# A Common Framework for Developing Robust Power-Flow Methods with High Convergence Rate
## 2 The Developed Solution Framework for PF Analysis
### Embedded Formulation  
The solution paradigm established in the previous section allows considering any explicit numerical method for developing robust PF solution techniques. However, not only explicit numerical integration techniques are available in the literature. The Embedded Runge-Kutta techniques arise as an alternative of Explicit methods. Such methodologies allow accurately estimating the local truncation error of a single step and properly adapting the step size to obtain high accuracy [19]. In this case, due to the step size having vanished in the developing formulation, its ability to estimate the local truncation error lacks importance; nonetheless, it is still relevant to rewrite (8) into its Embedded counterpart as follows:  
\[\left\{\begin{array}{c}\boldsymbol{y}_{2}^{(k)}\ =\ \boldsymbol{x}^{(k)}+a_{2| \mathbf{h}}\Big{(}\boldsymbol{x}^{(k)},\boldsymbol{x}^{(k)}\Big{)}\\ \boldsymbol{y}_{3}^{(k)}\ =\ \boldsymbol{x}^{(k)}+a_{3|\mathbf{1}}\mathbf{h} \Big{(}\boldsymbol{x}^{(k)},\boldsymbol{x}^{(k)}\Big{)}+a_{3|2}\mathbf{h} \Big{(}\boldsymbol{y}_{2}^{(k)},\boldsymbol{x}^{(k)}\Big{)}\\ \vdots\\ \boldsymbol{y}_{p}^{(k)}\ =\ \boldsymbol{x}^{(k)}+a_{p|1}\mathbf{h}\Big{(} \boldsymbol{x}^{(k)},\boldsymbol{x}^{(k)}\Big{)}+\sum\limits_{i=2}^{p}\Big{\{} a_{p|i}\mathbf{h}\Big{(}\boldsymbol{y}_{i}^{(k)},\boldsymbol{x}^{(k)}\Big{)} \Big{\}}\\ \boldsymbol{\hat{x}}^{(k)}\ =\ \boldsymbol{x}^{(k)}+b_{1}^{*}\mathbf{h} \Big{(}\boldsymbol{x}^{(k)},\boldsymbol{x}^{(k)}\Big{)}+\sum\limits_{i=2}^{m }\Big{\{}b_{i}\mathbf{h}\Big{(}\boldsymbol{y}_{i}^{(k)},\boldsymbol{x}^{(k)} \Big{)}\Big{\}}\\ \boldsymbol{x}^{(k+1)}\ =\ \boldsymbol{\hat{x}}^{(k)}+\big{(}b_{1}-b_{1}^{*} \big{)}\mathbf{h}\Big{(}\boldsymbol{x}^{(k)},\boldsymbol{x}^{(k)}\Big{)}+ \sum\limits_{i=2}^{m}\Big{\{}\big{(}b_{i}-b_{i}^{*}\big{)}\mathbf{h}\Big{(} \boldsymbol{y}_{i}^{(k)},\boldsymbol{x}^{(k)}\Big{)}\Big{\}}\end{array}\right. \tag{13}\]  
where \(b^{*}\) values \(\in\mathbb{R}\). Unlike paradigm (8), each weight in (13) is described by a pair of parameters (\(b\) and \(b^{*}\)). In this case, two numerical methods are nested, each contributing their own characteristic weights. In numerical analysis, this arrangement is very useful for achieving higher accuracy, while in our case, these parameters have been introduced just for copying the standard structure of the Embedded Runge-Kutta formulas [19] (p. 204). Thus, instead of directly calculating the value of the state vector for the \(k+1\) iteration through a single step arrangement, formulation (13) firstly calculates an approximation of the state vector \(\boldsymbol{\hat{x}}\) that is posteriorly refined by means of other numerical methods defined by the \(b^{*}\) values (see [20] for further details). This way, any PF solver based on the developed paradigm (13) could be defined by two matrixes as follows:  
\[\boldsymbol{A}\ =\ \left[\begin{array}{cccc}0&&&&\\ a_{2|1}&a_{2|2}&&\\ \vdots&a_{p|1}&\ldots&\\ a_{p|1}&\ddots&a_{p|p-1}&a_{p|p}\end{array}\right]\in\mathbb{R}^{p\times p} \tag{14}\]  
\[\boldsymbol{B}\ =\ \left[\begin{array}{cc}b_{1}^{*},b_{2}^{*},\ldots,b_{m}^{*}\\ b_{1},\ b_{2},\ldots,b_{m}\end{array}\right]\in\mathbb{R}^{2\times m}. \tag{15}\]  
It is worth noting that unlike the techniques developed in [1; 7; 8; 9; 10; 11; 12], any truncation of the Newton's increment vector based on the influence of the step size is considered in the developed solution frameworks. It means that the developed solution techniques are (as shown in Section 4) intrinsically robust. This point is relevant, since the influence of the step size vanishes, facilitating its codification and application in industry tools.