A Seventh Order Family of Jarratt Type Iterative Method for Electrical Power Systems

Abstract

A load flow study referred to as a power flow study is a numerical analysis of the electricity that flows through any electrical power system. For instance, if a transmission line needs to be taken out of service for maintenance, load flow studies allow us to assess whether the remaining line can carry the load without exceeding its rated capacity. So, we need to understand about the voltage level and voltage phase angle on each bus under steady-state conditions to keep the bus voltage within a specific range. In this paper, our goal is to present a higher order efficient iterative method to carry out a power flow study to determine the voltages (magnitude and angle) for a specific load, generation and network conditions. We introduce a new seventh-order three-step iterative scheme for obtaining approximate solution of nonlinear systems of equations. We attain the seventh-order convergence by using four function evaluations which makes it worthy of interest. Moreover, we show its applicability to the electrical power system for calculating voltages and phase angles. By calculating the bus angle and voltage level, we conclude that the performance of the power system is assessed in a more efficient manner using the new scheme. In addition, dynamical planes of the methods applied on nonlinear systems of equations show global convergence.

Graphical Insights into Dynamical Behavior: Advanced Iterative Methods for Root Finding in Computational Mathematics

“…Finally, we discuss the effectiveness of N4 by comparing it in Table 4 with existing family of seventh-order method introduced by: Yaseen et al [27], namely S1 Basins of attraction of all these methods are shown in figures (16)(17)(18)(19)(20). N4 has basins of attraction compatible with S1, S2 and S3 and better in some cases.…”

“…The first approach is based on Halley's method and Taylor's expansion, while the second method employs second derivative approximations to enhance the efficiency of the first method. Many researchers [24][25][26][27][28][29][30] introduced different new two-step, three-step, or four-step iterative methods of higher orders by utilizing different methods.…”

Stability Analysis of a New Fourth-Order Optimal Iterative Scheme for Nonlinear Equations

“…Forcing the coefficients of e k , e 2 k , and e 3 k to be zero, we obtain H(1) = 1, along with H ′ (1) = 3 4 , H ′′ (1) = 3 4 , and β = 2 3 . By replacing them in (5), we have…”

“…It is widely known that fixed-point iterative methods play a fundamental role in scientific disciplines such as Celestial Mechanics (see for example [1,2]), Electrical Power Systems [3], Chemistry [4], Hydraulic Engineering [5], and Civil Engineering [6]. These algorithms provide approximate solutions when exact solutions are challenging to obtain or when problems are ill-conditioned, while offering computational efficiency in terms of both time and computational resources.…”

A Novel Higher-Order Numerical Scheme for System of Nonlinear Load Flow Equations

“…Some researchers have also developed fifth-and 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 Steffensen's method for solving nonlinear systems, following which some seventh-order derivative-free schemes were designed [7][8][9][10][11].…”