## 1 Introduction
minimizes an objective function (e.g., the deviation of the nodal voltages from their reference values, the network losses, and/or the DG active power curtailments); (_ii_.) is subject to equality constraints (i.e., power flow (PF) equations) and to inequality constraints (i.e., limits on bus voltage magnitudes, operation range of conventional, and innovative voltage control devices); and (_iii_.) determines the optimal set-points of the local controllers (i.e., tap positions of OLTC/SVRs, optimal status of CBs, active, and or reactive powers of DER local controllers). The centralized VOP is a mixed-integer nonlinear programming. It is difficult to solve for two reasons: firstly, it is not convex due to the quadratic relationship between bus voltages and powers; then, it is of large dimension in dependence on the size of the network and the number of variables to be controlled [5]. To make the VOP convex and reduce its complexity, linearization techniques [6; 7; 8], semi-definite relaxation (SDR) [5; 9], and second-order conic (SOC) [10; 11] programming have been proposed. Once the non-convex centralized VOP is transformed into a convex one, to some extent, it is easier solving it also according to a decentralized approach. To this aim, the original centralized VOP is decomposed into a number of simple optimization sub-problems (equal to the number of VCZs obtained by the network partitioning); the consistency between centralized and decentralized optimization is assured by the local information exchanges among VCZs; this coordinating action can be implemented in various ways. In [12], the voltage of the dynamic subcommunity that presents the worst violations of the nodal voltage boundaries is independently controlled by employing a particle swarm optimization method; then, to address the interaction between different sub-communities, an overall network power flow calculation is conducted: if all the network nodal voltages are in the allowable range,the regulation is stopped; otherwise, the process is repeated for all the subcommunities. Despite of the limited processing time at subcommunity level, such an approach requires a heavy centralized coordination. In [13], a distributed sequential quadratic programming is used for optimizing DGs in each zone; coordination is based on a master-slave approach and requires an iterative boundary data exchange and power flow recalculation; the drawback is that no convergence conditions are provided. One of the most popular and effective method to perform a distributed optimization control between clusters is based on the alternating direction method of multipliers (ADMM) [7; 8; 14; 15; 16; 17; 18; 19]. The solution is based upon a cluster-based optimization process, where a limited amount of information is exchanged between neighboring clusters; the main benefits of ADMM are the absence of centralized coordination or load flow recalculation, and robust convergence characteristics. Recently, in [20], a new method for the decentralized voltage optimization of balanced active distribution systems with radial topology has been proposed. Firstly, a centralized VOP, minimizing the distance of the voltages from their reference values, is formulated as a strictly-convex constrained quadratic program to determine the optimal set-points of the reactive power controllers of DGs. To this end, a new linearized model formulation of the DistFlow equations is exploited [21]. Then, the centralized VOP is reformulated by partitioning the network into two VCZs with pilot nodes (PNs); the PN voltage variation best represents the variation of the voltages in the VCZ [22]. However, the new objective function, accounting only for voltage magnitude at PNs, keeps the centralized VOP convex but not in a strict sense.