# A Novel Higher-Order Numerical Scheme for System of Nonlinear Load Flow Equations
###### Abstract  
Power flow problems can be solved in a variety of ways by using the Newton-Raphson approach. The nonlinear power flow equations depend upon voltages \(|V_{i}|\) and phase angle \(\delta\). An electrical power system is obtained by taking the partial derivatives of load flow equations which contain active and reactive powers. In this paper, we present an efficient seventh-order iterative scheme to obtain the solutions of nonlinear system of equations, with only three steps in its formulation. Then, we illustrate the computational cost for different operations such as matrix-matrix multiplication, matrix-vector multiplication, and LU-decomposition, which is then used to calculate the cost of our proposed method and is compared with the cost of already seventh-order methods. Furthermore, we elucidate the applicability of our newly developed scheme in an electrical power system. The two-bus, three-bus, and four-bus power flow problems are then solved by using load flow equations that describe the applicability of the new schemes. system of nonlinear equations; Jarratt method; higher order of convergence; electrical power systems +
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sixth-order techniques in an effort to obtain faster algorithms, as seen, for instance, in [2; 3; 4]. However, there are only a few Jarratt-type seventh-order iterative schemes for solving nonlinear systems [5; 6] that have less computational cost. Another line of research is based on [PERSON]'s method for solving nonlinear systems, following which some seventh-order derivative-free schemes were designed [7; 8; 9; 10; 11]. It is evident that while efforts are being made by the researchers to enhance the order of convergence of an iterative approach, most of the time, this results in an increase in the computational cost per iteration, for example, the seventh- and eighth-order methods developed recently [12; 13; 14; 15; 16]. Therefore, even when we create new iterative techniques, we ought to make an effort to minimize the computing expense. Highly convergent multipoint iterative schemes can be categorized in two different ways: first, by developing a new scheme for scalar equations and then extending the same scheme for multidimensional cases with the same convergence order as that for the scalar equations; and second, developing the iterative schemes for system of equations with the help of different approaches like divided difference approach, quadratic formula, etc. Optimal Jarratt fourth-order two-step method and schemes by [PERSON] et al. [5], [PERSON] et al. [6], [PERSON] et al. [2], and [PERSON] and [PERSON] [4] are extendable for a multidimensional case in a way that is described in the first category. On the other hand, [PERSON] et al. [3] proposed the fourth-order method for multivariate case and [PERSON] et al. [17] developed some fourth- and sixth-order schemes by using the weight function approach. It seems like the first technique is an easy way to develop some multidimensional iterative methods, but this is not the case. The key objective is to retain the order of convergence of the described scheme while extending it to the multidimensional case. Nonlinear systems of equations are commonly used to describe scientific and engineering challenges. There are more and more applications for these systems, and the majority of the techniques, now in use, have limitations and drawbacks. Thus, it is crucial to create novel numerical techniques that are computationally efficient, fast and reliable. Among the problems of electrical engineering, load flow studies are important in planning and designing the future expansion of power systems [18; 19; 20; 21]. Planning the expansion of power systems and figuring out how to operate the current systems most effectively require a load-flow study. Additionally, it serves as the foundation for a number of studies that demand quick processing times, such as those on online applications, optimal power flow, and continuation power flow.