## 5 Conclusions  
A distributed algorithm based on the auxiliary problem principle (APP) has been proposed in this paper to solve the voltage optimization problem (VOP) in distribution networks with distributed energy resources (DERs) according to a decentralized approach. To this aim, the distribution network is decomposed in a reduced number of voltage control zones (VCZs) with pilot nodes (PNs) and a new linearized DistFlow method for radially-operated distribution networks is adopted to make the voltage optimization problem convex. The reformulation of the VOP that accounts only for PNs, allows for acquiring voltage measurements only at the PNs and does not require a system state-estimation. The main advantage of the proposed distributed algorithm is that it achieves  
Figure 14: Coupling error of the _Decentralized APP_ for the partition in seven VCZs: (**a**) Case 1, (**b**) Case 2, (**c**) Case 3. Figure 13: Objective function of the _Decentralized APP_ for the partition in 7 VCZs: (**a**) Case 1, (**b**) Case 2, (**c**) Case 3.  
the global optimal solution, although, of an approximate VOP, by iteratively solving strictly-positive definite quadratic subproblems of small dimension in each VCZ and exchanging only scalar values among VCZs. The proposed algorithm has been tested on the IEEE 123-bus system. The conducted analysis has been demonstrated that the adoption of the linearization technique assures high accuracy of the results with respect to classical optimization adopting nonlinear power flow equations. Furthermore, the optimization of the nodal voltages at the only PNs has given results comparable with the ones provided by a fully centralized approach. In particular, it has been shown that, even though not all the network voltages are optimized, the voltage amplitudes are polarized around the reference values, limiting the possibility that some nodal voltages may exceed normal operating bounds. Finally, the convergence characteristics of the distributed algorithm have been analyzed evidencing that the number of iterations to convergence increases with the number of the network partitions. Numerical results have shown that the choice of four VCZs is a good trade-off between accuracy and requirements for a practical implementation, keeping in mind that the best choice is strongly system-dependent. Future research will compare from both a theoretical and numerical point view the APP with other distributed techniques. The application of the proposed method to unbalanced distribution systems will also be investigated, by adopting an adequate three-phase linear model of the network. Figure 15: The network voltage profile for the _no VOP_, _Benchmark VOP_ and _Decentralized APP_ in _Case 1_. **Author Contributions:** Conceptualization, A.R.D.F. and M.R. ; methodology, A.R.D.F. and M.D.S. ; software, M.D.S. and C.R. ; validation, C.R. and M.D.S. ; formal analysis, A.R.D.F. ; investigation, A.R.D.F., C.R. and M.D.S. ; data curation, M.D.S. ; writing--original draft preparation, C.R. and M.D.S. ; writing--review and editing, A.R.D.F., M.D.S. and M.R. ; visualization, M.D.S. ; supervision, A.R.D.F. ; project administration, M.R. ; funding acquisition, M.R. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the Italian Ministry of University and Research by the special grant \"Dipartimenti di eccellenza\". **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. Figure 16: The network voltage profile for the _no VOP, Benchmark VOP,_ and _Decentralized APP_ in _Case 2_. Figure 17: The network voltage profile for the _no VOP, Benchmark VOP,_ and _Decentralized APP_ in _Case 3_. Data Availability Statement: Data are contained within the article and in referenced papers. Conflicts of Interest: The authors declare no conflict of interest. Sample Availability: Samples of the compounds are available from the authors.