Single Phase LinDistFlow

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 $j$

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

Three Phase LinDistFlow

From [2]

\[\begin{aligned} P_{ij,\phi} + p_{j,\phi} = \sum_{k:j\rightarrow k} P_{jk,\phi} \ \forall j \in \mathcal{N}^+, \forall \phi \in [1,2,3] \\ Q_{ij,\phi} + q_{j,\phi} = \sum_{k:j\rightarrow k} Q_{jk,\phi} \ \forall j \in \mathcal{N}^+, \forall \phi \in [1,2,3] \\ \boldsymbol{w}_j = \boldsymbol{w}_i + \boldsymbol{M}_{P,ij} \boldsymbol{P}_{ij} + \boldsymbol{M}_{Q,ij} \boldsymbol{Q}_{ij} \\ (\boldsymbol{v}_{j,\min})^2 \le \boldsymbol{w}_j \le (\boldsymbol{v}_{j,\max})^2 \ \forall j \in \mathcal{N}^+ \\ \boldsymbol{M}_{P,ij} = \begin{bmatrix} -2r_{11} & r_{12}-\sqrt{3}x_{12} & r_{13}+\sqrt{3}x_{13} \\ r_{21}+\sqrt{3}x_{21} & -2r_{22} & r_{23}-\sqrt{3}x_{23} \\ r_{31}-\sqrt{3}x_{31} & r_{32}+\sqrt{3}x_{32} & -2r_{33} \end{bmatrix} \\ \boldsymbol{M}_{Q,ij} = \begin{bmatrix} -2x_{11} & x_{12}+\sqrt{3}r_{12} & x_{13}-\sqrt{3}r_{13} \\ x_{21}-\sqrt{3}r{21} & -2x_{22} & x_{23}+\sqrt{3}r_{23} \\ x_{31}+\sqrt{3}r_{31} & x_{32}-\sqrt{3}r_{32} & -2x_{33} \end{bmatrix} \end{aligned}\]

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

[2]

Arnold, Daniel B., et al. "Optimal dispatch of reactive power for voltage regulation and balancing in unbalanced distribution systems." 2016 IEEE Power and Energy Society General Meeting (PESGM). IEEE, 2016.