## Appendix A  
Let us define \(\overline{\
abla}\) the vector of phase voltages and \(\bar{\mathbf{I}}\) the vector of the injected phase currents. The three-phase power flow equations are based on the bus voltage equations [47] expressed as:  
\[\bar{\mathbf{I}}=\dot{\mathbf{Y}}\overline{\
abla} \tag{11}\]  
where \(\dot{\mathbf{Y}}\) is the network admittance matrix whose dimension is \(3n_{g}x3n_{g}\), being \(n_{g}\) the number of buses. For a generic bus \(\mathbf{i}\), the definition of phase voltages as a function of the sequence voltages is given by:  
\[\left[ \begin{array}{c}\overline{\mathbf{V}}_{\mathbf{i}}^{1}\\ \overline{\mathbf{V}}_{\mathbf{i}}^{2}\\ \overline{\mathbf{V}}_{\mathbf{i}}^{3}\end{array} \right]=\left[ \begin{array}[ ]{ccc}1&1&1\\ 1&\mathbf{\alpha}^{2}&\mathbf{\alpha}\\ 1&\mathbf{\alpha}&\mathbf{\alpha}^{2}\end{array} \right]\left[ \begin{array}{c} \overline{\mathbf{V}}_{\mathbf{i}}^{\mathbf{zero}}\\ \overline{\mathbf{V}}_{\mathbf{i}}^{\mathbf{inv}}\\ \overline{\mathbf{V}}_{\mathbf{i}}^{\mathbf{inv}}\end{array} \right]=\bar{ \mathbf{T}}\left[ \begin{array}{c}\overline{\mathbf{V}}_{\mathbf{i}}^{ \mathbf{zero}}\\ \overline{\mathbf{V}}_{\mathbf{i}}^{\mathbf{inv}}\end{array} \right] \tag{12}\]  
and an analogous expression can be derived for the currents, as:  
\[\left[ \begin{array}{c}\bar{\mathbf{I}}_{\mathbf{i}}^{1}\\ \bar{\mathbf{I}}_{\mathbf{i}}^{2}\\ \bar{\mathbf{I}}_{\mathbf{i}}^{3}\end{array} \right]=\bar{\mathbf{T}}\left[ \begin{array}{c}\bar{\mathbf{I}}_{\mathbf{i}}^{\mathbf{zero}}\\ \bar{\mathbf{I}}_{\mathbf{i}}^{\mathbf{dir}}\\ \bar{\mathbf{I}}_{\mathbf{i}}^{\mathbf{inv}}\end{array} \right] \tag{13}\]  
If the definition of phase voltages and currents as a function of the sequence voltages is applied to the vectors of phase voltages \(\overline{\
abla}\), it can be obtained:  
\[\overline{\
abla}=\dot{\mathbf{T}}_{\mathbf{i}}\overline{\
abla}^{\mathbf{ seq}} \tag{14}\]  
where \(\overline{\
abla}^{\mathbf{seq}}\) is the vector collecting all the sequence phase voltages and \(\dot{\mathbf{T}}_{\mathbf{t}}\) is a \(3n_{g}x3n_{g}\) matrix, calculated by applying the Kronecker product (denoted by \(\otimes\)), to the identity matrix \(\mathbf{I}_{\mathbf{n_{g}}}\) (of dimension \(\mathbf{n_{g}}\)) and \(\dot{\mathbf{T}}\), as:  
\[\dot{\mathbf{T}}_{\mathbf{t}}=\mathbf{I}_{\mathbf{n_{g}}}\otimes\dot{ \mathbf{T}} \tag{15}\]  
For the vector of currents, it results:  
\[\bar{\mathbf{I}}=\dot{\mathbf{T}}_{\mathbf{t}}\bar{\mathbf{I}}^{\mathbf{seq}} \tag{16}\]  
with a definition of \(\bar{\mathbf{T}}^{\mathbf{seq}}\) analogous to \(\overline{\
abla}^{\mathbf{seq}}\). The bus voltage Equation (11) can, then, be rewritten as:  
\[\bar{\mathbf{T}}^{\mathbf{seq}}=\dot{\mathbf{Y}}^{\mathbf{seq}}\overline{ \
abla}^{\mathbf{seq}} \tag{17}\]  
being:  
\[\dot{\mathbf{Y}}^{\mathbf{seq}}=\left(\dot{\mathbf{T}}_{\mathbf{t}}\right)^{- 1}\dot{\mathbf{Y}}\dot{\mathbf{T}}_{\mathbf{t}} \tag{18}\]  
If the vectors \(\overline{\
abla}^{\mathbf{seq}}\) and \(\bar{\mathbf{I}}^{\mathbf{seq}}\) are rearranged by grouping the voltages and currents according to the sequences, it results:  
\[\left[ \begin{array}{c}\bar{\mathbf{I}}^{\mathbf{zero}}\\ \bar{\mathbf{I}}^{\mathbf{dir}}\\ \bar{\mathbf{I}}^{\mathbf{inv}}\end{array} \right]=\left[ \begin{array}{ccc} \dot{\mathbf{Y}}^{\mathbf{zero}-\mathbf{zero}}&\dot{\mathbf{Y}}^{\mathbf{ zero}-\mathbf{dir}}&\dot{\mathbf{Y}}^{\mathbf{zero}-\mathbf{inv}}\\ \dot{\mathbf{Y}}^{\mathbf{dir}-\mathbf{zero}}&\dot{\mathbf{Y}}^{\mathbf{ dir}-\mathbf{dir}}&\dot{\mathbf{Y}}^{\mathbf{dir}-\mathbf{inv}}\\ \dot{\mathbf{Y}}^{\mathbf{inv}-\mathbf{zero}}&\dot{\mathbf{Y}}^{\mathbf{inv}- \mathbf{dir}}&\dot{\mathbf{Y}}^{\mathbf{inv}-\mathbf{inv}}\\ \dot{\mathbf{Y}}^{\mathbf{inv}}\end{array} \right]\left[ \begin{array}{c} \bar{\mathbf{V}}^{\mathbf{zero}}\\ \overline{\mathbf{V}}^{\mathbf{dir}}\\ \overline{\mathbf{V}}^{\mathbf{inv}}\end{array} \right]=\dot{\mathbf{Y}}_{ \mathbf{g}}^{\mathbf{seq}}\left[