## 4 Case Study
### Impact of the Network Partitioning in the Centralized VOP  
The zoning methodology in [22] is applied to obtain a simplified representation of the distribution network suitable for voltage control. Two partitions of the grid are performed: the first one considers four VCZs and, the second one, seven VCZs, as shown in Figures 5 and 6, respectively. They are performed taking into account for the sensitivities of the nodal voltages to the only reactive power, being in an MV network the reactive part of the line impedance bigger than its resistive counterpart. Tables 4 and 5 report the PNs and the DER nodes in the four VCZs and the seven VCZs network partition, respectively. Figure 4: The histogram of the grid voltage amplitudes in _Case 3_ for the _LinearDistFlow VOP_. Figure 3: The histogram of the grid voltage amplitudes in _Case 3_ for the _Benchmark VOP_. Figure 5: The partition of the IEEE 123-bus system in 4 VCZs. Figure 6: The partition of the IEEE 123-bus system in 7 VCZs. Hereafter, the impact of the different partitioning on the results provided by the solution of the problem (3) is analyzed. Firstly, the comparison is carried out by using the VPI values provided by the optimization with four VCZs and seven VCZs; results are reported in Table 6. In _Case 1_, the partition in four VCZs provides a VPI value larger than the one obtained by partitioning the network in seven VCZs; in _Case 2_, opposite results are obtained; in _Case 3_, the partition in four VCZs provides a VPI value three times smaller than that obtained by the partition in seven VCZs. From the analysis of Table 6, the partition in four VCZs gives results that are comparable with and in some cases even better than the partition in seven VCZs in terms of VPI. The above results are also confirmed by the statistical parameters that are reported in Table 7.  
Then, the comparison is carried out by using the VPI values provided by the _LinearDistFlow VOP_ in a network without partitioning and reported in Table 2. Comparing Table 6 with Table 2, it appears that, in both _Case 1_ and _Case 2_, the VPI values are almost one order of magnitude larger than the VPI values calculated by solving the _LinearDistFlow VOP_; on the contrary, in _Case 3_, the VPI value for the partition in four VCZs is similar to the one obtained by solving the _LinearDistFlow VOP_. These results give evidence that, since the objective function adopted for a network partitioned in four VCZs (or in seven VCZs) is a reduced representation of the voltages of the network, the effect of the optimization in terms of VPI is strongly dependent of the operating conditions and cannot be predicted a priori. The above results are also confirmed by comparing the statistical parameters for four VCZs and seven VCZs reported in Table 7 with the ones for the _LinearDistFlow VOP_ in Table 3. The mean values confirm that, in _Case 1_ and _Case 2_, the obtained results for four VCZs and seven VCZs are worse than the ones provided by the _LinearDistFlow VOP_, whereas, in _Case 3_, while the partitioning in seven VCZs is still the worst, the partitioning in four VCZ gives a result comparable with the one provided by the _LinearDistFlow VOP_. Similar considerations can be done for the maximum and minimum values. Concerning the standard deviations, it is interesting to highlight that they present the same small order of magnitude in all the considered cases for the VOP performed in a network clustered in VCZs and in the network without partitioning. Generally speaking, _LinearDistFlow VOP_ flattens all the nodal voltages to values that are very close to the reference values; the same behavior is shown by the VOP in four VCZs and seven VCZs, although around a value of the nodal voltage slightly lower than 1.0 p.u. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Case \#** & **4 VCZs** & **7 VCZs** \\
**(10\({}^{-3}\) p.u.)** & **(10\({}^{-2}\) p.u.)** \\ \hline
1 & 2.05 & 1.16 \\
2 & 1.02 & 1.55 \\
3 & 0.51 & 1.52 \\ \hline \hline \end{tabular}
\end{table}