## 4 Case Study
### Decentralized VOP  
In this subsection, the performance of the decentralized VOP based on the APP algorithm, referred to in the following as _Decentralized APP_, is tested by solving the problem (14)-(16) for the partitioning in both four and seven VCZs. Table 8 shows the final values of the objective function obtained by the APP algorithm for the two considered partitioning and in the three considered cases. For the sake of comparison, Table 8 reports also the results obtained by the centralized solution of the original problem (3), referred to as _Centralized VCZ-based VOP_. From the results, it is apparent that the values of the objective function obtained by decentralized and centralized algorithms are comparable for both the network partitions and in all the cases. As expected, these results give evidence that the _Decentralized APP_ converges to solutions that are equal or slightly worse to the ones given by _Centralized VCZ-based VOP_ in terms of objective function. However, the optimal solutions may correspond to different set-points of active and reactive powers of DER controllers. Consequently, the resulting voltage profiles could be different, as evidenced by the values assumed by the VPI. To this aim, Table 9 shows the values of the VPI obtained by the _Decentralized APP_ for four VCZs and seven VCZs in the three considered cases. By comparing Table 9 with Table 6, it is apparent that the values of VPI for the _Decentralized APP_ are higher or lower than that for the _Centralized VCZ-based VOP_. For sake of completeness, Table 10 shows the statistical parameters of the nodal voltages distribution obtained by the _Decentralized APP_ for both the partitions and in the three operating conditions. In addition, in this case, the values assumed by the standard deviations are almost equal to the ones obtained by the centralized approach and always very small. With reference only to _Case 3_, the histograms of the nodal voltage distribution calculated by the _Decentralized APP_ for both the partitions are reported in Figures 9 and 10, respectively. \begin{table}
\begin{tabular}{l c c c c} \hline \hline  & \multicolumn{2}{c}{**Decentralized APP**} & \multicolumn{2}{c}{**Centralized VCZ-Based VOP**} \\
**Case** & \multicolumn{2}{c}{**(\(\mathbf{10^{-4}}\) p.u)**} & \multicolumn{2}{c}{**(\(\mathbf{10^{-4}}\) p.u)**} \\  & **4 VCZ** & **7 VCZ** & **4 VCZ** & **7 VCZ** \\ \hline
1 & 1.16 & 1.36 & 0.84 & 1.01 \\
2 & 0.75 & 0.86 & 0.75 & 0.84 \\
3 & 1.24 & 1.38 & 1.16 & 1.38 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Final values of the objective function for _Decentralized APP_ and _Centralized VCZ-based VOP_ with the partitioning in four and seven VCZs. \begin{table}
\begin{tabular}{l c c} \hline \hline  & **4 VCZs** & **7 VCZs** \\  & **(\(\mathbf{10^{-2}}\) p.u.)** & **(\(\mathbf{10^{-2}}\) p.u.)** \\ \hline
1 & 1.96 & 1.41 \\
2 & 0.77 & 1.50 \\
3 & 2.45 & 1.98 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Voltage performance index for the partitioning in four VCZs and seven VCZs provided by _Decentralized APP_. In summary, comparing decentralized and centralized solutions, although the objective functions are very similar, the optimal set-points of the active and reactive power controllers of DERs are different because the problem (3) is not strictly convex. In practice, there are infinite combinations of DERs active and reactive powers that give, as solutions, the same objective functions. However, different power injections cause different voltage profiles, as confirmed by the VPI and statistical parameters of the voltages distribution. Anyway, the amplitudes of the nodal voltages are always polarized around the reference values.