# The Equivalence between Successive Approximations and
## 3 MBF Load Flow  
The MBF load flow formulation is a compact generalization of the classical iterative sweep load flow, where the incidence matrix is employed. To obtain this compact load flow formulation, the incidence matrix is defined as follows [8; 9]:  
[Incidence matrix \(\mathcal{A}\)] An electrical distribution network can be represented by a connected graph composed of \(n\) vertices and \(b\) branches using a rectangular matrix \(\mathcal{A}\in\mathbb{R}^{b\times n}\) called the incidence matrix. To construct this matrix, arbitrary directions associated with the current flows are assumed in all the branches as follows:  
* \(\mathcal{A}_{i,j}=1\) when branch \(i\) connects bus \(j\) and its current is leaving bus \(i\);
* \(\mathcal{A}_{i,j}=-1\) when branch \(i\) connects bus \(j\) and its current is arriving bus \(j\);
* \(\mathcal{A}_{i,j}=0\) when branch \(i\) has no connection with bus \(j\). For an electrical distribution network, the incidence matrix, can be split into two submatrices that contain information of the slack and demand nodes and their relationswith all the branches as follows (for the sake of simplicity, we assume that the slack node is located at bus 1):  
\[\mathcal{A}=\big{[}\mathcal{A}_{g}\ \mathcal{A}_{d}\big{]}, \tag{8}\]  
where \(\mathcal{A}_{g}\) represents the first column of the matrix \(\mathcal{A}\) (i.e., voltage-controlled source), and \(\mathcal{A}_{d}\) is the remaining columns of the matrix \(\mathcal{A}\), i.e., columns associated with all the load buses. Now, we define the voltage drop at branch \(i\) as \(\mathbf{E}_{i}\) and the voltages on its sending and receiving nodes \(\mathbf{V}_{j}\). Then, the following result is achieved:  
\[\mathbf{E}=\mathcal{A}_{g}\mathbf{V}_{g}+\mathcal{A}_{d}\mathbf{V}_{d}, \tag{9}\]  
where \(\mathbf{E}\) is a vector that has all the voltage drops in all the links of the distribution network. In addition, to determine the relation between the branch's currents and their voltage drops, the [PERSON]'s law is applied to each one of them, which allows the finding of the following:  
\[\mathbf{E}=\mathbb{Z}_{p}\mathbf{J}, \tag{10}\]  
where \(\mathbb{Z}_{p}\) corresponds to the diagonal invertible matrix that holds in all the branch' impedances, i.e., \(\mathbb{Z}=\mathbf{diag}(Z_{1}\ Z_{2}\,...,Z_{b}\ )\). Now, to define the mathematical relation between the nodal currents (i.e., \(\mathbf{I}\)) and all the lines' currents, we apply the second [PERSON]'s law on each bus, which produces, in a compact form, the following result, split for the slack and demand nodes:  
\[\mathbf{I}_{g} =\mathcal{A}_{g}^{T}\mathbf{J}, \tag{11}\] \[\mathbf{I}_{d} =\mathcal{A}_{d}^{T}\mathbf{J}. \tag{12}\]  
Observe that if Expressions (9), (10), and (12) are combined, considering that \(\mathbb{Y}_{p}=\mathbb{Z}^{-1}\), then the result is as follows:  
\[\mathbf{I}_{d}=\mathcal{A}_{d}^{T}\mathbb{Y}_{p}\mathcal{A}_{g}\mathbf{V}_{g} +\mathcal{A}_{d}^{T}\mathbb{Y}_{p}\mathcal{A}_{d}\mathbf{V}_{d}. \tag{13}\]  
To determine the relationship between the nodal currents and the net powers, from the [PERSON]'s theorem defined in (2), we can obtain the following:  
\[\mathbf{I}_{g}=\mathbf{diag}^{-1}\Big{(}\mathbf{V}_{g}^{*}\Big{)} \mathbf{S}_{g}^{*}, \tag{14}\] \[\mathbf{I}_{d}=-\mathbf{diag}^{-1}(\mathbf{V}_{d}^{*})\mathbf{S} _{d}^{*}. \tag{15}\]  
Now, if Equations (1) and (15) are mixed to attain \(\mathbf{V}_{d}\), then, the next result yields:  
\[\mathbf{V}_{d}=-\mathbf{Z}_{dd}\big{[}\mathbf{diag}^{-1}(\mathbf{V}_{d}^{*}) \mathbf{S}_{d}^{*}+\mathcal{A}_{d}^{T}\mathbb{Y}_{p}\mathcal{A}_{g}\mathbf{V} _{g}\big{]}, \tag{16}\]  
being \(\mathbf{Z}_{dd}\) defined as \(\big{[}\mathcal{A}_{d}^{T}\mathbb{Y}_{p}\mathcal{A}_{d}\big{]}^{-1}\). **Remark 4**. : _The solution of Equation (16) is found in the same form as described for Equation (7) by adding an iterative counter and starting with plane voltages.