## 4 Case Study
### Impact of the Linearization on the Centralized VOP  
To evaluate the approximations introduced by the linearization of the PF equations, in this subsection, the results obtained by executing the centralized VOP (1) subject to the linearized PF equations (referred to as _LinearDistFlow VOP_) are compared with the results obtained by a VOP subject to the PF equations (referred in the following to as the _Benchmark VOP_). In the reminder, the performance of the optimization methods is evaluated by a voltage performance index (VPI) that is calculated as the sum of the squared deviations of the squared voltages of all the nodes of the MV feeders and subfeeders with respect to the reference values, fixed at 1.0 p.u. in all the analyzed cases; it is worth noting that the VPI does not include neither the HV slack bus nor the MV substation busbar. Table 2 compares the values of the VPI obtained by the _LinearDistFlow VOP_ and the _Benchmark VOP_ in the three considered cases. From the results reported in Table 2, it is evident that the VPI values are in the same order of magnitude for the two VOP methodologies: it is obvious that, in all the three operating conditions, the _Benchmark VOP_ performs better than the _LinearDistFlow VOP_. Particularly, in _Case 1_, the VPI values are only 0.3\(\cdot\)10\({}^{-3}\) p.u. different one with the other, in _Case 2_, the distance increases to 0.9\(\cdot\)10\({}^{-3}\) p.u., finally, in _Case 3_, the performance of the _LinearDistFlow VOP_ is about 3\(\cdot\)10\({}^{-3}\) p.u. different from the one of the _Benchmark VOP_, but the two values are still comparable. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline
**Power** & **\(\mathbf{ DER_{i}}\)** & **\(\mathbf{ DER_{3}}\)** & **\(\mathbf{ DER_{i}}\)** & **\(\mathbf{ DER_{5}}\)** & **\(\mathbf{ DER_{6}}\)** & **\(\mathbf{ DER_{7}}\)** & **\(\mathbf{ DER_{8}}\)** & **\(\mathbf{  
To further confirm the accuracy of the _LinearDistFlow VOP_, some statistical parameters related to the distribution of the voltage amplitudes into the grid have been evaluated for both _Benchmark VOP_ and _LinearDistFlow VOP_. Table 3 reports the mean value \(V\), the standard deviation \(\sigma\), the maximum value \(V_{M}\), and the minimum value \(V_{m}\) of the voltage amplitude of all the network nodes, derived by applying _benchmark VOP_ and _LinearDistFlow VOP_ for the three considered cases. Similarly to the VPI, hereafter, the statical parameters are evaluated with reference to nodes of the MV feeders and subfeeders, excluding the HV slack bus and the MV substation busbar. From Table 3, it is apparent that, for all the three operating conditions, the reported values for the two algorithms are very close. The histograms of the distributions of the network node voltages for the _Benchmark VOP_ and the _LinearDistFlow VOP_ are reported in Figures 3 and 4, respectively, with reference to the only _Case 3_, which is the worst case according to Table 2. Both histograms show bell-shaped curves; the entire range of values, that goes from 0.99 to 1.01 p.u., is divided into 20 intervals. The peaks of both bell-shape histograms show more or less the same height, between 30 and 40, which represents the number of voltages falling into the most populated interval. For both Figures 3 and 4, the largest number of voltages are concentrated in the range between 0.995 and 1 p.u., slightly closer to 1 p.u. for the _Benchmark VOP_. As a final consideration, it is important to underline that, in all the considered cases, the linearization of the PF equations uses the same matrix \(\Gamma\), which, as previously explained, has been obtained in the same operating conditions as _Case 1_, but in absence of any DER. It is evident that the larger the difference between the considered case and the operating conditions in which linearization is performed, the larger the approximation introduced by the _LinearDistFlow VOP_; this is the reason why the results in _Case 1_ are the best and the ones in _Case 3_ are the worst.