1. Introduction
In numerical analysis, we investigate, develop, and analyze numerous methods and algorithms for numerically solving real-life problems in the diverse domains of science such as physics, chemistry, mechanical engineering, chemical engineering, electrical engineering, and other applied sciences. Various kinds of efficient iterative methods are constructed to approximate the roots of system of nonlinear equations of the form
where
with
for multivariate function. One of the powerful and simplest root finding methods to solve the system of nonlinear equations is Newton–Raphson technique, expressed as
where
is the Jacobian matrix evaluated in the iterate
. This involves one function and one Jacobian evaluation at each step. Over the passage of time, the higher-order and computationally efficient variants are developed to solve the large-scale real-world problems.
Many researchers have introduced multipoint iterative schemes as a solution to the limitations of one-point iterative approaches. For example, in 1969, Jarratt [
1] developed a fourth-order two-step optimal method. 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]. 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 Abad et al. [
5], Yaseen et al. [
6], Behl et al. [
2], and Lee and Kim [
4] are extendable for a multidimensional case in a way that is described in the first category. On the other hand, Hueso et al. [
3] proposed the fourth-order method for multivariate case and Sharma 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. Energy passes from the generator to the load in a power system through numerous networks. The flow of active power
P and reactive power
Q is referred to as the load flow. In a steady-state analysis of a power system, power flow analysis is an effective approach, and many iterative strategies for solving power flow equations were developed by researchers; in 1956, the first automated digital solution to the power flow problem was given by Ward and Hale [
22]. The study of power flow analysis gives different techniques for determining various bus components such as active power, reactive power, voltage magnitude
, and phase angle
in a power grid. The resulting equations are known as power flow equations. A power flow study’s goal is to determine the voltages (magnitude and angle) for a specific load, generation, and network state. Line flows and losses can be computed after the voltages for each bus are known. Determining the known and unknown factors in the system is the first step towards solving power flow issues.
Table 1 illustrates the three categories of buses that are created based on these variables: reference/slack/swing bus, generator/
bus, and load/
bus. Each bus specifies active and reactive power in a three-phase system. To solve the power flow equations, each bus consists of two defined and two unidentified variables.
To provide the mismatch between scheduled generation, total system load (including losses), and total generation, the slack bus is necessary. Each generator bus has a predetermined value of real power P excluding a slack bus. As a result of the specification of both voltage magnitude and angles, the slack bus, also known as the swing bus, is sometimes regarded as the reference bus. Because the net real power is specified and the voltage magnitude is regulated, the remaining generator buses are referred to as regulated or PV buses. Practical power systems have many load buses. Due to the specification of both net real and reactive power loads, load buses are also known as PQ buses. For PQ buses, both voltage magnitudes and angles are unknown, whereas for PV buses, only the voltage angle is unknown. For slack bus, voltage magnitude and angles are already known so there are no variables that are unknown. In a system with n buses and g generators, there are unknowns. To solve these unknowns, real and reactive power balance equations are used. To obtain these equations, the transmission network is modeled using the admittance matrix (Y-bus). Under certain conditions, the magnitude of voltage V and phase angle at each bus in a power flow problems are calculated by using different iterative techniques.
Nodal power balancing equations must be solved in order to conduct power flow analysis. Due to the nonlinear nature of these equations, iterative approaches like the Newton–Raphson, Gauss–Seidel, fast-decoupled, modified Newton, DC load flow methods, and sparse matrix techniques are frequently employed to resolve this issue. The following are the advantages and disadvantages of these techniques:
Each technique has its own trade-offs in terms of convergence speed, accuracy, computational complexity, and suitability for different system sizes and nonlinearities. The choice of method depends on the specific requirements and characteristics of the power system being analyzed. Thus, development of new algorithms is required to meet the needs, including those related to speed, storage, dependability, calculation time, convergence characteristics, etc. On the other hand, Newton–Raphson and its higher-order variants are reliable and give robust solutions. Higher-order Jarratt methods offer several merits for solving load flow equations in power system analysis. These methods exhibit faster convergence compared to traditional methods like Gauss–Seidel or Newton–Raphson, especially for power systems with highly nonlinear characteristics, improved stability properties, making them more robust in handling systems with large mismatches between generation and consumption or systems with voltage stability issues, enhanced robustness against initial guess selection and system parameter variations, reducing the likelihood of convergence failures and improving overall solution reliability. Higher-order Jarratt methods offer higher-order accuracy in approximating the solutions to load flow equations, leading to more accurate results compared to lower-order methods. Jarratt methods, particularly higher-order variants, are efficient for solving load flow equations in large-scale power systems, where traditional methods may suffer from slow convergence or computational inefficiencies. These methods typically require fewer iterations to converge compared to lower-order methods, resulting in reduced computational burden and faster solution times, which is crucial for real-time and large-scale power system analysis. With the increasing complexity and nonlinearity of modern power systems, higher-order Jarratt methods offer a viable solution approach that can effectively handle the intricacies of these systems, ensuring accurate and efficient load flow analysis.
The nonlinear power flow equations are influenced by voltages and phase angle . Newton–Raphson type methods are widely used for the power flow analysis which comprises the bus admittance matrix. In order to solve the power system, the derivative of a function is expressed by a matrix, and the Jacobian is computed. Furthermore, we obtain the partial derivatives of power flow equations containing active and reactive powers. The basic procedure to address the power flow problem is described as follows:
Constructing a mathematical model that illustrates the relationship between voltages and powers in an interconnected system.
Specifying the voltage and power conditions for each network bus.
Calculating the voltage magnitude and phase angle at each bus in a power system under some balanced steady-state conditions
Each power system has one slack bus containing two known quantities, voltage magnitude and phase angle. The power station in a power system refers to the generator bus, sometimes known as the bus or generation bus. In the bus, the known quantities are voltage magnitude and active power P. The load bus, also known as the bus, is a type of bus in the network that holds both active and reactive power.
The main purpose of this paper is to develop and examine a new Jarratt-type method with higher convergence order in order to solve the systems of nonlinear equations, so that we can achieve higher convergence order as well as better computational efficiency for solving the load flow problem. The outline is as follows.
Section 2 consists of the development of an efficient seventh-order method in order to solve the system of nonlinear equations. The structure and various cases of the weight functions are also introduced in this section. The computational cost for various operations is also calculated such as matrix–vector multiplication, matrix–matrix multiplication, and LU decomposition. Then, the total computational cost for our newly developed three-step seventh-order method is calculated and compared with the cost of some existing methods. Furthermore, in
Section 3, the power flow equations in an electrical power systems are described and converted into systems of nonlinear equations by using our newly introduced family of seventh-order method. For this, power flow problems are considered and the results are compared with a seventh-order method proposed by Yaseen et al. [
6]. Conclusions are given in
Section 4.