# A Novel Higher-Order Numerical Scheme for System of Nonlinear Load Flow Equations
## 3 Numerical Experiments
### Formation of Load Flow Equations  
In terms of the real power \((P)\), reactive power \((Q)\), and the voltage \((V)\), the general power flow equation is as follows:  
\[S_{i}=P_{i}+jQ_{i}, \tag{25}\]  
where \(P_{i}\) represents real power injection, \(Q_{i}\) represents reactive power injection, and \(S_{i}\) is the complex power injection into the \(i\)th bus from the generating source. Thus,  
\[S_{i}=P_{i}+jQ_{i}=V_{i}I_{i}^{*}, \tag{26}\]  
with  
\[I_{i}^{*}=(\sum\limits_{k=1}^{n}Y_{jk}V_{k})^{*},\text{ and }I^{*}=(\bar{I})^{ t}.\]  
By taking the complex conjugate of Equation (26), we obtain  
\[S_{i}^{*}=P_{i}-jQ_{i}=V_{i}^{*}I_{i}=V_{i}^{*}\sum\limits_{k=1}^{n}Y_{ik}V_{ k},i=1,2,\ldots,k. \tag{27}\]  
In polar coordinate system, the Equation (27) for \(i\)th bus is described as  
\[S_{i}^{*}=P_{i}-jQ_{i}=(V_{i}\angle-\delta_{i})\sum\limits_{k=1}^{n}(Y_{ik} \angle\theta_{ik})(V_{k}\angle\delta_{k}),\]  
\[S_{i}^{*}=\sum\limits_{k=1}^{n}\mid V_{i}V_{k}Y_{jk}\mid\angle(\theta_{ik}+ \delta_{k}-\delta_{i}).\]  
Now, for real power \(P_{i}\) of \(i\)th bus, we have  
\[P_{i}=\sum\limits_{k=1}^{n}\mid V_{i}V_{k}Y_{jk}\mid\cos(\theta_{ik}+\delta_{ k}-\delta_{i}). \tag{28}\]  
Similarly, for the reactive power \(Q_{i}\) of \(i\)th bus, we have  
\[Q_{i}=-\sum\limits_{k=1}^{n}\mid V_{i}V_{k}Y_{jk}\mid\sin(\theta_{ik}+\delta_ {k}-\delta_{i}). \tag{29}\]  
Therefore,  
\[P_{i}=V_{i}V_{i}Y_{ii}\cos(\theta_{ii})+\sum\limits_{ \begin{subarray}{c}k=1\\ k\
eq i\end{subarray} }^{n}\mid V_{i}V_{k}Y_{ik}\mid\cos(\theta_{ik}+\delta_{ k}-\delta_{i}). \tag{30}\]  
and  
\[Q_{i}=-V_{i}V_{i}Y_{ii}\sin(\theta_{ii})-\sum\limits_{ \begin{subarray}{c}k=1\\ k\
eq i\end{subarray} }^{n}\mid V_{i}V_{k}Y_{ik}\mid\sin(\theta_{ik}+\delta_{ k}-\delta_{i}). \tag{31}\]  
Four quantities at each bus are involved in order to solve the power flow equations, i.e., the real power \(P_{i}\), reactive power \(Q_{i}\), voltage \(V_{i}\), and phase angle \(\delta_{i}\). \[P_{i}=g_{1}(\delta,V)\text{ and }Q_{i}=g_{2}(\delta,V).\]We suppose that  
\[\Delta P_{i} = P_{i}^{s}-P_{i}(\delta_{2},\delta_{3},\ldots,\delta_{k},V_{2},V_{3},\ldots,V_{k}),\] \[\Delta Q_{i} = Q_{i}^{s}-Q_{i}(\delta_{2},\delta_{3},\ldots,\delta_{k},V_{2},V_{3 },\ldots,V_{k}),\]  
where \(P_{i}^{s}\) and \(Q_{i}^{s}\) are the defined values of \(P_{i}\) and \(Q_{i}\), respectively. By taking partial derivatives of \(P_{i}\) and \(Q_{i}\) with respect to \(\delta_{j}\) and \(V_{j}\), we obtain  
\[\frac{\partial\Delta P_{i}}{\partial\delta_{j}} = \frac{\partial(P_{i}^{s}-P_{i}(\delta,V))}{\partial\delta_{j}}=- \frac{\partial P_{i}}{\partial\delta_{j}}\] \[\frac{\partial\Delta P_{i}}{\partial V_{j}} = -\frac{\partial P_{i}}{\partial V_{j}}.\]  
Similarly,  
\[\frac{\partial\Delta Q_{i}}{\partial\delta_{j}} = \frac{\partial(Q_{i}^{s}-Q_{i}(\delta,V))}{\partial\delta_{j}}= -\frac{\partial Q_{i}}{\partial\delta_{j}}\] \[\frac{\partial\Delta Q_{i}}{\partial V_{j}} = -\frac{\partial Q_{i}}{\partial V_{j}}.\]  
The Jacobian matrix is partitioned into the submatrices indicated below:  
\[J= \begin{bmatrix}J_{11}&J_{12}\\ J_{21}&J_{22}\end{bmatrix} .\]  
\[ \begin{bmatrix}\Delta P\\ \Delta Q\end{bmatrix} = \begin{bmatrix}J_{11}&J_{12}\\ J_{21}&J_{22}\end{bmatrix}\begin{bmatrix}\Delta\delta\\ \Delta V\end{bmatrix} ,\]  
where  
\[\Delta\delta= \begin{bmatrix}\Delta\delta_{2}\\ \Delta\delta_{3}\\ \vdots\\ \Delta\delta_{n}\end{bmatrix} ,\Delta V= \begin{bmatrix}\Delta V_{2}\\ \Delta V_{3}\\ \vdots\\ \Delta V_{n}\end{bmatrix} ,\Delta P= \begin{bmatrix}\Delta P_{2}\\ \Delta P_{3}\\ \vdots\\ \Delta P_{n}\end{bmatrix} ,\Delta Q= \begin{bmatrix}\Delta Q_{2}\\ \Delta Q_{3}\\ \vdots\\ \Delta Q_{n}\end{bmatrix} .\]  
The off-diagonal and diagonal elements of Jacobian submatrices are obtained by taking the partial derivatives of Equations (28) and (29) with respect to \(V\) and \(\delta\). \[J_{11}=\frac{\partial P}{\partial\delta^{\prime}},J_{12}=\frac{\partial P}{ \partial V},J_{21}=\frac{\partial Q}{\partial\delta^{\prime}},J_{22}=\frac{ \partial Q}{\partial V}.\]  
**Example 1**.