS installed at the phase \(\xi_{i}\) of the bus \(i\) \\ \(\eta^{g^{\prime}}_{L_{\langle y,d\rangle}}\) & discharging efficiencies of the DESS installed at the phase \(\xi_{i}\) of the bus \(i\) \\ \(\theta^{g^{\prime}}_{L_{\langle y,d\rangle}}\) & voltage argument at bus \(i\), phase \(p\) during the time interval \(k\), the typical day \(d\), year \(y\) \\ \(\lambda_{i}\) & \(i\)th eigenvalue of \(\dot{Y}\) \\ \(\lambda_{k}\) & eigenvalue of minimum modulus of \(\dot{Y}\) \\ \(\lambda_{1}^{\rm inv-inv}\) & \(i\)th eigenvalue of \((\dot{\mathbf{Z}}^{\rm inv-inv})^{-1}\) \\ \(\lambda_{k}^{\rm inv-inv}\) & eigenvalue of minimum modulus of \((\dot{\mathbf{Z}}^{\rm inv-inv})^{-1}\) \\ \(g_{i}\) & phase of bus \(i\) at which DESS or DG unit is connected \\ \(\Delta t\) & duration of the time interval \\ \(\Omega^{g^{\prime}}_{L_{\langle y,d\rangle}}\) & set of time intervals in which the DESS at bus \(i\) phase \(\xi_{i}\) can charge in the typical day \(d\) year \(y\) \\ \(\Omega^{g^{\prime}}_{L_{\langle y,d\rangle}}\) & set of time intervals in which the DESS at bus \(i\) phase \(\xi_{i}\) can discharge in the typical day \(d\) year \(y\) \\ \(\Omega_{ABS}\) & set of candidate buses \\ \(\Omega_{DESS}\) & set of buses where DESSs are connected \\ \(\Omega_{DG}\) & set of buses where DG units are located \\ \(\Omega_{i}\) & sets of selected buses (\(i=1,\ldots,5\)) \\ \(\dot{\Gamma}^{i}_{\dot{\mathbf{i}}}\) & left eigenvector associated to \(\dot{\lambda}_{\dot{\mathbf{i}}}\) \\ \(\dot{\Gamma}^{i}_{\dot{\mathbf{i}}}\) & left eigenvector associated to \(\dot{\lambda}^{\rm inv-inv}_{\dot{\mathbf{i}}}\) \\ \(\dot{\Gamma}^{i}_{\dot{\mathbf{i}}}\) & left eigenvector associated to \(\dot{\lambda}^{\rm inv-inv}_{\dot{\mathbf{i}}}\) \\ \(\dot{\Gamma}^{i}_{\dot{\mathbf{i}}}\) & left eigenvector associated to \(\dot{\lambda}^{\rm inv-inv}_{\dot{\mathbf{i}}}\) \\ \(\dot{\Gamma}_{\dot{\mathbf{k}}}\) & left eigenvector associated to \(\dot{\lambda}^{\rm inv}_{\dot{\mathbf{k}}}\) \\ \(\overline{\Delta\mathbf{I}}\) & vector of the variation of the injected currents \\ \(\overline{\Delta\mathbf{I}}^{\rm zero}\) & vector of the variation of the zero sequence injected currents \\ \(\overline{\Delta\mathbf{I}}^{\rm dir}\) & vector of the variation of the direct sequence injected currents \\ \(\overline{\Delta\mathbf{I}}^{\rm inv}\) & vector of the variation of the inverse sequence injected currents \\ \(\overline{\Delta\mathbf{V}}\) & vector of the variation of the phase voltages \\ \(\overline{\Delta\mathbf{V}}^{\rm inv}\) & vector of the variation of the inverse-sequence voltages \\ \(\dot{\mathbf{X}}_{\dot{\mathbf{i}}}\) & right eigenvector associated to \(\dot{\lambda}_{\dot{\mathbf{i}}}\) \\ \(\dot{\mathbf{X}}_{\dot{\mathbf{i}}}\) & right eigenvector associated to \(\dot{\lambda}_{\dot{\mathbf{i}}}\) \\ \(\dot{\mathbf{X}}_{\dot{\mathbf{k}}}\) & right eigenvector associated to \(\dot{\lambda}^{\rm inv-inv}_{\dot{\mathbf{k}}}\) \\ \(\dot{\mathbf{X}}_{\dot{\mathbf{k}}}\) & right eigenvector associated to \(\dot{\lambda}^{\rm inv-inv}_{\dot{\mathbf{k}}}\) \\ \(\dot{\mathbf{Y}}\) & nodal admittance matrix \\ \(\dot{\mathbf{Z}}^{\rm seq\ k-seq\ m}\) & coupling impedance sub-matrices between sequences k and m (k, m \(\in[\rm zero,dir,inv)\)). \\ \end{tabular}