# Parameterized Modeling and Planning of Distributed Energy Storage in Active Distribution Networks
## 4 Mixed-Integer Second-Order Cone Programming (MISOCP) Model Conversion  
In this section, to solve the MINLP model more effectively, the original model is converted into a MISOCP model by convex relaxation. First, let \(u_{s,t,i}\) and \(i_{s,t,ij}\) denote the quadratic terms \(U_{s,t,i}^{2}\) and \(l_{s,t,ji}^{2}\) Linearized functions are expressed as follows:  
\[\sum\
olimits_{ik\in\Omega_{\rm b}}P_{s,t,ik}=\sum\
olimits_{ji\in\Omega_{ \rm b}}\left(P_{s,t,ji}-r_{ji}i_{s,t,ij}\right)+P_{s,t,j} \tag{26}\]  
\[\sum\
olimits_{ik\in\Omega_{\rm b}}Q_{s,t,ik}=\sum\
olimits_{ji\in\Omega_{ \rm b}}\left(Q_{s,t,ji}-x_{ji}i_{s,t,ij}\right)+Q_{s,t,i} \tag{27}\]  
\[u_{s,t,i}-u_{s,t,j}-2\left(r_{ij}P_{s,t,ij}+x_{ij}Q_{s,t,ij}\right)+\left(r_{ ij}^{2}+x_{ij}^{2}\right)i_{s,t,ij}=0 \tag{28}\]  
\[i_{s,t,ij}=\frac{P_{s,t,ij}^{2}+Q_{s,t,ij}^{2}}{u_{s,t,i}} \tag{29}\]  
\[\left(U_{i}^{\rm min}\right)^{2}\leq u_{s,t,i}\leq\left(U_{i}^{\rm max}\right) ^{2} \tag{30}\]  
\[0\leq i_{s,t,ij}\leq\left(l_{ij}^{\rm max}\right)^{2} \tag{31}\]  
The operation constraints of DES in (15) and (16) can be transformed into the rotated quadratic cone constraints:  
\[\left(P_{s,t,i,m}^{\rm DES}\right)^{2}+\left(Q_{s,t,i,m}^{\rm DES}\right)^{2} \leq 2\frac{y_{i,m}S_{m}^{\rm unit}}{\sqrt{2}}\frac{y_{i,m}S_{m}^{\rm unit }}{\sqrt{2}} \tag{32}\]  
\[\left(P_{s,t,i,m}^{\rm DES}\right)^{2}+\left(Q_{s,t,i,m}^{\rm DES}\right)^{2} \leq 2\frac{P_{s,t,i,m}^{\rm DES,L}}{\sqrt{2}A_{m}^{\rm DES}}\frac{P_{s,t,i,m}^{ \rm DES,L}}{\sqrt{2A_{m}^{\rm DES}}} \tag{33}\]Although the absolute value term \(\left|P_{s,t,j,m}^{\text{DES}}\right|\) in constraint (19) is nonlinear, it can be replaced by a linear program according to Chebyshev approximation problem [32]. An auxiliary variable \(P_{s,t,j,m}^{\text{AUX}}\) is introduced to replace \(\left|P_{s,t,j,m}^{\text{DES}}\right|\), and constraint (19) can be transformed to constraint (34). Constraints (35)-(37) are added to bound the value of \(P_{s,t,j,m}^{\text{AUX}}\), making constraint (34) linearized. Then, the constraint (19) is transformed into constraints (34)-(37), which are expressed as:  
\[\frac{\sum_{t\in\text{N}_{\text{T}}}P_{s,t,j,m}^{\text{AUX}}\Delta t}{2z_{i,m} E_{m}^{\text{unit}}}\leq\frac{1  
Assuming that the annual load curve, annual WT output curve and PV output curve in the distribution network are shown in Figures 2-4, respectively. The typical scenarios for DES planning are generated by the clustering technique, containing the patterns of load demand, wind turbine output and photovoltaic output, which are shown in Figure A1 in the Appendix A. It is assumed that the maximum power and energy capacity of DES in planning are 1 MVA/4 MWh. The total number of nodes that DES can be installed with is up to 4.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameter** & \multicolumn{5}{c}{**Wind Turbines**} & \multicolumn{5}{c}{**Photovoltaic Generators**} \\ \hline Location & 10 & 16 & 17 & 30 & 33 & 7 & 13 & 27 \\ Capacity (kVA) & 500 & 300 & 200 & 200 & 300 & 500 & 300 & 400 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of distributed generators (DGs).  
Figure 3: Annual power output of wind turbines (WTs).  
Figure 2: Annual load curve.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **On-Peak** & **Mid-Peak** & **Off-Peak** \\ \hline Time span & 16:00–22:00 & 8:00–15:00 & 1:00–7:00, 23:00–24:00 \\ Electricity price (\$/kWh) & 0.173 & 0.104 & 0.050 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters of time-of-use electricity price.  
The proposed model is implemented with the General Algebraic Modeling System (GAMS) optimization software and is solved by the commercial solver GUROBI [35]. GUROBI is an optimization package which has been widely applied in solving the MISOCP problem. The calculation is carried out on a PC with Intel Core i5 3.20 GHz CPU and 4 GB RAM.