# A Common Framework for Developing Robust Power-Flow Methods with High Convergence Rate
## 2 The Developed Solution Framework for PF Analysis  
### Analogy between NRJ and the Explicit Midpoint Method  
Let us consider the PF problem in polar coordinates as a set of nonlinear equations as follows:  
\[\mathbf{g}(\mathbf{x})\ =\ 0 \tag{1}\]  
where \(\mathbf{g}:\mathbb{R}^{n}\longmapsto\mathbb{R}^{n}\) are the PF equations and \(\mathbf{x}\in\mathbb{R}^{n}\) is the PF state vector. Due to strong nonlinearities in (1), its solution must be achieved by using iterative techniques. The most popular one in PF analysis is NR, which is performed as follows:  
\[\mathbf{x}^{(k+1)}\ =\ \mathbf{x}^{(k)}-\left[\mathbf{g}^{{}^{\prime}}\Big{(} \mathbf{x}^{(k)}\Big{)}\right]^{-1}\mathbf{g}\Big{(}\mathbf{x}^{(k)}\Big{)} \tag{2}\]  
where \(\left[*\right]^{-1}:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}\) is the inverse operator, the superscript denotes the \(k^{\text{th}}\) iteration of the iterative procedure, and \(\mathbf{g}^{{}^{\prime}}\in\mathbb{R}^{n\times n}\) is the Jacobian matrix of the PF equations, which is formed by the first partial derivatives of \(\mathbf{g}\) w.r.t. \(\mathbf{x}\). In contrast to (2), NRJ poses a two-step iterative scheme for solving the set in Equation (1) described by:  
\[\left\{\begin{array}{l}\mathbf{y}_{2}^{(k)}\ =\ \mathbf{x}^{(k)}-\frac{1}{2} \Big{[}\mathbf{g}^{{}^{\prime}}\Big{(}\mathbf{x}^{(k)}\Big{)}\Big{]}^{-1} \mathbf{g}\Big{(}\mathbf{x}^{(k)}\Big{)}\\ \mathbf{x}^{(k+1)}\ =\ \mathbf{x}^{(k)}-\Big{[}\mathbf{g}^{{}^{\prime}}\Big{(} \mathbf{y}_{2}^{(k)}\Big{)}\Big{]}^{-1}\mathbf{g}\Big{(}\mathbf{x}^{(k)} \Big{)}\end{array}\right.. \tag{3}\]  
In [10], the Explicit Midpoint rule [19] was exploited for solving the PF equations on the basis of the Continuous Newton's principle. This technique poses a two-stage iterative solver, which is given by the mapping in (4).  
\[\left\{\begin{array}{l}\mathbf{y}_{2}^{(k)}\ =\ \mathbf{x}^{(k)}+\Big{(} \frac{1}{2}\Delta t\Big{)}\mathbf{f}\Big{(}\mathbf{x}^{(k)}\Big{)}\\ \mathbf{x}^{(k+1)}\ =\ \mathbf{x}^{(k)}+\Delta t\mathbf{f}\Big{(}\mathbf{y}_{2}^{(k)} \Big{)}\end{array}\right. \tag{4}\]  
where \(\Delta t\in\mathbb{R}^{+}\) is the step size (or truncation) and \(\mathbf{f}:\mathbb{R}^{n}\longmapsto\mathbb{R}^{n}\) is given by:  
\[\mathbf{f}(\mathbf{x})\ =\ -\Big{[}\mathbf{g}^{{}^{\prime}}(\mathbf{x})\Big{]}^{-1} \mathbf{g}(\mathbf{x}). \tag{5}\]  
Further insight on mappings (3) and (4) reveals a clear analogy between these two solvers. In fact, the iterative map (3) could be rewritten as follows:  
\[\left\{\begin{array}{l}\mathbf{y}_{2}^{(k)}\ =\ \mathbf{x}^{(k)}+\frac{1}{2} \mathbf{h}\Big{(}\mathbf{x}^{(k)},\mathbf{x}^{(k)}\Big{)}\\ \mathbf{x}^{(k+1)}\ =\ \mathbf{x}^{(k)}+\mathbf{h}\Big{(}\mathbf{y}_{2}^{(k)}, \mathbf{x}^{(k)}\Big{)}\end{array}\right. \tag{6}\]  
where the step size has vanished and the auxiliary function \(\mathbf{h}:\mathbb{R}^{n}\longmapsto\mathbb{R}^{n}\) is defined by:  
\[\mathbf{h}(\mathbf{y},\mathbf{x})\ =\ -\Big{[}\mathbf{g}^{{}^{\prime}}(\mathbf{y}) \Big{]}^{-1}\mathbf{g}(\mathbf{x}). \tag{7}\]The analogy described above reveals that NRJ and the Explicit Midpoint rule are actually equivalent mappings. Keeping this in mind, NRJ may be seen as the application of the Explicit Midpoint method for solving the function \(\mathbf{h}\), whereas the different methodologies proposed in [1; 10] are devoted to solving \(\mathbf{f}\). In the same way, NR could be assimilated as the direct application of the Explicit Euler methodology for solving (7).