## 1 Introduction
Therefore, for overcoming the lack of strict convexity, a dual approach is considered; the dual VOP works on the augmented Lagrangian function (ALF), and it is iteratively solved by the method of multipliers (MM) [14]; this latter is a two-level iterative method which solves at each step the VOP to obtain the values of the optimization variables that are used in the second level to update the Lagrange multipliers. Unfortunately, quadratic terms in the ALF make the VOP still not separable among the VCZs. Eventually, the VOP is solved in a fully decentralized form by using the auxiliary problem principle (APP) [23; 24]. Generally speaking, the APP involves linearizing the non-separable terms in the ALF and adding convex terms which can be chosen to be separable [25], so as to decompose the VOP into zone VOPs. In each iteration step, the zone VOPs are solved in parallel, each one by the solver assigned to the related VCZ; then, an information exchange allows for updating the Lagrange multipliers so as to re-formulate the zone VOPs for the next step. The iterative process continues until convergence is reached. In general, the APP technique has convergence conditions [26] and a better convergence behavior than the ADMM in the case of non-convex problems [27]. According to the authors' best knowledge, this is the first time that APP has been applied to the decentralized voltage optimization of distribution networks by varying active and reactive powers of DERs. This paper extends and improves the method outlined in [20]. First of all, the proposed decentralized VOP is generalized to the case of a distribution systems partitioned into an appropriate number of VCZs. Moreover, since, in distribution networks, the nodal voltage magnitudes are often fairly sensitive to variations of active rather than reactive power injections; in this paper, the VOP determines not only the reactive power generation of DGs but also the active and reactive powers provided by electric storage systems. Finally, since DER controllers are much faster then conventional voltage control devices, the optimal setting of the slow time-scale actuators are evaluated a priori and assigned as parameters in the formulation of the VOP linearized around a particular initial operating point [8]. The main features of the proposed decentralized VOP are summarized hereafter. * In the centralized formulation of VOP, the linearized DistFlow equations, based on a Jacobian method proposed in [21], assures an improvement of the accuracy of the results with respect to other linearization techniques. In fact, the adoption of a peculiar set of variables for the analytical derivatives does not require to neglect or to approximate as constants the nonlinear terms of the DistFlow equations as in [7; 8; 16; 19]. Consequently, the adoption of such technique limits the approximation of the solution with respect to classical VOP using nonlinear PF equations. * To decompose the centralized VOP into subproblems, the adopted partitioning method takes into account the dependence of the nodal voltages on both active and reactive powers injected in all the nodes of the grid [22; 28]; in such a way, the network partitioning considers the R/X ratio which characterizes the distribution networks. The method is based on the definition of the electrical distance, which is evaluated by using once again the linear model in [21] that provides in a closed form the sensitivity of the voltage magnitudes to the power variations; consequently, the partition is extracted from structural characteristics of the network independently from the number and position of the DERs in a specific configuration. * The reformulation of the VOP that accounts only for PNs, allows for acquiring a reduced number of voltage measurements from the field (i.e., one for each VCZ) and does not require a system state-estimation; consequently, it can be easily implemented in practice.