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1. Introduction:
The main function of electric power systems is to provide electrical energy to its loads efficiently and economically.
The steady-state performance of an interconnected electric system during normal operation can be analyzed based on nonlinear nodal analysis to form power flow equations and must be solved by efficient iterative methods [1-9]. Power flow analysis is commonly used as part of the operation of the power system and planning. As AC power flow solution methods were first developed more than half a century ago, there are two widely used numerical methods (Gauss-Seidel: GS and Newton-Raphson: NR) to solve this problem and are therefore referred to as GS and NR power flow solution methods, respectively. As widely known, the NR method has been successfully developed and recognized as the most powerful algorithm for power flow analysis in electrical networks. In large-scale energy systems with hundreds or up to a thousand buses, the standard NR method gives slow execution time due to a large Jacobian matrix that must be recalculated and factored at each iteration [10,11]. Hence, the decoupled and fast coupling versions of power flow [12-13] have been released. As a result, the power flow solution can be obtained more quickly. This method is very useful in practical power system analyses, for example, contingency analysis, online power flow control, etc. [4-14].
With a long development history, power flow algorithms are numerous and have a variety of applications. Improvement of the computational efficiency of power flow algorithms has been achieved through various different approaches in subsystems. Therefore, the overall system power flow solution can be obtained by directly coupling the solutions of distinct subsystems [15] based on the GS method. This concept is very useful for parallelizing power flow algorithms to implement both parallel and sequential power flow runs on a cluster of computers if the GS method or other numerical methods such as successive over-relaxation (SOR) are used as the main solver [16-17]. From some perspectives, an initial guess solution for power flow calculation is one of the major factors causing slow computation. In [18-19], an initial line solution based on the decoupling principle of active and reactive power decomposition was used as a starting point for power flow calculations. Additionally, there are modified versions of the NR power flow method to handle poorly conditioned power systems [20-21]. The computational algorithm has been continuously developed by several researchers around the world. A complex form of power flow calculation has been introduced for a three-phase unbalanced power flow solution [22-23]. A power flow solution based on a local search method has been claimed [24] to be robust and applicable to cases where the classic power flow method has failed. Due to advancements in FACTS technology, the power flow equations have been modified and rewritten in current injection forms for incorporating FACTS devices and all sorts of control strategies [25-26]. Moreover, studies on power flow solution methods for specific applications, such as economic dispatch [27], optimal power flow [28], FACTS devices [29-30], and AC/DC Power Systems [31], have been reported.
For several decades, electric power systems have been characterized using nodal analysis to solve a set of voltage solution equations. In general, electricity demands are defined in constant power, leading to the non-linearity of the nodal voltage equation. To date, the standard NR method of power flow is one of the most powerful algorithms, which has a long development history and is widely used to develop commercial power flow solution software. Although the standard NR method of power flow is very efficient and commonly used for power flow calculations in several power system texts [1-9], formulating the Jacobian updated iterative matrix equation requires complex expressions and lengthy terms. In this article, the iterative NR method is still employed as the main solution framework. The essential difference is that the proposed algorithm seeks the roots of the current inadequacy equations instead of those of the power mismatch equations. This approach allows for simplifying a very long and complicated mathematical formula to a very simple and short mathematical expression. With this simplification, the overall execution time is expected to be improved. To achieve this goal, expressions for obtaining the Jacobian updated matrix elements must be derived.
This document consists of six sections. Section 2 presents the formulation of the proposed NR power flow problem. Derivation of the updated Jacobian matrix elements is included in this section. Section 3 presents floating-point operation counting to evaluate its computational effort. A simple 3-bus power system [1] has been chosen as a numerical example in Section 4. Section 5 presents simulation results based on the 5-bus [2], 6-bus [4], 24-bus [32], 30-bus [1], and 57-bus [33] standard IEEE test systems for benchmarking. The final section, Section 6, provides the conclusion of the work.

2. Simplified Newton-Raphson Power Flow:

The power flow problem to determine the solutions of the nonlinear power system voltage equations [1]. If other nonlinear equations of current inadequacy are selected and used according to the estimated roots. Given an n-bus power system, bus 1 is assigned a constant voltage magnitude and a zero phase angle. Considering the KTH bus, the balance equations currently characterizing this bus can be expressed as follows.
(igen,k-idem,k)- ...........................................(1)

Where I(g, k) denotes the generator current at bus k (dem, k) denotes the load current at bus k, k is the phasor vk denotes voltage, Yki denotes the k-th row and i-th column element of the system bus admittance matrix. In practice, loads in electric networks are in the form of power; therefore, it is convenient to rewrite (1) in terms of powers as follows.
fk=( ) .............................................(2)
Define Fk = Gk + jHk to be the current inadequacy at bus k.
Replacing the above expression in (2), we have

fk= ... ... ... ... .... ... .. (3)
Gh= ............... (4)
Hk= ... ... ... ... ... .. (5)
(4) and (5) are the real and imaginary parts of the current inadequacy at bus k. The inadequacy will be zero when all unknown bus voltages are successfully resolved to find a set of voltage solution using the NR method [34,35]. These two equations must be expanded using Taylor series as follows.
Gk= .... ... ... ... ... ... ... ... .. (6)

Hk= ... ... ... ... ... ... ... ... .. (7)
Where S denotes the bus declining.
With n-1 unknown complex variables and n-1 complex current inadequacy equations, a compact matrix form used to update the voltage solution can be expressed as follows.

... ... ...... ... ... .. (8)
Therefore, the elements of the Jacobian matrices can be derived similarly to those of the standard NR method and are summarized in (9) - (16).
Sub-matrix J1
... ... ... ... ...if k?i ... ... .........(9).
. .. ... ..... .. (10)
Sub-matrix J2

... ... ... ... ...if k?i ... ... .........(11).
. .. ... ..... .. (12)
Sub-matrix J3
... ... ... ... ...if k?i ... ... .........(13).
. .. ... ..... .. (14)
Sub-matrix J4
... ... ... ... ...if k?i ... ... .........(15).
. .. ... ..... .. (16)


.....................................(17)
To compare the efficacy of the proposed NR method against the standard NR method, the expressions for the Jacobian matrix elements J1, J2, J3 and J4, the real and imaginary current elements of matrix G and H, and the active and reactive power calculated elements of matrix Cal and Cal must be evaluated as described in the following section.

3. Evaluation of Floating-Point Operation Counting:

The execution time of power flow calculation depends on the number of floating-point operations (flops) [34-35]. Assuming that the other steps of the NR iterative methods are exactly the same, the Jacobian update step dominates the overall execution time. Generally, the time spent performing multiplication and division is significantly the same. But is higher than addition and subtraction. Consequently, the counting of additional flops is negligible. In this document, flops, it is always the flops of multiplication for brevity and is used to evaluate the computational effort of the proposed algorithm.
As is often written in power system analysis textbooks [1-9], the Jacobian updated matrix formula of the standard NR method is described in (18).

. .... ... ... ... ... ... ... ... ... ... (18).

Recall (9), it represents the updated formula for the off-diagonal elements of J1 in the proposed NR method, while (19) describes the update formula for the off-diagonal element of J1 in the standard NR method [1] for comparison.

if k?i ... ... ... ... ... (19).

Each off-diagonal element of J1 for either the standard or proposed NR methods can be expressed in a single term. (9) requires n trailing multiplications, while three multiplications are needed to execute (19). There are (n-1) x (n-2) diagonal elements in the J1 sub-matrix. Thus, the standard NR method requires 3 x (n-1) x (n-2) FLOPS per iteration or 3n² + o(n). O(n) means "small terms of the order n" [34]. Remarkably, the proposed NR method requires 2 * (n-2) flops.
For the diagonal elements of J1, (10) is the expression used for the proposed NR method, while (20) is used for the standard NR method [1].





Table 1:
STANDARD NR SUB-MATRIX PROPOSED NR
J1 diagonal
Off-diagonal
total 3*(n-1)
3n²+o(n)
3n²+o(n) 4*(n-2)
2*(n-2)
6*(n-2)
J1 diagonal
Off-diagonal
total 2*(n-1)+3
2n²+o(n)

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