# A Common Framework for Developing Robust Power-Flow Methods with High Convergence Rate
## Appendix A Proof of Cubic Convergence of the Developed Solvers
Considering the following inverse definition:_  
\[\Big{[}\mathbf{g}^{{}^{\prime}}\Big{(}\mathbf{x}^{(k)}\Big{)}\Big{]}^{-1}\mathbf{g}^{{}^{ \prime}}\Big{(}\mathbf{x}^{(k)}\Big{)}\ =\ \mathbf{g}^{{}^{\prime}}\Big{(}\mathbf{x}^{(k)}\Big{)}\Big{[}\mathbf{g}^{{}^{\prime}} \Big{(}\mathbf{x}^{(k)}\Big{)}\Big{]}^{-1}\ =\ \mathbf{I} \tag{27}\]  
_and solving the resulting linear system, one obtains:_  
\[\Big{[}\mathbf{g}^{{}^{\prime}}\Big{(}\mathbf{x}^{(k)}\Big{)}\Big{]}^{-1}\ =\ \Big{[}\mathbf{I}-2\mathbf{C}_{2}\mathbf{e}^{(k)}+ \Big{(}4\mathbf{C}_{2}^{2}-3\mathbf{C}_{3}\Big{)}\mathbf{e}^{(k)^{2}}\Big{]}\Big{[}\mathbf{g}^{ {}^{\prime}}(\mathbf{r})\Big{]}^{-1}+\mathbf{O}\Big{(}\mathbf{e}^{(k)^{4}}\Big{)}\text{.} \tag{28}\]  
_Now, defining \(\mathbf{e}^{(k)}_{\mathbf{y}_{2}}\ =\ \mathbf{y}_{2}^{(k)}-\mathbf{r}\), we can write:_  
\[\mathbf{e}^{(k)}_{\mathbf{y}_{2}}\ =\ \mathbf{x}^{(k)}-\Big{[}\mathbf{g}^{{}^{\prime}}\Big{(}\mathbf{x}^ {(k)}\Big{)}\Big{]}^{-1}\mathbf{g}\Big{(}\mathbf{x}^{(k)}\Big{)}-\mathbf{r}\ =\ \mathbf{e}^{(k)}-\Big{[}\mathbf{g}^{{}^{\prime}}\Big{(}\mathbf{x}^{(k)}\Big{)}\Big{]}^{-1} \mathbf{g}\Big{(}\mathbf{x}^{(k)}\Big{)}\ =\ \Big{(}2\mathbf{C}_{3}-2\mathbf{C}_{2}^{2}\Big{)}\mathbf{e}^{(k)^{3}}+\mathbf{C}_{2}\mathbf{e}^{(k) ^{2}}+\mathbf{O}\Big{(}\mathbf{e}^{(k)^{4}}\Big{)}\text{.} \tag{29}\]  
_Therefore, as in (24) and (28), the Taylor expansions of \(\mathbf{g}\Big{(}\mathbf{y}_{2}^{(k)}\Big{)}\) and \(\Big{[}\mathbf{g}^{{}^{\prime}}\Big{(}\mathbf{y}_{2}^{(k)}\Big{)}\Big{]}^{-1}\) about \(\mathbf{r}\) are given by:_  
\[\mathbf{g}\Big{(}\mathbf{y}_{2}^{(k)}\Big{)}\ =\ \mathbf{g}^{{}^{\prime}}(\mathbf{r})\Big{[} \mathbf{e}^{(k)}_{\mathbf{y}_{2}}+\mathbf{C}_{2}\mathbf{e}^{(k)^{2}}_{\mathbf{y}_{2}}+\mathbf{C}_{3}\mathbf{e }^{(k)^{3}}_{\mathbf{y}_{2}}\Big{]}+\mathbf{O}\Big{(}\mathbf{e}^{(k)^{4}}\Big{)}\ =\ \mathbf{g}^{{}^{\prime}}(\mathbf{r})\Big{[}\mathbf{C}_{2}\mathbf{e}^{(k)^{2}}+\mathbf{C}_{2}^{3}\mathbf{e }^{(k)^{4}}+\mathbf{C}_{2}^{2}\mathbf{C}_{3}\mathbf{e}^{(k)^{6}}\Big{]}+\mathbf{O}\Big{(}\mathbf{e} ^{(k)^{7}}\Big{)} \tag{30}\]  
\[\Big{[}\mathbf{g}^{{}^{\prime}}\Big{(}\mathbf{y}_{2}^{(k)}\Big{)}\Big{]}^{-1}\ =\ \Big{[}\mathbf{I}-2\mathbf{C}_{2}^{2}\mathbf{e}^{(k)^{2}}+ \Big{(}4\mathbf{C}_{2}^{4}-3\mathbf{C}_{2}^{2}\mathbf{C}_{3}\Big{)}\mathbf{e}^{(k)^{4}}\Big{]} \Big{[}\mathbf{g}^{{}^{\prime}}(\mathbf{r})\Big{]}^{-1}+\mathbf{O}\Big{(}\mathbf{e}^{(k)^{5}} \Big{)}\text{.} \tag{31}\]  
_Finally, we calculate the error function at \(k+1\) as follows:_  
\[\mathbf{e}^{(k+1)}-\mathbf{x}^{(k)}-\mathbf{x}^{(k)}-\frac{1}{2}\Big{[}\Big{[}\mathbf{g}^{{}^{ \prime}}\big{(}\mathbf{x}^{(k)}\Big{)}\Big{]}^{-1}\mathbf{g}^{(k)}\Big{]}-\mathbf{x}^{(k)} \mathbf{e}^{(k)}\mathbf{e}^{(k)}\mathbf{e}^{(k)}\mathbf{e}^{(k)}\mathbf{e}^{(k)}\mathbf{e}^{(k)}\mathbf{e}^{ (k)}\mathbf{e}^{(k)}\mathbf{e}^{(k)}\mathbf{e}^{(k)}\mathbf{e}^{(k)}\mathbf{e}^{(k)}\  
**Theorem A2**. : _Let \(\mathbf{g}\) be sufficiently differentiable at each point of an open neighborhood \(D\) of \(\mathbf{r}\in\mathbb{R}^{n}\); this is a solution of the system \(\mathbf{g}(\mathbf{x})~{}=~{}0\). Let us suppose that \(\mathbf{g}(\mathbf{x})\) is continuous and nonsingular in \(\mathbf{x}\). Then, the developed method (24) converges to \(\mathbf{r}\) with order three._  
**Proof of Theorem A2. **_Taking the Taylor expansions of \(\mathbf{g}\Big{(}\mathbf{x}^{(k)}\Big{)}\) and \(\Big{[}\mathbf{g}^{\prime}\Big{(}\mathbf{x}^{(k)}\Big{)}\Big{]}^{-1}\) about \(\mathbf{r}\) from (A1) and (A2), respectively, let us define \(\mathbf{e  
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