Single Phase Bus Injection Model (Unrelaxed)

Rectangular voltage variables

Notation:

$s_j$ net complex power injection at node $j$
$\mathcal{N}$ set of all nodes in network
$v_j$ voltage at node $j$

\[\begin{aligned} &s_j = \sum_{k: j \sim k} Y[j,k]^* ( |v_j|^2 - v_j v_k^*) \ \forall j \in \mathcal{N} \\ &v_{\text{substation bus}} = v_0 \end{aligned}\]

Polar voltage variables

Notation:

$G_{ij}$ real entry of admittance matrix at row $i$, column $j$
$B_{ij}$ imaginary entry of admittance matrix at row $i$, column $j$
$|v_j|$ voltage magnitude at bus $j$
$\angle v_j$ voltage angle at bus $j$
$p_j$ real power injection at bus $j$
$q_j$ reactive power injection at bus $j$

\[\begin{aligned} &p_j = |v_j| \sum_{i \in 1\dots\mathcal{N}} |v_i| \left[ G_{ij} \cos(\angle v_j - \angle v_i) + B_{ij} \sin(\angle v_j - \angle v_i) \right] \\ &q_j = |v_j| \sum_{i \in 1\dots\mathcal{N}} |v_i| \left[ G_{ij} \sin(\angle v_j - \angle v_i) - B_{ij} \cos(\angle v_j - \angle v_i) \right] \\ &|v_{\text{substation bus}}| = v_0 \\ &\angle v_{\text{substation bus}} = 0 \end{aligned}\]

Multi-Phase Bus Injection Model (Unrelaxed)

Rectangular voltage variables

Notation:

$\boldsymbol s_j$ net complex power injection at bus $j$, vector of phases
$\Phi_{j}$ phases connected to bus $j$ (take sub-set of vector)
$\boldsymbol v_j^{\Phi_{jk}}$ complex voltage vector at bus $j$ for the phases connected to bus $k$
$k: j \sim k$ set of busses connected to bus $j$
$H$ conjugate transpose
$\boldsymbol Y_{jk}^H$ phase admittance matrix for busses $j$ and $k$

\[\begin{aligned} \boldsymbol s_j^{\Phi_{j}} = \sum_{k: j \sim k} \text{diag} \left( \boldsymbol v_j^{\Phi_{jk}} \left[ \boldsymbol v_j^{\Phi_{jk}} - \boldsymbol v_k^{\Phi_{jk}} \right]^H \boldsymbol Y_{jk}^H \right) \quad \forall j \in \mathcal{N} \end{aligned}\]