## 3 Decentralized Voltage Optimization Problem
### Auxiliary Problem Principle  
The APP [23; 26] is used to decompose VOP (6) and (7) into separable subproblems to be solved in each VCZ and to coordinate the solutions of subproblems toward the solution of the overall problem. The basic idea of the APP is to iteratively solve an auxiliary minimization problem, which is built around an auxiliary function (AF). To this aim, APP minimizes at each step a new objective function composed of:  
* the AF = \(\sum_{i=1}^{N_{z}}K_{i}(\mathbf{z}_{i})\) which is strictly convex and additive with respect to the decomposition variables;
* the first-order linearization of the ALF = \(\sum_{i=1}^{N_{z}}LA_{i}(\mathbf{z}_{1},\dots,\mathbf{z}_{N_{z}},\lambda_{12},\dots,\lambda_{N_{z}-1\,N_{z}})\) around the solution at the previous step; and
* the first-order derivative of AF = \(\sum_{i=1}^{N_{z}}K_{i}(\mathbf{z}_{i})\) around the solution at the previous step. The new auxiliary minimization problem is formulated according to  
\[\min_{i=1,...,N_{z}}\sum_{i=1}^{N_{z}}K_{i}(\mathbf{z}_{i})+\sum_{i=1}^{N_{z} }\left(\epsilon ALF_{i}(\mathbf{z}_{1},\dots,\mathbf{z}_{N_{z}},\lambda_{12},\dots,\lambda_{N_{z}-1\,N_{z}})^{\prime}|_{\mathbf{z}^{\prime},\lambda_{ij}^ {*}}-(K_{i}(\mathbf{z}_{i}))^{\prime}|_{\mathbf{z}^{\prime},\mathbf{z}_{i}}\right) \tag{8}\]  
subject to  
\[y_{i_{\min}}\leq\mathbf{c}_{i}^{\mathsf{T}}\mathbf{z}_{i}\leq y_{i_{\max}} \quad i=1,...,N_{z}\]  
followed by updating the Lagrange multipliers \(\lambda_{ij}\) as  
\[\lambda_{ij}=\lambda_{ij}^{*}+\rho(\mathbf{e}_{ii}^{\mathsf{T}}\mathbf{z}_{i} +\mathbf{e}_{ij}^{\mathsf{T}}\mathbf{z}_{j})\quad\;i,j=1,...,N_{z},\;j\
eq i, \tag{9}\]  
where \(\epsilon\) is a positive parameter and \(\left\langle\cdot,\cdot\right\rangle\) stands for scalar product. The choice of an AF strictly convex assures the existence and uniqueness of the solution of (8) and (9). Moreover, the additive properties of AF together with the first-order linearization of ALF (in particular of the ALF quadratic terms) permit to split (8) into independent subproblems. Furthermore, it is easy to prove that such an auxiliary problem converges to a solution which forces the same first-order conditions of the original VOP (6) and (7). Finally, it is worth noticing that convergence characteristics of (8) and (9) are strictly related to the choice of the parameters \(\epsilon,\rho\) (together with \(c\), which is present in the expression of the \(ALF_{i}\) in (6)) [26]. Choosing an AF  
\[AF=\sum_{i=1}^{N_{z}}K_{i}(\mathbf{z}_{i})=\sum_{i=1}^{N_{z}}\tfrac{1}{2} \mathbf{z}_{i}^{\mathsf{T}}\mathbf{K}_{i}\mathbf{z}_{i} \tag{10}\]  
where \(\mathbf{K}_{i}\) are suitably-chosen positive-definite matrices, and recalling the form (8)-(9), the following two-level iterative auxiliary VOP is obtained  
\[\min_{ \begin{subarray}{c}\mathbf{z}_{i}\\ i=1,...,N_{z}\end{subarray} }\sum_{i=1}^{N_{z}}\tfrac{1}{2}\mathbf{z}_{i}^{ \mathsf{T}}\mathbf{K}_{i}\mathbf{z}_{i}+\mathbf{k}_{i}^{\mathsf{T}}\mathbf{z }_{i}\]  
where  
\[\mathbf{k}_{i}^{\mathsf{T}}=\epsilon(\mathbf{z}_{i}^{\mathsf{T}}\mathbf{A}_{i }+\mathbf{b}_{i}^{\mathsf{T}})+\epsilon\sum_{ \begin{subarray}{c}j=1\\ j\
eq i\end{subarray} }^{N_{z}}\{[\lambda_{ij}^{\ast}+c(\mathbf{e}_{i}^{\mathsf{ T}}\mathbf{z}_{i}^{\ast}+\mathbf{e}_{j}^{\mathsf{T}}\mathbf{z}_{j}^{\ast})] \mathbf{e}_{ii}^{\mathsf{T}}+[\lambda_{ji}^{\ast}+c(\mathbf{e}_{j}^{\mathsf{ T}}\mathbf{z}_{j}^{\ast}+\mathbf{e}_{j}^{\mathsf{T}}\mathbf{z}_{i}^{\ast})] \mathbf{e}_{ji}^{\mathsf{T}}\}-\mathbf{z}_{i}^{\mathsf{T}}\mathbf{K}_{i} \tag{11}\]  
subject to  
\[y_{i_{\min}} \leq\mathbf{c}_{i}^{\mathsf{T}}\mathbf{z}_{i}\leq y_{i_{\max}} i=1,...,N_{z}\] \[\mathbf{z}_{i_{\min}} \leq\mathbf{D}_{i}\mathbf{z}_{i}\leq\mathbf{z}_{i_{\max}} i=1,...,N_{z} \tag{12}\]  
followed by  
\[\lambda_{ij}=\lambda_{ij}^{\ast}+\rho(\mathbf{e}_{ii}^{\mathsf{T}}\mathbf{z }_{i}+\mathbf{e}_{ij}^{\mathsf{T}}\mathbf{z}_{j}) i,j=1,...,N_{z},\ j\
eq i \tag{13}\]  
VOP (12)-(13) can be split into \(N_{z}\) subproblems that can be solved in parallel by each VCZ.