# A Common Framework for Developing Robust Power-Flow Methods with High Convergence Rate
## 2 The Developed Solution Framework for PF Analysis
### Stability of the Developed Solution Paradigms  
In order to study the stability of the mappings (8) and (13), let us introduce the following definition:  
**Definition 1:**_Hyperbolic points: a fixed point \(\mathbf{x}^{*}\) of a map \(\mathbf{G}\ :\mathbf{C}\subset\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) is hyperbolic, if the Jacobian of the equation \(\hat{\mathbf{G}}\ =\ \mathbf{G}(\mathbf{x})-\mathbf{x}\) at \(\mathbf{x}^{*}\) has no eigenvalues with a zero real part. In addition, \(\mathbf{x}^{*}\) is asymptotically stable if all the eigenvalues of the Jacobian of \(\hat{\mathbf{G}}\) at \(\mathbf{x}^{*}\) have a modulus \(\leq 1\). A hyperbolic point can be:_  
* Sink: if all the eigenvalues of the Jacobian of \(\hat{\mathbf{G}}\) have a negative real part.
* Source: if at least one of the eigenvalues of the Jacobian of \(\hat{\mathbf{G}}\) has a positive real part.  
_Contrarily, a fixed point is said to be unstable if it is not stable._  
By Definition 1, one requires a PF solution (i.e., \(\mathbf{x}^{*}\) such that \(\mathbf{g}(\mathbf{x}^{*})\ =\ 0\)) to be asymptotically stable. In addition, we desire from \(\mathbf{x}^{*}\) to be a sink. By using definition 1, we can study the numerical stability of the developed solution frameworks, which is performed in the following lemma.  
**Lemma 1:**_a solution of the PF equations (i.e., \(\mathbf{x}^{*}\) such that \(g(\mathbf{x}^{*})\ =\ 0\)) is a sink for the nonlinear solution paradigms (8) and (13), if \(\sum_{i\ =\ 1}^{m}b_{i}\geq 0\) and \(\sum_{i\ =\ 1}^{m}\big{(}b_{i}-b_{i}^{*}\big{)}\geq 0\), respectively._  
**Proof of Lemma 1.**_By differentiating (7) with respect \(\mathbf{x}\) one obtains:_  
\[\
abla_{\mathbf{x}}\mathbf{h}\ =\ -\
abla_{\mathbf{x}}\Big{[}\mathbf{g}^{{}^{\prime}}( \mathbf{y})\Big{]}^{-1}\mathbf{g}(\mathbf{x})-\Big{[}\mathbf{g}^{{}^{\prime}}(\mathbf{y} )\Big{]}^{-1}\mathbf{g}^{{}^{\prime}}(\mathbf{x}). \tag{16}\]  
Figure 1: Flowchart of the considered strategy for handling with reactive limits in generation units.  
One can easily check that at an equilibrium point (i.e., \(\mathbf{x}^{*}\) such that \(\mathbf{g}(\mathbf{x}^{*})\ =\ 0\)), the developed solution paradigms (8) and (13) take the form:  
\[\left.\mathbf{y}_{i}\right|_{\mathbf{x}^{*}}\ =\ \mathbf{x}^{*},\ \forall i. \tag{17}\]  
Substituting (17) into (16), one obtains:  
\[\left.\
abla_{\mathbf{x}}\mathbf{h}\right|_{\mathbf{x}^{*}}\ =\ -\
abla_{\mathbf{x}}\Big{[} \mathbf{g}^{{}^{\prime}}(\mathbf{x}^{*})\Big{]}^{-1}\mathbf{g}(\mathbf{x}^{*})-\Big{[} \mathbf{g}^{{}^{\prime}}(\mathbf{x}^{*})\Big{]}^{-1}\mathbf{g}^{{}^{\prime}}(\mathbf{x }^{*})\ =\ -\mathbf{I} \tag{18}\]  
where \(\mathbf{I}\in\mathbb{R}^{n\times n}\) is the identity matrix. Then, one can replace (18) into the differential equation associated with the mapping (8) obtaining:  
\[\left.\
abla_{\mathbf{x}}\Big{(}\mathbf{x}^{(k+1)}-\mathbf{x}^{(k)}\Big{)}\right|_{\mathbf{x} ^{*}}\ =\ -\sum_{i\ =\ 1}^{m}\{b_{i}\mathbf{I}\}. \tag{19}\]  
While for the paradigm (13), one has:  
\[\left.\
abla_{\mathbf{x}}\Big{(}\mathbf{x}^{(k+1)}-\mathbf{\hat{x}}^{(k)}\Big{)}\right|_{ \mathbf{x}^{*}}\ =\ -\ \sum_{i\ =\ 1}^{m}\{(b_{i}-b_{i}^{*})\mathbf{I}\}. \tag{20}\]  
Equations (19) and (20) show that at the equilibrium point, all the eigenvalues of the Jacobian of the differential equations associated with the developed solution paradigms do not have imaginary parts. In addition, their real parts are equal to \(-\sum_{i\ =\ 1}^{m}\{b_{i}\}\) in the case of (8) and \(-\sum_{i\ =\ 1}^{m}\{b_{i}-b_{i}^{*}\}\) for the mapping (13). Therefore, the equilibrium point \(\mathbf{x}^{*}\) is asymptotically stable for the paradigm (8) if:  
\[\sum_{i\ =\ 1}^{m}b_{i}\geq 0. \tag{21}\]  
While for the mapping (13), the following inequality must hold:  
\[\sum_{i\ =\ 1}^{m}\left(b_{i}-b_{i}^{*}\right)\geq 0 \tag{22}\]  
\(\square\)