Single Phase BranchFlowModel

From [1]

Notation:

• $P_{ij}$ real power flow from node $i$ to node $j$
• $p_j$ real power injection on node $j$
• `\\mathcal{N}^+ set of all nodes in network except the source
• $w_j$ voltage magnitude squared on node $j$
• $\ell_{ij}$ current magnitude squared from node $i$ to node $j$

\[\begin{aligned} P_{ij} - r_{ij} \ell_{ij} + p_j = \sum_{k:j\rightarrow k} P_{jk} \ \forall j \in \mathcal{N}^+ \\ Q_{ij} - x_{ij} \ell_{ij} + q_j = \sum_{k:j\rightarrow k} Q_{jk} \ \forall j \in \mathcal{N}^+ \\ w_j = w_i - 2 r_{ij} P_{ij} - 2 x_{ij} Q_{ij} + (r_{ij}^2 + x_{ij}^2) \ell_{ij} \ \forall j \in \mathcal{N}^+ \\ w_i \ell_{ij} = P_{ij}^2 + Q_{ij}^2 \forall (i,j) \in \mathcal{E} \\ (v_{j,\min})^2 \le w_j \le (v_{j,\max})^2 \ \forall j \in \mathcal{N}^+ \end{aligned}\]

Three Phase BranchFlowModel

TODO

References

[1]

Baran, Mesut E., and Felix F. Wu. "Optimal capacitor placement on radial distribution systems." IEEE Transactions on power Delivery 4.1 (1989): 725-734. Chicago