# A Novel Higher-Order Numerical Scheme for System of Nonlinear Load Flow Equations
## 3 Numerical Experiments
### Formation of Load Flow Equations
: _Suppose we have a two-bus power system as shown in the Figure 1. Bus 1 is taken as slack bus having \(\delta_{1}=0\), \(|V_{1}|=1.0\) pu, and Bus 2 is a load bus with \(P_{2}=-1.0\) pu, \(Q_{2}=-0.5\) pu. A load of \(100\) mw and \(50\) Mvar is taken for Bus 2. The line impedance is \(Z_{12}=0.12+j0.16\) pu. Let us suppose an initial estimate of voltage magnitude \(|V_{2}|=1\) pu and phase angle \(\delta_{2}=0\)._  
As the numbers of buses in this system are two, so \(Y_{bus}\) can be written as  
\[Y_{bus}=\left[ \begin{array}{cc}Y_{11}&Y_{12}\\ Y_{21}&Y_{22}\end{array} \right],\]  
\[Y_{bus}=\left[ \begin{array}{cc}5\angle-53.13^{0}&5\angle 126.87^{0}\\ 5\angle 126.87^{0}&5\angle-53.13^{0}\end{array} \right].\]  
By expanding power flow equations, (28) and (29) for \(n=2\), we obtain  
\[P_{2}=V_{2}\sum_{k=1}^{2}Y_{2k}V_{k}\cos(\theta_{2k}+\delta_{k}-\delta_{2}), \tag{32}\]\[Q_{2}=-V_{2}\sum_{k=1}^{2}Y_{2k}\sin(\theta_{2k}+\delta_{k}-\delta_{2}). \tag{33}\]  
A \(2\times 2\) system of nonlinear equations is obtained by substituting values of \(Y_{bus}\) in Equations (32) and (33):  
\[P_{2}=5V_{2}\cos(126.87^{0}-\delta_{2})+5V_{2}^{2}\cos(-53.13^{0}),\]  
\[Q_{2}=-5V_{2}\sin(126.87^{0}-\delta_{2})-5V_{2}^{2}\sin(-53.13^{0}).\]  
The initial guess is taken as \(X^{(0)}=(1,1)^{t}\). Numerical results are shown in Table 4.  
It can be noted from Table 4, that our method \(HM_{1}\) dominates all the methods in terms of less error and less computational time. **Example 2**. : _Consider a general three-bus power system [6]. Bus \(1\) is considered as a slack bus with voltage magnitude \(|V_{1}|=1.05\ pu\), Bus \(2\) is a PQ bus with a load of \(P_{2}=-4.0\ pu\) and \begin{table}
\begin{tabular}{c c c c c} \hline \hline _Cases_ & \(k\) & \(\parallel X^{(k)}-X^{(k-1)}\parallel_{\infty}\) & \(\parallel F(X^{(k)})\parallel_{\infty}\) & \(t^{(k)}\) \\ \hline \(HM_{1}\) & \(1\) & \(9.89838070\ (-1)\) & \(4.78059596\ (-2)\) & \\ \hline  & \(2\) & \(1.01619291\ (-2)\) & \(2.07001814\ (-12)\) & \\ \hline  & \(3\) & \(4.43800774\ (-13)\) & \(4.21224910\ (-55)\) & \(1.353641\) \\ \hline \(HM_{2}\) & \(1\) & \(9.90403466\) & \(4.447803895\ (-2)\) & \\ \hline  & \(2\) & \(9.59653379\ (-3)\) & \(6.44704740\ (-13)\) & \\ \hline  & \(3\) & \(6.71636130\ (-5)\) & \(1.84188372\ (-43)\) & \(1.409686\) \\ \hline \(HM_{3}\) & \(1\) & \(9.86538347\ (-1)\) & \(6.33730798\ (-2)\) & \\ \hline  & \(2\) & \(1.34616521\ (-2)\) & \(8.90751822\ (-12)\) & \\ \hline  & \(3\) & \(1.91374962\ (-12)\) & \(1.49111311\ (-56)\) & \(2.496505\) \\ \hline  & \(1\) & \(9.82754355\ (-1)\) & \(8.02911898\ (-2)\) & \\ \hline  & \(2\) & \(1.724564463\ (-2)\) & \(7.46401068\ (-11)\) & \\ \hline  & \(3\) & \(1.62714696\ (-11)\) & \(4.50928147\ (-54)\) & \(1.454692\) \\ \hline  & \(1\) & \(9.84331415\ (-1)\) & \(7.30705632\ (-2)\) & \\ \hline  & \(2\) & \(1.56685847\ (-2)\) & \(1.70343570\ (-11)\) & \\ \hline  & \(3\) & \(3.70209774\ (-12)\) & \(2.20545208\ (-55)\) & \(1.40545\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of seventh-order schemes for load flow problem. Figure 1: Two-bus system. \(Q_{2}=-2.5\) pu, and Bus 3 is a generation bus having magnitude of voltage as \(|V_{3}|=1.04\) pu and \(P_{3}=2.0\) pu. The impedances are shown in Figure 2.  
\(Y_{bus}\) matrix for this system is written as  
\[Y_{bus}=\left[ \begin{array}{ccc}Y_{11}&Y_{12}&Y_{13}\\ Y_{21}&Y_{22}&Y_{23}\\ Y_{31}&Y_{32}&Y_{33}\end{array} \right],\]  
\[|Y_{bus}|=\left[ \begin{array}{ccc}53.8516&22.3606&31.6228\\ 22.3606&58.1378&35.7771\\ 31.6228&35.7771&67.2309\end{array} \right]\text{and }|Y_{Bus}|=\left[ \begin{array}{ccc}68.1986&116.5651&108.4349\\ 116.5651&63.4349&116.5651\\ 108.4349&116.5631&67.2490\end{array} \right].\]  
Thus, in this case, \(Y_{Bus}\) admittance matrix is  
\[Y_{bus}=\left[ \begin{array}{ccc}53.8516\angle-68.1986^{0}&22.3606\angle 116.5651^{0}&31.6228\angle 1 08.4349^{0}\\ 22.3606\angle 116.5651^{0}&58.1378\angle-63.4349^{0}&35.7771\angle 116.5651^{0}\\ 31.6228\angle 108.4349^{0}&35.7771\angle 116.5631^{0}&67.2309\angle-67.2490^{0} \end{array}