# A Common Framework for Developing Robust Power-Flow Methods with High Convergence Rate
## 2 The Developed Solution Framework for PF Analysis
### The Novel Family of Robust PF Techniques  
In the previous section, we showed that NR and NRJ can be seen as the application of the Explicit Euler and Midpoint techniques for solving (7). This analogy suggests that any other member of the family of Explicit Runge-Kutta methods can be adapted to be applied to the PF problem. Similar to the generic form of the Runge-Kutta methods (e.g., see [10] (Equation (11)), we could develop a generic form for the application of this family of numerical methods to the PF problem, as follows:  
\[\left\{\begin{array}{c}\boldsymbol{y}_{2}^{(k)}\ =\ \boldsymbol{x}^{(k)}+a_{2 |1}\mathbf{h}\Big{(}\boldsymbol{x}^{(k)}.\boldsymbol{x}^{(k)}\Big{)}\\ \boldsymbol{y}_{3}^{(k)}\ =\ \boldsymbol{x}^{(k)}+a_{3|1}\mathbf{h}\Big{(} \boldsymbol{x}^{(k)},\boldsymbol{x}^{(k)}\Big{)}+a_{3|2}\mathbf{h}\Big{(} \boldsymbol{y}_{2}^{(k)}.\boldsymbol{x}^{(k)}\Big{)}\\ \vdots\\ \boldsymbol{y}_{p}^{(k)}\ =\ \boldsymbol{x}^{(k)}+a_{p|1}\mathbf{h}\Big{(} \boldsymbol{x}^{(k)},\boldsymbol{x}^{(k)}\Big{)}+\sum\limits_{i\ =\ 2}^{p}\Big{\{}a_{p|i}\mathbf{h}\Big{(} \boldsymbol{y}_{i}^{(k)},\boldsymbol{x}^{(k)}\Big{)}\Big{\}}\\ \boldsymbol{x}^{(k+1)}\ =\ \boldsymbol{x}^{(k)}+b_{1}\mathbf{h}\Big{(} \boldsymbol{x}^{(k)},\boldsymbol{x}^{(k)}\Big{)}+\sum\limits_{i\ =\ 2}^{m}\Big{\{}b_{i}\mathbf{h}\Big{(} \boldsymbol{y}_{i}^{(k)},\boldsymbol{x}^{(k)}\Big{)}\Big{\}}\end{array}\right. \tag{8}\]  
where the \(a\) and \(b\) values \(\in\mathbb{R}\). In essence, these parameters are introduced to copy the standard structure of the Runge-Kutta methods, which are broadly defined by the total number of stages involved (\(p\)), the value of the elements of the Runge-Kutta matrix (\(a\) values) and weights (\(b\) values) [19]. Conventionally, the Runge-Kutta methods are presented by their so-called Butcher tableaus [19] (p. 95), which succinctly collects all the necessary information to completely define a Runge-Kutta technique, as follows:  
\[\boldsymbol{A}\ =\ \left[\begin{array}{cccc}0&&&\\ a_{2|1}&a_{2|2}&&\\ \vdots&&\ddots&\\ a_{p|1}&\cdots&a_{p|p-1}&a_{p|p}\end{array}\right]\in\mathbb{R}^{p\times p} \tag{9}\]  
\[\boldsymbol{b}\ =\ [b_{1},b_{2},\ldots,b_{m}]\in\mathbb{R}^{m}. \tag{10}\]  
It is worth mentioning that (9) is lower triangular, since we are dealing with explicit Runge-Kutta formulas (the implicit formulation of (8), although possible, may be computationally expensive, since it would require the factorization of the inverse of (7) [1]). In addition, it is worth noting that nodes [19] are not included in the formulation developed in this paper. This is because the application of the Runge-Kutta formulas to PF analysis is developed from the assimilation of the set of equations (1) as an autonomous set of differential equations. In such formulations, the nodes of a numerical method lack importance [19].  
One can note that (8) can be adapted to any Explicit Runge-Kutta formula. At this point, readers can easily check that both NR and NRJ are actually special cases of (8). Thus, for NR, one has:  
\[\boldsymbol{A}\ =\ \left[\begin{array}{cc}0&0\\ 1&0\end{array}\right]\text{, }\boldsymbol{b}\ =\ [1,\ 0]. \tag{11}\]  
While NRJ is fully described by:  
\[\boldsymbol{A}\ =\ \left[\begin{array}{cc}0&0\\ 1/2&0\end{array}\right]\text{, }\boldsymbol{b}\ =\ [0,\ 1]. \tag{12}\]Interested readers can easily check in the vast literature that (11) and (12) correspond with the Runge-Kutta matrix and weight vectors of the Explicit Euler and Midpoint techniques, respectively.