**1. Introduction**

Power Flow (PF) is the backbone of power system analysis. From a mathematical point of view, PF is a nonlinear problem in which the operational steady state of a power system is obtained. Traditional methods for tackling this problem are the iterative NR [1] and decoupled techniques [2–4].

Although PF is customarily solved in polar coordinates form, other formulations have been studied. A PF formulation based on current injections instead of power injections has been proposed by da Costa et al. [5] and posteriorly embellished by Garcia et al. in [6]. Saleh has developed a formulation of the PF problem in the well-known d-q framework in [7,8]. More recently, PF formulation in complex variables has been exploited in [9], using Witinger Calculus.

Ill-conditioned systems bring some issues for traditional PF solution techniques. This topic has been profusely studied for decades. For example, the reader can be referred to the works of Iwamoto and Tamura [10], Tripathy et al. [11] or Braz et al. [12]. More recently, these kinds of problems have been tackled using the Continuous Newton's paradigm by Milano in [13] or by some of the authors in several recent papers [14–16]. The works of Pourbagher and Derakhshandeh have been focused on the solution of ill-conditioned power systems using the Levenberg–Marquardt technique [17,18]. Alternatively, a novel paradigm has been proposed by the authors in [19], which studies the application of the Gauss–Newton method for PF analysis.

High-order Newton-like methods have also been studied for PF analysis. In [20], Pourbagher and Derakhshandeh studied the application of Newton-like techniques of

**Citation:** Tostado-Véliz, M.; Kamel, S.; Taha, I.B.M.; Jurado, F. Exploiting the S-Iteration Process for Solving Power Flow Problems: Novel Algorithms and Comprehensive Analysis. *Electronics* **2021**, *10*, 3011. https://doi.org/10.3390/ electronics10233011

Academic Editors: Marinko Barukˇci´c, Nebojša Raiˇcevi´c and Vasilija Šarac

Received: 2 November 2021 Accepted: 29 November 2021 Published: 2 December 2021

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3rd, 4th, and 5th order to PF analysis. On the other hand, a Newton-like technique with a superquadratic convergence rate has been proposed to solve the PF problem in wellconditioned systems by some of the authors in [21].

Regarding continuation, Homotopic and Holomorphic techniques have also been exploited for PF analysis. The well-known Continuation Power Flow [22] may be the greatest exponent of this kind of methodology. This approach is traditionally used to determine the stability margin of a power system by calculating its Maximum Loadability Point. Some recent efforts have been made for adapting Continuation Power Flow to distribution systems [23,24]. The Homotopic principle has been applied to PF analysis by Yang and Zhou in [25]. Posteriorly, a family of robust and efficient PF solution techniques based on a combined Newton–Homotopic approach has been developed by some of the authors in [26]. The PF solution by the Holomorphic Embedding method was firstly studied in [27]. Recently, the PF solution by this principle has been further studied in [28,29].

The application of the S-iteration process (SIP) [30] to PF analysis has been recently tackled by the authors in [31]. In this regard, an iterative algorithm based on a combined Newton-SIP approach developed in [32] was adapted for solving either well or ill-conditioned systems. The developed solver turned out to be very efficient, since only an LU decomposition is required in the whole iterative process. This was reflected in very promising results, frequently outperforming NR or the decoupled methods. In addition, it turned out to be quite robust, efficiently handling some large and very large ill-conditioned systems. However, due to the linear convergence characteristic of this method, it suffers from slow convergence in heavy loading cases. In order to overcome this drawback, a Jacobian updated mechanism has also been proposed. Definitely, the PF solution technique proposed in [31] and its variant can be widely used in industry tools due to its capacity for managing well and ill-conditioned equations and its simplicity and efficiency. However, the application of SIP for PF analysis is still far from being fully studied. For example, several topics still need to be further analyzed:


In order to respond to the issues above, the authors strongly believe that further analysis of the SIP applied to PF analysis is still required. This paper aims to fill this gap by profusely studying the Newton-like methods developed in [32,33]. Two schemes are considered. Firstly, we take the constant Jacobian matrix, which corresponds with the standard form of the techniques developed in [32,33]. This mechanism brings linear algorithms; hence, the Newton-SIP methods are also studied for a fully updated scheme in which the Jacobian matrix involved is updated each iteration. The developed methods are compared in terms of efficiency and convergence rate. Finally, we study several numerical experiments in order to analyze the performance of the different Newton-SIP methods in well and ill-conditioned systems, comparing their results with those obtained by NR and analyzing the influence of the *s*-parameters in the overall performance of the Newton-SIP approaches.

The remainder of the paper is organized as follows. Firstly, the Newton-SIP methods developed in [32,33] are presented and adapted to the PF problem in Section 2. A convergence study of the considered PF solution techniques is provided in Section 3. Section 4 compares the studied methodologies in terms of efficiency. Section 5 describes the different numerical experiments considered, and the results obtained are interpreted and discussed. Finally, Section 6 concludes the paper.

#### **2. Newton-SIP Methods Applied to PF Analysis**
