## 3 Solution Method
### Selection of Candidate Buses for the Reduction of Unbalances  
As it is well-known, the unbalance factor is linked to the inverse-sequence voltage values. Therefore, to obtain the greatest reduction of the unbalance factor, the DESS has to be allocated in such a way that the inverse-sequence voltages at all buses will experience the greatest reduction. Let us now suppose that, starting from a reference operating point, a modification of the current vector is made. The variation of the inverse-sequence voltages \(\|\overline{\Delta\mathbf{V}}^{\mathbf{inv}}\|\) (at all buses except the slack bus) can be evaluated as (see Appendix A):  
\[\|\overline{\Delta\mathbf{V}}^{\mathbf{inv}}\|=\|\dot{\mathbf{Z}}^{\mathbf{ inv-zero}}\overline{\Delta\mathbf{I}}^{\mathbf{zero}}+\dot{\mathbf{Z}}^{\mathbf{ inv-dir}}\overline{\Delta\mathbf{I}}^{\mathbf{dir}}+\dot{\mathbf{Z}}^{ \mathbf{inv-inv}}\overline{\Delta\mathbf{I}}^{\mathbf{inv}}\|, \tag{13}\]  
where \(\overline{\Delta\mathbf{I}}^{\mathbf{zero}}\), \(\overline{\Delta\mathbf{I}}^{\mathbf{dir}}\) and \(\overline{\Delta\mathbf{I}}^{\mathbf{inv}}\) are the vectors of the variation of the injected currents at zero, direct, and inverse sequence and \(\dot{\mathbf{Z}}^{\mathbf{inv-zero}}\), \(\dot{\mathbf{Z}}^{\mathbf{inv-dir}}\), \(\dot{\mathbf{Z}}^{\mathbf{inv-inv}}\) are the coupling impedance sub-matricesbetween the inverse and zero sequences, the inverse and direct sequences and the inverse and the inverse sequences, respectively (the definitions are reported in Appendix A). As it is well known [44], it can be stated that:  
\[ \begin{split}\|\dot{\mathbf{Z}}^{\mathbf{inv}-\mathbf{zero}}\overline{ \Delta\mathbf{I}}^{\mathbf{zero}}+\dot{\mathbf{Z}}^{\mathbf{inv}-\mathbf{dir}} \overline{\Delta\mathbf{I}}^{\mathbf{dir}}+\dot{\mathbf{Z}}^{\mathbf{inv}- \mathbf{inv}}\overline{\Delta\mathbf{I}}^{\mathbf{inv}}\|\\ \leq\|\dot{\mathbf{Z}}^{\mathbf{inv}-\mathbf{zero}}\overline{ \Delta\mathbf{I}}^{\mathbf{zero}}\|+\|\dot{\mathbf{Z}}^{\mathbf{inv}-\mathbf{ dir}}\overline{\Delta\mathbf{I}}^{\mathbf{dir}}\|+\|\dot{\mathbf{Z}}^{\mathbf{inv}- \mathbf{inv}}\overline{\Delta\mathbf{I}}^{\mathbf{inv}}\|.\end{split}  \tag{14}\]  
If only inverse-sequence currents are supposed to change, the variation of the inverse-sequence voltages (13) is simplified as:  
\[\|\overline{\Delta\mathbf{V}}^{\mathbf{inv}}\|=\|\dot{\mathbf{Z}}^{\mathbf{ inv}-\mathbf{inv}}\overline{\Delta\mathbf{I}}^{\mathbf{inv}}\|. \tag{15}\]  
Assuming once again that the eigenvalues of \(\mathbf{inv}(\mathbf{Z}^{\mathbf{inv}-\mathbf{inv}})\) are all distinct and non-zero, the relation above can be exploited as:  
\[\|\overline{\Delta\mathbf{V}}^{\mathbf{inv}}\|=\|\sum_{\mathrm{i=1}}^{\mathrm{ ng-1}}\frac{1}{\dot{\lambda}_{\mathrm{i}}^{\mathrm{inv}-\mathrm{inv}}}\dot{ \mathbf{X}}_{\mathrm{i}}^{\mathbf{inv}-\mathbf{inv}}(\dot{\Gamma}_{\mathrm{i} }^{\mathbf{inv}-\mathbf{inv}})^{\prime}\overline{\Delta\mathbf{I}}^{\mathbf{ inv}}\|, \tag{16}\]  
with the same assumptions and meaning of the quantities as in (9), but in this case the eigensystem refers to the matrix \((\dot{\mathbf{Z}}^{\mathbf{inv}-\mathbf{inv}})^{-1}\). Let \(\dot{\lambda}_{\mathrm{k}}^{\mathrm{inv}-\mathrm{inv}}\) be the eigenvalue with minimum modulus; the 2-norm of the inverse-sequence voltage variation vector (16) can be approximated as:  
\[\|\overline{\Delta\mathbf{V}}^{\mathbf{inv}}\|\approx\frac{1}{\dot{\lambda}_{ \mathrm{k}}^{\mathrm{inv}-\mathrm{inv}}}|(\dot{\Gamma}_{\mathrm{k}}^{\mathbf{ inv}-\mathbf{inv}})^{\prime}\overline{\Delta\mathbf{I}}^{\mathbf{inv}}|. \tag{17}\]  
In (17), it has been assumed, once again, that the 2-norm of the vector \(\dot{\mathbf{X}}_{\mathrm{k}}^{\mathbf{inv}-\mathbf{inv}}\) is unitary.