# A Novel Higher-Order Numerical Scheme for System of Nonlinear Load Flow Equations
## 4 Concluding Remarks  
We introduced a new efficient three-step seventh-order scheme to obtain the solutions of nonlinear system of equations. The computational cost for different operations is expressed and is used to calculate the cost of our proposed method, and is compared with the cost of some existing seventh-order methods. The applicability of our scheme is interpreted in an electrical power system, and two-bus, three-bus, and four-bus power flow problems are solved by using load flow equations. In all the examples, we observe that the newly proposed methods are offering less computational time even when the number of buses increases. Moreover, our class of iterative schemes has only one inverse operator. Thus, by using the LU factorization for the linear systems involved per iteration, the computational cost is reduced, even for large systems in relation to other schemes with more than one inverse operator. Overall, we conclude that the higher-order Jarratt-type methods provide a valuable alternative for solving load flow equations in power system analysis, offering faster convergence, improved accuracy, and enhanced stability, particularly in the context of highly nonlinear power systems. In addition, one of the main drawbacks of these techniques is the J matrix singularity, due to which the methods are sometimes unable to produce the solution. The cause could be anything from a sudden big load addition or subtraction to a change in the line characteristics to an overvoltage or undervoltage at a specific bus. These demerits may be handled by combining the higher-order Jarratt methods and evolutionary/intelligent algorithms.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline _Cases_ & \(\mathbf{k}\) & \(\parallel\mathbf{X^{(k)}}-\mathbf{X^{(k-1)}}\parallel_{\infty}\) & \(\parallel\mathbf{F(X^{(k)})}\parallel_{\infty}\) & \(\mathbf{t^{(k)}}\) \\ \hline \(HM_{1}\) & 1 & 3.9406592707102 & 5.585285030714 & 2.617419 \\ \hline  & 2 & 2.3539462998374 & 1.185340119973 & \\ \hline  & 3 & 3.93041986150 \((-1)\) & 2.70471496585 \((-2)\) & \\ \hline \(HM_{2}\) & 1 & 1.12028592195 \((+1)\) & 1.69904638491 & 2.672200 \\ \hline  & 2 & 6.059912678220 & 4.67290040442 \((+2)\) & \\ \hline  & 3 & 5.615825776275 & 7.552107353633 & \\ \hline \(HM_{3}\) & 1 & 1.153695471270 & 3.35717069246 \((-1)\) & 2.561838 \\ \hline  & 2 & 3.77060518279 \((-1)\) & 3.22883832204 \((-4)\) & \\ \hline  & 3 & 2.63676291247 \((-4)\) & 1.80483165050 \((-20)\) & \\ \hline \(SF_{1}\) & 1 & 1.2238519878510 & 1.0000236962 & 2.713051 \\ \hline  & 2 & 1.238095340347 & 4.5105399975 \((-1)\) & \\ \hline  & 3 & 3.87777272650 \((-1)\) & 1.01937440334 \((-3)\) & \\ \hline  & 1 & 1.80701993172 & 1.78530748656 & 2.898047 \\ \hline  & 2 & 6.23700907210 & 1.527984713366 & \\ \hline  & 3 & 5.81002994904 & 1.277471753569 & \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of seventh-order schemes for power flow problem.  
**Author Contributions:** Conceptualization, F.Z.; Methodology, F.Z.; Software, A.C. and H.M.; Validation, A.C. and J.R.T.; Formal analysis, F.Z.; Writing--original draft, H.M.; Writing--review & editing, J.R.T.; Visualization, A.C.; Supervision, J.R.T. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Data Availability Statement:** No new data were created or analyzed in this study. Data sharing is not applicable to this article.  
**Conflicts of Interest:** The authors declare no conflict of interest.