## 3 Decentralized Voltage Optimization Problem
### Centralized VOP for Network Partitioned in VCZs  
By adopting the network partitioning method in [22], the distribution grid is divided into \(N_{z}\) VCZs; then, for each VCZ, a PN is identified [30]. The VOP (1) can be split into a set of reduced-size subproblems written for each VCZ according to  
\[\min_{\begin{subarray}{c}\mathbf{x}_{i}\\ i=1,...,N_{z}\end{subarray}}\sum_{i=1}^{N_{z}}\frac{1}{2}\,y_{i}^{2}+y_{ \text{ref}}\,y_{i} \tag{3}\]  
subject to  
\[y_{i}=\gamma_{ii}\mathbf{x}_{i}+\sum_{\begin{subarray}{c}j=1\\ j\
eq i\end{subarray}}^{N_{z}}\gamma_{ij}\mathbf{x}_{j}\qquad i=1,...,N_{z}\] \[y_{\text{imin}}\leq y_{i}\leq y_{\text{imax}}\qquad i=1,...,N_{z}\] \[\mathbf{x}_{\text{min}}\leq\mathbf{x}_{i}\leq\mathbf{x}_{\text{ imax}}\qquad i=1,...,N_{z}\]  
where \(y_{i}\) and \(y_{\text{ref}}\) are the variations of, respectively, the voltage and the voltage reference value at the \(PN_{i}\); \(\mathbf{x}_{i}\) the vector of the variations of the set-points of the active and reactive powers controllers of the DERs connected to \(VCZ_{i}\); \([y_{i_{min}},y_{i_{max}}]\) and \([\mathbf{x}_{i_{min}},\mathbf{x}_{i_{max}}]\) are the admissible ranges of variation of, respectively, \(y_{i}\) and \(\mathbf{x}_{i}\); \(\boldsymbol{\gamma}_{ii}\) and \(\boldsymbol{\gamma}_{ij}\) are submatrices of \(\boldsymbol{\Gamma}\) that relate the squared nodal voltage at \(PN_{i}\) to the powers injected by DERs connected, respectively, to the same VCZ (i.e., \(VCZ_{i}\)) and to the other VCZs (i.e., \(VCZ_{j}\) with \(j=1,\ldots,N_{z}\) and \(j\
eq i\)).  
Although the objective function in (3) has a form that trivially allows its distribution among VCZs, equality constraints present terms that cannot be assigned to a single VCZ since they couple \(VCZ_{i}\) with the other control zones. To easier management of coupling constraints, overlapping variables \(w_{ij}\) coupling \(VCZ_{i}\) with \(VCZ_{j}\) are introduced, and the VOP (3) is rewritten by introducing supplement equality constraints (\(w_{ij}=\boldsymbol{\gamma}_{ij}\mathbf{x}_{j}\)) that ensure the problem consistency and help the VOP to be solved in a distributed manner that is  
\[\min_{\begin{subarray}{c}\mathbf{x}_{i}\\ i=1,...,N_{z}\end{subarray}}\sum_{i=1}^{N_{z}}\frac{1}{2}\,y_{i}^{2}+y_{i \text{ref}}\,y_{i} \tag{4}\]  
subject to  
\[y_{i}=\boldsymbol{\gamma}_{ii}\mathbf{x}_{i}+\sum_{\begin{subarray}{c}j=1 \\ j\
eq i\end{subarray}}^{N_{z}}w_{ij}\qquad\qquad i=1,...,N_{z}\] \[w_{ij}=\boldsymbol{\gamma}_{ij}\mathbf{x}_{j}\qquad\qquad\qquad i,j=1,...,N_{z},\quad j\
eq i\] \[y_{i_{\min}}\leq y_{i}\leq y_{i_{\max}}\qquad\qquad i=1,...,N_{z}\] \[\mathbf{x}_{i_{\min}}\leq\mathbf{x}_{i}\leq\mathbf{x}_{i_{\max}} \qquad\qquad i=1,...,N_{z}\]  
Finally, VOP (4) can be rewritten in a matrix form by substituting the linearized model of the network in the objective function and by introducing the new vectors of controllable variables \(\mathbf{z}_{i}^{\text{T}}=\left(\mathbf{x}_{i}^{\text{T}}\ w_{ij}\right)\) with \(i,j=1,...,N_{z}\) and \(j\
eq i\), as  
\[\min_{\begin{subarray}{c}\mathbf{z}_{i}^{\text{T}}\\ i=1,...,N_{z}\end{subarray}}\sum_{i=1}^{N_{z}}\tfrac{1}{2}\mathbf{z}_{i}^{ \text{T}}\mathbf{A}_{i}\mathbf{z}_{i}+\mathbf{b}_{i}^{\text{T}}\mathbf{z}_{i} \tag{5}\]  
subject to  
\[\mathbf{e}_{ii}^{\text{T}}\mathbf{z}_{i}+\mathbf{e}_{ij}^{\text{T}} \mathbf{z}_{j}=0\qquad\quad i,j=1,...,N_{z},\quad j\
eq i\] \[y_{i_{\min}}\leq\mathbf{c}_{i}^{\text{T}}\mathbf{z}_{i}\leq y_{i _{\max}}\qquad\qquad\quad i=1,...,N_{z}\] \[\mathbf{z}_{i_{\min}}\leq\mathbf{D}_{i}\mathbf{z}_{i}\leq\mathbf{ z}_{i_{\max}}\qquad\qquad i=1,...,N_{z}\]  
Expressions of matrices and vectors in (5) are in Appendix A.  
VOP (5) cannot be distributed among VCZs because the equality constraints are not yet separable. Furthermore, with respect to the VOP (1), it is turned into a convex constrained quadratic programming problem providing local minimum points; this is due to the objective function that, considering only PNs, remains convex but not in a strictly sense.