# The Equivalence between Successive Approximations and
## 1 Introduction  
Load flow is one of the most recognized studies in electrical networks from high- to low-voltage levels [1, 2]. This problem relates voltages, currents, and powers under steady state conditions, which produce a set of nonlinear non-convex equations that need numerical methods to be resolved [3]. In the case of power systems, the Newton-Raphson (NR) and the Gauss-Seidel (GS) methods are the most classical approaches, the former being the most used method in commercial applications such as DigSILENT [4] and ETAP [5], as it converges speedily to the solution of the problem using the information of the derivatives in the Jacobian matrix [6, 7]. However, for the analyses of electrical distribution networks in load flow studies, there are multiple approaches in the literature that take advantage of the grid structure to propose alternative formulations that can work directly in the complex domain, such as Gauss-Seidel derived methods. These methods are classical backward/forward and its matricial equivalent [8, 9], the triangular-based load flow method [3], the hyperbolic and product-derived Taylor methods [10, 11], the successive approximations (SA) method [12], and linear approximations based on the logarithmic transformation of the voltage variables [13], and the Laurent series expansion approach [14], among others. Based on the multiple possibilities for efficiently solving the load flow problem in electrical networks, typically the preferred methods avoid the use of derivatives in their formulation due to the sparsity nature of electrical distribution grids [15]. The two most recommended methods are the recently developed SA [12] and the matricial backward/forward (MBF) methods [8, 9], as they can work with mesh and radial distribution grid topologies.  
From a simplistic point of view, these methods look different, as SA is a compact representation of the Gauss-Seidel method using a matricial representation derived from the admittance matrix representation, while the MBF load flow method is derived using the topology of the network based on the incidence matrix [9], however, it is possible to demonstrate that the recursive formulas derived for these methods are completely equivalent, i.e., both methodologies are exactly equal. The aim of this study is to prove that these methods are exactly the same but derived from two different points of view.  
One of the most important advantages of these load flow representations is that the convergence of both approaches can be demonstrated using the Banach fixed-point theorem as was demonstrated in [8; 9] for the the MBF method and in [12] for the SA approach. To validate the proposed comparative study, we employ two classical distribution networks composed of 33 and 69 nodes with radial topologies in medium voltage levels [16] to compare it with other literature approaches. The radial configuration is selected to the tests, as this configuration produces worsens operative conditions when compared with meshed topologies.  
The remaining sections of this brief are organized as follows: Section 2 describes the complete formulation of the SA load flow approach, while Section 3 presents the complete derivation of the MBF load flow approach. Section 4 demonstrates via incidence matrix representation that both methodologies are equal based on the definition of the nodal admittance matrix. Section 5 presents the configuration of the test feeders employed to compare the two methodologies. Section 6 shows the computational validations, including comparisons with other classical load flow methods applicable to distribution networks. Finally, Section 7 presents the main concluding remarks of this study.