# Parameterized Modeling and Planning of Distributed Energy Storage in Active Distribution Networks
## Nomenclature  
### Sets  
\(\Omega_{\rm b}\) & set of all branches  
\(\Omega_{\rm sub}\) & set of substation nodes  
\(\Omega_{\rm s}\) & set of scenarios  
\(\Omega_{\rm type}\) & set of DES types  
_Indices_  
\(i\), \(j\) & indices of nodes  
\(t\) & indices of time periods  
\(s\) & indices of scenarios  
\(m\) & indices of DES types  
_Variables_  
\(P_{s,k,ij}\) & active power flow of branch \(ij\) in the \(s\)th scenario  
\(Q_{s,t,ij}\) & reactive power flow of branch \(ij\) in the \(s\)th scenario  
\(P_{s,k,i}\) & total active power injection at node \(i\) in the \(s\)th scenario  
\(Q_{s,t,i}\) & total reactive power injection at node \(i\) in the \(s\)th scenario  
\(I_{t,ij}\), \(i_{t,ij}\) & branch current magnitude and its square  
\(U_{t,j}\), \(u_{t,i}\) & node voltage magnitude and its square  
\(\mathrm{PKC}\) & active power injection by DG at node \(i\) in the \(s\)th scenario  
\(Q_{s,t,i}^{\mathrm{PKC}}\) & reactive power injection by DG at node \(i\) in the \(s\)th scenario  
\(\mathrm{PKS}\) & active power at the substation in the \(s\)th scenario  
\(\mathrm{PKS}\) & active power injection by the \(m\)th DES at node \(i\) in the \(s\)th scenario  
\(\mathrm{PKS}\) & reactive power injection by the \(m\)th DES at node \(i\) in the \(s\)th scenario  
\(\mathrm{PKS}\) & reactive power injected by the \(m\)th DES at node \(i\) in the \(s\)th scenario  
\(\mathrm{PKS}\) & reactive power losses of the \(m\)th DES at node \(i\) in the \(s\)th scenario  
\(\mathrm{PKS}\) & energy stored of the \(m\)th DES at node \(i\) in the \(s\)th scenario  
\(y_{i,m}\) & total number of power unit of the \(m\)th DES at node \(i\)  
\(z_{i,m}\) & total number of energy unit of the \(m\)th DES at node \(i\)  
_Parameters_  
\(N_{\rm T}\) & total periods of the time horizon  
\(N_{\rm N}\) & total number of the nodes  
\(p_{s,t,i}^{\rm LOAD}\) & active power load at node \(i\) in the \(s\)th scenario  
\(Q_{s,t,i}^{\rm LOAD}\) & reactive power load at node \(i\) in the \(s\)th scenario  
\(r_{ij}\) & resistance of branch \(ij\)  
\(x_{ij}\) & reactance of branch \(ij\)  
\(U_{i}^{\rm max}\) & upper limit of statutory voltage at node \(i\)  
\(U_{i}^{\rm min}\) & lower limit of statutory voltage at node \(i\)  
\(I_{ij}^{\rm max}\) & upper limit of statutory current at branch \(ij\)  
\(S_{m}^{\rm unit}\) & unit power capacity of the \(m\)th DES  
\(E_{m}^{\rm count}\) & unit energy capacity of the \(m\)th DES  
\(\mathrm{CPW}\) & capital cost for 1 kW power capacity of the \(m\)th DES  
\(C_{m}^{\rm ENE}\) & capital cost for 1 kWh energy capacity of the \(m\)th DES  
\(A_{m}^{\rm DES}\) & loss coefficient of the \(m\)th DES  
\(\mathrm{SOC}_{m}^{\rm max}\) & maximum state of charge limit of the \(m\)th DES  
\(\mathrm{SOC}_{m}^{\rm min}\) & minimum state of charge limit of the \(m\)th DES  
\(\lambda_{t}\) & time-of-use electricity price  
\(\lambda_{f}\) & cost of DG power fluctuation  
\(N_{m}^{M}\) & life cycle (cycles) of the \(m\)th DES  
\(y_{m}\) & lifetime (years) of the \(m\)th DES  
\(d\) & discount rate of DES  
\(DoD_{m}^{\rm max}\) & maximum depth of discharge of the \(m\)th DES  
\(\Delta t\) & time interval  
\(P_{n}\) & probability of the \(s\)th scenario  
\(\mathrm{SPGT}\) & maximum power capacity of DES in planning  
\(E^{\rm BGT}\) & maximum energy capacity of DES in planning  
\(\delta_{i}\) & binary variable of DES installing at node \(i\)  
\(n^{\rm DES}\) & maximum number of DES installing nodes  
\(P_{n}\) & probability of the \(s\)th scenario