## 3 Solution Method
### Selection of Candidate Buses for the Improvement of Voltage Profile  
Let us consider initially the allocation of one single-phase DESS that will cause the variation of the phase voltages at all buses of the network except the slack bus (\(\overline{\Delta\mathbf{V}}\)). Having in mind that the network has been represented with a three-phase model, this variation can be evaluated by:  
\[\|\overline{\Delta\mathbf{V}}\|=\|(\dot{\mathbf{Y}})^{-1}\overline{\Delta \mathbf{I}}\|, \tag{8}\]  
where \(\dot{\mathbf{Y}}\) is the nodal admittance matrix, \(\overline{\Delta\mathbf{I}}\) is the vector of the variation of the injected currents and the symbol \(\|\cdot\|\) indicates the 2-norm of the vector. In this case, the vector \(\overline{\Delta\mathbf{I}}\) has only one  
Figure 1: Flow chart of the proposed procedure. GA: genetic algorithm. ABS: allocation bus set.  
non-zero element, that is the one corresponding to the phase of the bus where the DESS is connected. The non-zero element is the current injected by the single-phase DESS.  
Based on the ISTN applied to unbalanced systems [36], if the eigensystem of the admittance matrix is known and assuming that the eigenvalues are all distinct and non-zero and an adequate normalization of eigenvalues is applied, (8) can be exploited as:  
\[\|\overline{\Delta\mathbf{V}}\|=\|\sum_{i=1}^{3(\mathrm{ng}-1)}\frac{1}{ \dot{\lambda_{i}}}\dot{\mathbf{x}_{i}}\dot{\mathbf{r}_{i}}^{\prime}\overline{ \Delta\mathbf{I}}\|, \tag{9}\]  
where \(\dot{\Gamma}_{1}^{\prime}\) (\(\dot{\mathbf{X}_{i}}\)) is the left (right) eigenvector associated to the ith eigenvalue of \(\dot{\mathbf{Y}}\) (\(\dot{\lambda_{i}}\)).  
As it is well known [32; 33; 34], the decomposition in (9) can be simplified as:  
\[\|\overline{\Delta\mathbf{V}}\|\approx\|\frac{1}{\dot{\lambda_{k}}}\dot{ \mathbf{X}_{k}}\dot{\Gamma}_{k}^{\prime}\overline{\Delta\mathbf{I}}\|, \tag{10}\]  
where \(\dot{\lambda_{k}}\) is the eigenvalue of minimum modulus of \(\dot{\mathrm{Y}}\). When the normalization of the eigenvector is used to have unitary norm, (10) can be further simplified as:  
\[\|\overline{\Delta\mathbf{V}}\|\approx\frac{1}{\dot{\lambda_{k}}}\Big{|}\dot{ \Gamma}_{k}^{\prime}\overline{\Delta\mathbf{I}}\Big{|}. \tag{11}\]  
The DESS should be allocated at a bus such that the maximum value of the voltage variation is obtained. Then, if we refer to (11), we can expect that the maximum value of \(\|\overline{\Delta\mathbf{V}}\|\) will be experimented if the single-phase DESS is allocated at the bus corresponding to the maximum element (in absolute value) of \(\dot{\Gamma}_{k}\).  
Since the selection is based on a simplified equation, that is (10), and it could be necessary to adopt the procedure to allocate more than one DESS (not only single-phase, but also three-phase systems), to determine the best allocations it is advisable to select a subset of candidate buses (instead of a single bus) corresponding to the highest elements of the left eigenvector \(\dot{\Gamma}_{k}\). Hereinafter, this subset will be indicated as \(\Omega_{1}\). The dimension of the subset \(\Omega_{1}\) depends on the number of buses included in it. The selection of the number of candidate buses can be performed by choosing the buses corresponding to the elements i (in absolute value) of the left eigenvector \(\dot{\Gamma}_{k}\) that meet the condition:  
\[\frac{\Gamma_{k,\mathrm{MAX}}-\mathbf{\Gamma}_{k,\mathrm{i}}}{\Gamma_{k, \mathrm{MAX}}}100\leq\mathrm{m}_{100}, \tag{12}\]  
where \(\Gamma_{k,\mathrm{MAX}}\) is the maximum element (in absolute value) of \(\dot{\Gamma}_{k}\) and \(\mathrm{m}_{100}\) is a proper percentage threshold value.