## Appendix A Three-Phase Power Flow Notation  
Appendix A documents the notation required to define the fundamentals of the mathematical formulations. The notation discussed here is used in the ac unbalanced three-phase power flow. These equations are used as the core power flow formulation dictating the physics of power flow in distribution feeders with networkable microgrids. It should be noted that not all of the following parameters and variables are used in the formulation presented herein, but are included for a general formulation that could be used in power flow and optimal power flow formulations.  
**Sets:**  
\(\mathcal{C}\)--set of complex numbers  
\(\mathcal{N}\)--set of buses (nodes), indexed by \(i\)  
\(\mathcal{E}\)--set of edges (lines and transformers), indexed by \((i,j,k)\); each edge is assigned an arbitrary direction from a bus \(i\) to a bus \(j\) and is uniquely identified with \(k\)  
\(\chi\in\mathcal{E}\)--set of three-phase transformers indexed by \((i,j,k)\)  
\(\mathcal{G}\)--set of generator, indexed by \(g\)  
\(\mathcal{G}_{i}\)--set of generator connected to bus \(i\in\mathcal{N}\), indexed by \(g\)  
\(\mathcal{B}\)--set of batteries, indexed by \(\beta\)  
\(\mathcal{B}_{i}\)--set of batteries connected to bus \(i\in\mathcal{N}\), indexed by \(\beta\)  
\(\mathcal{E}_{i}^{+}\)--set of outgoing edges from \(i\), indexed by \((i,j,k)\)  
\(\mathcal{E}_{i}^{-}\)--set of incoming edges to \(i\), indexed by \((j,i,k)\)  
\(\Phi\) =--phase set  
\(\Phi_{i}\subseteq\{a,b,c\}\)--phase set for bus \(i\)  
\(\Phi_{ijk}\subseteq\{a,b,c\}\)--phase set for line \((i,j,k)\)  
**Parameters:**  
\(\mathbf{Y}_{ijk}^{\Phi}=\begin{bmatrix}\mathbf{Y}_{ijk}^{an}&\mathbf{Y}_{ ijk}^{ab}&\mathbf{Y}_{ijk}^{ac}\\ \bullet&\mathbf{Y}_{ijk}^{ab}&\mathbf{Y}_{ijk}^{ac}\\ \bullet&\bullet&\mathbf{Y}_{ijk}^{ac}\end{bmatrix}\in\mathbb{C}^{6}\)--symmetric complex admittance matrix for edge \((i,j,k)\in E\)\(\mathbf{Z}_{ijk}^{\Phi}=\left(\mathbf{Y}_{ijk}^{\Phi}\right)^{-1}\)--symmetric complex impedance matrix for edge \((i,j,k)\in E\)  
\(\mathbf{Y}_{ijk}^{1},\mathbf{Y}_{ijk}^{2}\)--admittance matrix of three-phase transformer \((i,j,k)\)  
\(\mathbf{y}_{ijk}\)--admittance of transformer \((i,j,k)\)  
\(\eta_{ijk}\)--tap ratio of transformer \((i,j,k)\)  
\(W_{ijk}\)--connection matrix of three-phase transformer \((i,j,k)\)  
**Variables:**  
\(\mathbf{I}_{S}^{\Phi}=\left(\ell_{j}^{\Phi}+\mathbf{j}\mathbf{m}_{g}^{\phi},\ \phi\in\Phi\right)\in\mathbb{C}^{3}\)--vector of complex current generation at \(g\in\mathcal{G}\)  
\(\mathbf{I}_{\beta}^{\Phi}=\left(\ell_{\beta}^{\phi}+\mathbf{j}\mathbf{m}_{\beta}^{ \phi},\ \phi\in\Phi\right)\in\mathbb{C}^{3}\)--vector of complex current provided (injected) at \(\beta\in\mathcal{B}\)  
\(\mathbf{I}_{i}^{\Phi}=\left(\mathbf{I}_{i}^{\phi}+\mathbf{j}\mathbf{m}_{ijk}^{\phi},\ \phi\in\Phi\right)\in\mathbb{C}^{3}\)--vector of complex current consumed at \(i\in\mathcal{N}\)  
\(\mathbf{I}_{ijk}^{\Phi}=\left(\ell_{ijk}^{\phi}+\mathbf{j}\mathbf{m}_{ijk}^{\phi},\ \phi\in\Phi\right)\in\mathbb{C}^{3}\)--vector of complex current flow on line \((i,j)\in\mathcal{E}\)  
\(\mathbf{V}_{i}^{\Phi}\in\mathbb{C}^{3}\)--vector of complex voltage on bus \(i\in\mathcal{N}\)  
\(\mathbf{S}_{S}^{\Phi}=\left(\mathbf{p}_{\beta}^{\phi}+\mathbf{j}\mathbf{q}_{\beta}^{\phi}, \ \phi\in\Phi\right)\in\mathbb{C}^{3}\)--vector of complex power generation at \(g\in G\)  
\(\mathbf{S}_{\beta}^{\Phi}=\left(\mathbf{p}_{\beta}^{\phi}+\mathbf{j}\mathbf{q}_{\beta}^{ \phi},\ \phi\in\Phi\right)\in\mathbb{C}^{3}\)--vector of complex power provided (injected) at \(\beta\in\mathcal{B}\)  
\(\mathbf{S}_{i}^{\Phi}=\left(\mathbf{p}_{\beta}^{\phi}+\mathbf{j}\mathbf{q}_{i}^{\phi},\ \phi\in\Phi\right)\in\mathbb{C}^{3}\)--vector of complex power consumed at \(i\in\mathcal{N}\)  
\(\mathbf{S}_{ijk}^{\Phi}=\left(\mathbf{p}_{ijk}^{\phi}+\mathbf{j}\mathbf{q}_{ijk}^{\phi}, \ \phi\in\Phi\right)\in\mathbb{C}^{3}\)--vector of complex power flow on line \((i,j,k)\in\mathcal{E}\)