## Nomenclature  
_Aconyms_  
DN Distribution Network  
DG Distributed Generation  
BESS Battery Energy Storage System  
ENS Energy Not Supplied  
OPF Optimal Power Flow  
DSR Demand Side Response  
DER Distributed Energy Resource  
MG Microgrid  
VM Virtual Microgrid  
AND Active Distribution Network  
PSO Particle Swarm Optimization  
MINLP Mixed-integer Nonlinear Problem  
MILP Mixed-integer Linear Problem  
DSO Distribution System Operator  
TSO Transmission System Operator  
VMTS Virtual Microgrid Topology Scheduling  
PV Photovoltaic panels  
WT Wind Turbine  
SOC State of Charge  
OF Objective Function  
_Symbols_  
\(i\in\mathbb{B}\) Set of buses  
\(i\in\mathbb{T}\) Set of time periods  
\((i,j)\in\mathbb{L}\) Set of lines, such that \(\mathbb{L}=\{(i,j);i,j\in\mathbb{B}\}\)  
\(k\in\mathbb{G}_{type}\) Set of type of generation source  
\(l\in\mathbb{B}_{type}\) Set of type of batteries  
\(PG_{i,t}^{unit}\) Minimum active power output of bus \(i\) at period \(t\) in [%]  
\(PG_{i,t}^{unit}\) Maximum active power output of bus \(i\) at period \(t\) in [%]  
\(QG_{i,t}^{unit}\) Minimum reactive power output of bus \(i\) at period \(t\) in [%]  
\(QG_{i,t}^{max}\) Maximum reactive power output of bus \(i\) at period \(t\) in [%]  
\(V_{i}^{unit}\) Minimum voltage allowed for bus \(i\) [p.u.]  
\(V_{i}^{max}\) Maximum voltage allowed for bus \(i\) [p.u.]  
\(B_{i,j}\) Non-zero upper diagonal bus admittance matrix elements [p.u.]  
\begin{tabular}{l l} \(S_{i,i}^{max}\) & Allowed maximum power of the line between buses \(i\) and \(j\) [MVA] \\ \(G_{i,j}^{ij}\) & Conductance of the line between buses \(i\) and \(j\) such that \(i\
eq j[p.u.]\) \\ \(B_{i,j}^{ij}\) & Susceptance of the line between buses \(i\) and \(j\) such that \(i\
eq j[p.u.]\) \\ \(G_{i}^{l}\) & Conductance of the line between buses \(i\) and \(j\) such that \(i=j[p.u.]\) \\ \(B_{i}^{i}\) & Susceptance of the line between buses \(i\) and \(j\) such that \(i=j[p.u.]\) \\ \(sb\) & Base power [MVAr] \\ \(r_{i,j}\) & Line resistance between buses \(i\) and \(j\) [p.u.] \\ \(SOC_{i}^{min}\) & Minimum SOC of battery type \(l\) [\%] \\ \(SOC_{i}^{max}\) & Maximum SOC of battery type \(l\) [\%] \\ \(U_{k}^{R}\) & Active power capacity installed of generator type \(k\) [MW] \\ \(U_{k}^{R}\) & Reactive power capacity installed of generator type \(k\) [MW] \\ \(U_{l}^{H}\) & Active power capacity installed of battery type \(l\) [MWh] \\ \(\varphi^{l}\) & Efficiency of battery type \(l\) [\%] \\ \(M^{l}\) & Maximum charge/discharge for battery type \(l\) in a period [MWh] \\ \(Y_{i}^{l}\) & Binary parameter representing if a battery type \(l\) is installed in node \(i\) \\ \(P\)L\({}_{i,t}\) & Active load of bus \(i\) at period \(t\) [MW] \\ \(QL_{i,t}\) & Reactive load of bus \(i\) at period \(t\) [MVAr] \\ \(pg_{i,t}^{k}\) & Active power of generator type \(k\) of bus \(i\) at period \(t\) [MW] \\ \(qg_{i,t}^{k}\) & Reactive power of generator type \(k\) of bus \(i\) at period \(t\) [MVAr] \\ \(v_{i,t}\) & Voltage of bus \(i\) at period \(t\) [p.u] \\ \(\theta_{i,t}\) & Angle of bus \(i\) at period \(t\) [p.u] \\ \(p_{i,t}\) & Active power in line between buses \(i\) and \(j\) at period \(t\) [MW] \\ \(q_{i,j,t}\) & Reactive power in line between buses \(i\) and \(j\) at period \(t\) [MVAr] \\ \(ch_{i,t}^{l}\) & Power absorbed for the storage type \(l\) of bus \(i\) at period \(t\) [MW] \\ \(ds_{i,t}^{l}\) & Power injected for the storage type \(l\) of bus \(i\) at period \(t\) [MW] \\ \(soc_{i,t}^{l}\) & SOC of battery type \(l\) of bus \(i\) at period \(t\) [MWh] \\ \(w_{i,t}^{l}\) & Binary variable: 1 if the battery type \(l\) is charging in bus \(i\) at period \(t\), 0 otherwise \\ \(\lambda_{i,t}\) & Virtual active and reactive power of bus \(i\) at period \(t\) [MW] \\ \(\tau_{i,t}\) & Active and reactive power supplied on bus \(i\) at period \(t\) [MW] \\ \(\delta_{i,t}\) & Active and reactive power not supplied on bus \(i\) at period \(t\) [MW] \\ \end{tabular}