## 3 Convergence Analysis  
To prove that the recursive formula (12) guarantees the convergence of the power-flow problem in the case of distribution networks comprising radial and mesh structures formulated using branch-to-nodal incidence matrices, let us consider the following. **Assumption 1**. : _The amount of power consumed by the loads does produce voltage instabilities in the distribution network, i.e., the system operates under well-defined voltage conditions._  
**Assumption 2**. : _There exists a positive lower bound for all the voltage profiles, i.e., \(V^{\min}>0\), which is typically defined by the utility and regulatory policies._  
**Assumption 3**. : _The demand-to-demand impedance matrix, i.e., \(\mathbb{Z}_{dd}\), is diagonal dominant, which implies that, always, \(\left|\mathbb{Z}_{dd_{j}}\right|\geq\left|\mathbb{Z}_{dd_{jk}}\right|,\ \forall j\
eq k\)._  
To demonstrate the convergence properties of the recursive power flow approach defined by (12), let us present the general definition of the Banach fixed-point theorem as follows. **Theorem 1** (Banach fixed-point theorem). : _The iterative formula presented in (12) is stable and corresponds to a contraction map with the form_  
\[\mathbf{V}_{d}^{t+1}=g\left(\mathbf{V}_{d}^{t}\right), \tag{13}\]  
_for some \(\mathbf{V}\) that fulfills Assumption 1 irrespective of the starting point, i.e., \(\mathbf{V}^{0}\), such that_  
\[\left|\left|g\left(\mathbf{V}_{d}^{0}\right)-g\left(\mathbf{W}\right)\right| \right|\leq\theta\left|\left|\mathbf{V}^{0}-\mathbf{W}\right|\right| \tag{14}\]  
_where \(\mathbf{W}\) is the solution of the power-flow problem, and \(\theta\) is a real number between \(0\) and \(1\)._  
Proof. : The recursive formula for the iterative backward-forward power-flow method presented in (12) can be rewritten as follows:  
\[\mathbf{V}_{d}^{t+1}=g\left(\mathbf{V}_{d}^{t}\right)=-\mathbf{b}-\mathbf{Z} _{dd}\left[\frac{\mathbb{S}_{d_{i}}^{\star}}{\forall_{d_{i}}^{t,\star}}\right] _{i\in\Omega_{S}}^{T} \tag{15}\]  
where \(\Omega_{S}\) is the set that contains all the demand nodes. In addition, based on the structure of the Banach fixed-point theorem, it is possible to affirm that the solution of the power-flow problem, i.e., \(\mathbb{W}\), satisfies \(\mathbb{W}=g\left(\mathbb{W}\right)\), and it is unique if and only if \(g\left(\mathbb{W}\right)\) is a contraction mapping on \(\mathbb{V}_{d}\). \[ \begin{split}\left|\left|\mathbb{V}_{d}^{t+1}-\mathbb{W}\right| \right|&=\left|\left|g\left(\mathbb{V}_{d}^{t+1}\right)-g\left( \mathbb{W}\right)\right|\right|\\ &\left|\left|\mathbb{Z}_{dd}\mathbb{S}_{d}^{\star}\left[\frac{1}{ \mathbb{W}_{i}^{\star}}-\frac{1}{\mathbb{V}_{d_{i}}^{t,\star}}\right]_{i\in \Omega_{S}}^{T}\right|\right|\\ &\left|\left|\mathbb{Z}_{dd}\mathbb{S}_{d}^{\star}\right|\left| \left|\left|\left[\frac{\mathbb{V}_{d_{i}}^{t,\star}-\mathbb{W}_{i}^{\star}}{ \mathbb{W}_{i}^{\star}\mathbb{V}_{d_{i}}^{t,\star}}-\right]_{i\in\Omega_{S}}^ {T}\right|\right|\\ &\leq\theta\left|\left|\mathbb{V}_{d}^{t}-\mathbb{W}\right| \right|,\end{split}  \tag{16}\]  
where  
\[\theta=\frac{||\mathbb{Z}_{dd}\mathbb{S}_{d}^{\star}||}{\left(\mathbb{V}^{ \text{min}}\right)^{2}}. \tag{17}\]  
Now, on considering Assumption 2, which is regarding the positive definiteness of the impedance matrix, the expression (17) can be reduced to  
\[\theta=\max_{i\in\Omega_{S}}\left\{\frac{\left|\mathbb{Z}_{dd_{ii}}\right| \left|\mathbb{S}_{d_{i}}^{\star}\right|}{\left(\mathbb{V}^{\text{min}}\right) ^{2}}\right\}. \tag{18}\]  
It should be noted that, considering the mathematical structure presented in (18) and that \(\mathbb{Z}_{dd_{ii}}\) is the equivalent impedance at node \(i\), the following relation can be obtained.