## 1 Introduction  
In distribution networks, the large penetration of distributed energy resources (DERs), such as distributed generators (DGs), storage systems, electrical vehicles, and controllable loads, has a significant impact on voltage control. Then, the proper selection of the voltage control strategy and architecture, the adequate settings of local controllers and, in the near future, the coordination of the transmission system operators (TSOs) and distribution system operators (DSOs) are the many challenges that need to be faced for voltage control in smart grids [1]. Presently, the voltage control strategies can be divided into four categories on the basis of communication: local, centralized, distributed, and decentralized [2]. Optimizing the voltages of the distribution networks in a centralized way would seem the most natural approach to solve the problem of the voltage regulation as it projects the method used in the transmission system into the distribution ones. Centralized controls consist of collecting measurements from the field, solving a constrained optimization problem using a distribution management system (DMS) and sending the optimal set-pointsto the local voltage control devices. Conventional devices employed for voltage control in distribution networks are: on-load tap changer (OLTC) of the transformer in HV/MV substation, step voltage regulators (SVRs), and capacitor banks (CBs); innovative devices are inverter-interfaced DERs, which can respond much faster than the previous ones by varying their active and/or reactive generation and consumption. Although the centralized approach provides the optimal global solution to the problem, it is expected to encounter technical obstacles resultant from the distributed nature of the electrical system. In fact, the high number of nodes, the increasing integration of DERs, and their participation in the voltage regulation reflect on a large number of measurements to be acquired from the field, sent to, and processed by the DMS as well as on many signals to be dispatched to the control devices. It results in a high cost of investments for advanced measuring technologies, high-speed two-way communication infrastructures, and a powerful computing facility with high computation efficiency [3]. Furthermore, a centralized control architecture is unable to quickly respond to adverse events becoming vulnerable in the face of cyber and physical attacks to the DMS [4]. To mitigate the drawbacks of centralized strategies, a great potential is offered by decentralized controls. The basic idea is, firstly, to decompose the large distribution network into smaller voltage control zones (VCZs), in which the nodes are strongly coupled from an electrical point of view; and, then, to achieve the regulation of the voltage profiles of the entire distribution network through intra VCZ control and inter VCZ coordination. In particular, inside a VCZ, a centralized control optimizes the voltages of its cluster of nodes acting on only its local voltage control devices; among VCZs, a distributed control, consisting of a data exchanges among specified nodes of VCZs, is used to drive the VCZ solutions to the global optimization of the whole distribution system. By reducing the size of the problem to be solved in each VCZ, computing facilities will process a limited amount of data; furthermore, information exchange and communication infrastructure will cover limited network areas with the great advantage of lower investment costs and higher resilience to faults and cyber attacks. Despite of these undoubtable advantages, decentralised approaches provide a sub-optimal solution with respect to the centralized one. To formulate and solve the voltage optimization problem (VOP) in a decentralized manner, first of all, the VOP must be formulated according to a centralized approach. Typically, it is a constraint optimization program that: (_i_.)