## 4 Case Study
### Decentralized VOP
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multicolumn{2}{c}{\(V\) **(p.u.)**} & \multicolumn{2}{c}{\(\sigma\) **(p.u.)**} & \multicolumn{2}{c}{\(V_{M}\) **(p.u.)**} & \multicolumn{2}{c}{\(V_{m}\) **(p.u.)**} \\  & **4 VCZs** & **7 VCZs** & **4 VCZs** & **7 VCZs** & **4 VCZs** & **7 VCZs** & **4 VCZs** & **7 VCZs** \\ \hline
1 & 0.9955 & 0.9966 & 0.0042 & 0.0040 & 1.0041 & 1.0048 & 0.9883 & 0.9896 \\
2 & 0.9968 & 0.9960 & 0.0021 & 0.0037 & 1.0023 & 1.0033 & 0.9934 & 0.9888 \\
3 & 0.9945 & 0.9954 & 0.0041 & 0.0041 & 1.0032 & 1.0038 & 0.9866 & 0.9874 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Statistical parameters of the distribution of the nodal voltage amplitudes for the partitioning in four VCZs and seven VCZs provided by _Decentralized APP_. Figure 10: The histogram of the distribution of the nodal voltages for the _Decentralized APP_ in _Case 3_ with the partitioning in seven _VCZs_. Figure 9: The histogram of the distribution of the nodal voltages for the _Decentralized APP_ in _Case 3_ with the partitioning in four _VCZs_. In addition to the considerations related to the final solution achieved by the decentralized algorithm, it is important to also investigate the convergence characteristics. In particular, the number of iterations that the _Decentralized APP_ needs to reach convergence is different for the two network partitions. Convergence is achieved when the coupling error is smaller than a tolerance value, which is fixed to 2.5\(\cdot 10^{-5}\) p.u.. The coupling error is defined as the infinity norm of the vector composed of all the errors on the equalities in (5), which are calculated by each VCZ to update the Lagrangian multipliers according to (15)-(16). Obviously, the convergence rate also depends on the values of the parameters \(\epsilon\), \(c\) and \(\rho\) in the _decentralized APP_ algorithm. In this analysis, the set was equal to:  
* 4 VCZs partition: \(\epsilon=0.1\), \(c=0.15\) and \(\rho=0.29\);
* 7 VCZs partition: \(\epsilon=0.075\), \(c=0.15\) and \(\rho=0.29\). Figures 11 and 12 show the evolutions of, respectively, the objective function and the coupling error, for the partitioning in four VCZs in the three cases. It is evident that in all the cases the number of iterations does not exceed the value of 400. The worst performance is in _Case 3_, where the number of iterations to convergence is almost equal to 400. Figures 13 and 14 show the evolutions of the same quantities for the partitioning in seven VCZs. In comparison with the partitioning in four VCZs, it is evident that, for a network partitioned in seven VCZs, the _Decentralized APP_ nearly doubles the number of iterations to reach convergence, as expected from the theory. It is important to notice that the evolution of objective function always presents a rapid decrease in the first iterations. Consequently, if the algorithm stops after few tens of iterations, the value of the objective function is already close to the optimal solution. In addition, the evolution of the coupling error is also strongly decreasing after the first iterations, reaching in few tens of iterations the order of magnitude of \(10^{-3}\), which is already acceptable being expressed in voltage p.u. These considerations are important from a practical perspective point of view because they allow for relaxing the communication requirements of the decentralized solution. Figure 11: Objective function of the _Decentralized APP_ for the partition in four VCZs: (**a**) _Case 1_, (**b**) _Case 2_, (**c**) _Case 3_. Figure 12: Coupling error of the _Decentralized APP_ for the partition in four VCZs: (**a**) _Case 1_, (**b**) _Case 2_, (**c**) _Case 3_.