# A Novel Higher-Order Numerical Scheme for System of Nonlinear Load Flow Equations
###### Abstract
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 [PERSON] and [PERSON] [22]. The study of power flow analysis gives different techniques for determining various bus components such as active power, reactive power, voltage magnitude \(|V|\), and phase angle \(\delta\) 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/\(PV\) bus, and load/\(PQ\) 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 pre-determined 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 \(2(n-1)-(g-1)\) 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:  
1. Gauss-Seidel Method: * Advantages: Simple implementation, low memory requirement, suitable for small to medium-sized systems. * Disadvantages: Slow convergence for large and highly nonlinear systems, may not converge for certain network configurations, sensitive to initial guesses. 2. Newton-Raphson Method: * Advantages: Faster convergence compared to Gauss-Seidel, suitable for large and highly nonlinear systems, allows for simultaneous solution of multiple equations. * Disadvantages: Higher computational complexity, requires initial estimates for all variables, may encounter convergence issues for ill-conditioned systems. Moreover, it starts to lose its ability to converge fast with the increasing system size. 3. Fast Decoupled Method: * Advantages: Improves convergence speed compared to Newton-Raphson, less computational burden, suitable for medium to large systems with moderate nonlinearity.