# The Equivalence between Successive Approximations and
## 2 SA Load Flow  
The complex load flow formulation of electrical distribution grids is developed by applying the conventional nodal voltage method to all buses of the system under steady-state operative conditions [11]. The main complication of the load flow problem is that it is nonlinear non-convex because of the hyperbolic relation between voltages and currents, which is due to the presence of constant power loads. The basis of the load flow problem formulation is presented below [12; 17]. \[\mathbf{I}=\mathbb{Y}\mathbf{V}, \tag{1}\]  
\[\mathbf{S}=\mathbf{diag}(\mathbf{V})\mathbf{I}^{*}\Leftrightarrow\mathbf{S}^{ *}=\mathbf{diag}(\mathbf{V}^{*})\mathbf{I}, \tag{2}\]  
where \(\mathbf{V}\in\mathcal{C}^{n}\) and \(\mathbf{I}\in\mathcal{C}^{n}\) are column vectors with the nodal voltages and injected currents, respectively. \(\mathbb{Y}\in\mathcal{C}^{n\times n}\) is the bus admittance matrix of the distribution grid; \(\mathbf{diag}(\mathbf{V})\) being a diagonal matrix that has invertible properties, such that \(\mathbf{diag}_{ii}(\cdot)=\mathbf{V}_{i},\ i=1,2,...,n\) and \(\mathbf{diag}_{ij}(\cdot)=0,\ i\
eq j\). In addition, \(\mathbf{S}\in\mathcal{C}^{n}\) is a column vector that has apparent power injections in all the buses of the grid, and \(n\) is the total nodes of the distribution system. If Expressions (1) and (2) are combined, the classical load flow representation in the domain of the complex number is found to be formulated as in (3):  
\[\mathbf{S}^{*}=\mathbf{diag}(\mathbf{V}^{*})\mathbb{Y}\mathbf{V}, \tag{3}\]  
**Remark 1**. : _Note that the main interest in the solution of the load flow formulation defined in (3) corresponds to the determination of the unknown voltage variables associated with all the demand nodes, as in the slack node, this variable is perfectly known._  
Taking this into account, Expression (3) can be split as presented below. \[\mathbf{S}^{*}_{g}=\mathbf{diag}\Big{(}\mathbf{V}^{*}_{g}\Big{)} \Big{[}\mathbb{Y}_{gg}\mathbf{V}_{g}+\mathbb{Y}_{gd}\mathbf{V}_{d}\Big{]}, \tag{4}\] \[-\mathbf{S}^{*}_{d}=\mathbf{diag}(\mathbf{V}^{*}_{d})\Big{[} \mathbb{Y}_{dg}\mathbf{V}_{g}+\mathbb{Y}_{dd}\mathbf{V}_{d}\Big{]}, \tag{5}\]where \(\mathbf{S}_{d}\in C^{n-s}\) and \(\mathbf{S}_{g}\in C^{s}\) represent the complex power consumed in all load buses and the apparent power generation of voltage controlled buses, respectively; \(\mathbf{V}_{g}\in C^{s}\) corresponds to the voltage outputs in the slack buses, while \(\mathbf{V}_{d}\in C^{n-s}\) represents the unknowable voltages of all the load buses of the grid. Observe that \(\mathbf{diag}_{g}\big{(}\mathbf{V}_{g}\big{)}\in\mathbb{C}^{s\times s}\) and \(\mathbf{diag}_{d}(\mathbf{V}_{d})\in\mathbb{C}^{(n-s)\times(n-s)}\) can be interpreted in the same form that \(\mathbf{diag}(\mathbf{V})\). In addition, note that \(s\) defines the number of ideal voltage-controlled sources (e.g., slack nodes), which entails that \(s\geq 1\). In addition,  
\[\mathbb{Y}=\left[ \begin{array}{cc}\mathbb{Y}_{gg}&\mathbb{Y}_{gd}\\ \mathbb{Y}_{dg}&\mathbb{Y}_{dd}\end{array} \right],\]  
where we know that \(\mathbb{Y}_{dg}=\mathbb{Y}_{gd}^{T}\) due to the properties of symmetry of the admittance matrix. Note that in the load flow formulation given by (4) and (5), the expression associated with the slack nodes, i.e., Equation (4), is linear, as \(\mathbf{V}_{g}\) is perfectly known in magnitude and angle, and the variables \(\mathbf{V}_{d}\) and \(\mathbf{S}_{g}\) have a linear relation. Nonetheless, the set of Equations in (5) remain a nonlinear set of expressions in the complex domain due to the presence of the product between voltages in the demand nodes.