# A Common Framework for Developing Robust Power-Flow Methods with High Convergence Rate
## 5 Conclusions and Future Works  
Inspired by the similitude observed between the methodology described in [6] and the Explicit Midpoint method, we have developed a novel framework for developing efficient and robust PF solution approaches based on Runge-Kutta formulas. This solution paradigm allows adapting any member of the Runge-Kutta family to the PF problem. Although this idea is similar to that described by [PERSON] in [1], the developed framework presents two important advantages:  
* The influence of the step size vanishes from the formulation, which means that the developed solution paradigm is intrinsically robust.
* High convergence rates can be achieved, even higher than two.  
Explicit and Embedded formulations of the developed solution paradigm have been discussed, and two PF solution techniques based on the Explicit Heun and Embedded Heun-Euler's methods with cubic order of convergence have been developed.  
The developed techniques have been validated using the EU PEGASE systems considering different demanding scenarios. Results have been compared with those obtained with other well-known robust and conventional PF solutions approaches. Our proposals turned out to be more stable and efficient than the other tested techniques, which occasionally showed an unstable behaviour or a high computational cost.  
On the basis of the results obtained, it is concluded that the introduced solution paradigm constitutes a promising framework for developing novel PF solution techniques. This opens the door for further applying the developed solution paradigms. Future works should be focused on developing and validating other PF solution techniques, applying other members of the Runge-Kutta family to the developed solution framework. Similarly, analogies between the introduced solution paradigm and other families of numerical integration techniques such as [PERSON]'s methods [8] or [PERSON]'s scheme [9] can be found and, consequently, other high-order robust PF solution approaches can be developed. We expect the emergence of this kind of techniques.  
The developed solution techniques may be also adapted to other Power System tools such as the Continuation Power-Flow [26]. Here, their robustness along its high convergence order may bring very good results. Future works should study the application of the introduced PF solution techniques to this problem.  
Conceptualization, M.T.-V. and S.K.; methodology, M.T.-V., S.K. and A.E.; software, M.T.-V.; validation, M.T.-V. and S.K.; formal analysis, M.T.-V. and S.K.; investigation, M.T.-V., S.K., A.E. and D.V.; resources, S.K., D.V. and F.J.; data curation, M.T.-V. and A.E.; writing-original draft preparation, M.T.-V. and A.E.; writing--review and editing, S.K., D.V. and F.J.; visualization, M.T.-V., S.K. and A.E.; supervision, S.K., D.V. and F.J.; project administration, F.J.; funding acquisition, D.V. and F.J. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
This work has not been support by any financial support or grant.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **89-Bus** & **1354-Bus** & **2869-Bus** & **9241-Bus** & **13659-Bus** \\ \hline RBS & 13 & 13 & – & – & – \\ BEM & 14 & 15 & 13 & 15 & – \\ NRJ & 8 & 7 & 6 & – & – \\ Equation (23) & 6 & 6 & 5 & 6 & 8 \\ Equation (24) & 6 & 6 & 5 & 6 & 8 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Total number of iterations at the maximum loadability point.