The PowerModels Mathematical Model

\[ \begin{align} % \mbox{sets:} & \nonumber \\ & N \mbox{ - buses}\nonumber \\ & R \mbox{ - refrences buses}\nonumber \\ & E, E^R \mbox{ - branches, forward and reverse orientation} \nonumber \\ & G, G_i \mbox{ - generators and generators at bus $i$} \nonumber \\ & L, L_i \mbox{ - loads and loads at bus $i$} \nonumber \\ & S, S_i \mbox{ - shunts and shunts at bus $i$} \nonumber \\ % \mbox{data:} & \nonumber \\ & S^{gl}_k, S^{gu}_k \;\; \forall k \in G \nonumber \\ & c_{2k}, c_{1k}, c_{0k} \;\; \forall k \in G \nonumber \\ & v^l_i, v^u_i \;\; \forall i \in N \nonumber \\ & {S^d}_k \;\; \forall k \in L \nonumber \\ & Y^s_{k} \;\; \forall k \in S \nonumber \\ & Y_{ij}, Y^c_{ij}, Y^c_{ji}, {T}_{ij} \;\; \forall (i,j) \in E \nonumber \\ & {s^u}_{ij}, \theta^{\Delta l}_{ij}, \theta^{\Delta u}_{ij} \;\; \forall (i,j) \in E \nonumber \\ % \mbox{variables: } & \nonumber \\ & S^g_k \;\; \forall k\in G \nonumber \\ & V_i \;\; \forall i\in N \nonumber \\ & S_{ij} \;\; \forall (i,j) \in E \cup E^R \nonumber \\ % \mbox{minimize: } & \sum_{k \in G} c_{2k} (\Re(S^g_k))^2 + c_{1k}\Re(S^g_k) + c_{0k} \\ % \mbox{subject to: } & \nonumber \\ & \angle V_{r} = 0 \;\; \forall r \in R \\ & S^{gl}_k \leq S^g_k \leq S^{gu}_k \;\; \forall k \in G \\ & v^l_i \leq |V_i| \leq v^u_i \;\; \forall i \in N \\ & \sum_{\substack{k \in G_i}} S^g_k - \sum_{\substack{k \in L_i}} S^d_k - \sum_{\substack{k \in S_i}} Y^s_k |V_i|^2 = \sum_{\substack{(i,j)\in E_i \cup E_i^R}} S_{ij} \;\; \forall i\in N \\ & S_{ij} = \left( Y_{ij} + Y^c_{ij}\right)^* \frac{|V_i|^2}{|{T}_{ij}|^2} - Y^*_{ij} \frac{V_i V^*_j}{{T}_{ij}} \;\; \forall (i,j)\in E \\ & S_{ji} = \left( Y_{ij} + Y^c_{ji} \right)^* |V_j|^2 - Y^*_{ij} \frac{V^*_i V_j}{{T}^*_{ij}} \;\; \forall (i,j)\in E \\ & |S_{ij}| \leq s^u_{ij} \;\; \forall (i,j) \in E \cup E^R \\ & \theta^{\Delta l}_{ij} \leq \angle (V_i V^*_j) \leq \theta^{\Delta u}_{ij} \;\; \forall (i,j) \in E % \end{align}\]