# A New Method for Constructing the Static-Voltage-Stability Boundary with a Conversion Strategy
## 5 Case Studies  
### Single-Machine, Single-Load System  
The single-machine, single-load system and the system parameters are shown in Figure 10. The SVSRB is constructed as the active and reactive power of the load bus change. The conventional boundary tracking method proposed in reference [25] is deployed to construct the SVSRB of the single-machine, single-bus power system. The coordinate of the ground state point \(N\) is set as \([\Delta\)P, \(\Delta\)Q\(]^{\mathrm{T}}=[0.8660,\,0.5]^{\mathrm{T}}\). Thus, the initial power growth direction can be calculated as \(\beta=\arctan(\Delta\)P/\(\Delta\)Q). The CPF method is applied to initiate the boundary point \(S_{0}=[\theta,\,V,l_{0},\,l_{1},\,1,\,0]^{\mathrm{T}}\). Starting from \(S_{0}\) as the initial point, the proposed method in this paper and the conventional tracking method were used to search for each boundary point for comparison, as shown Figure 11.  
Figure 10: Single-machine, single-load power system model. Figure 9: The schematic diagram of algorithm flowchart. As is shown in Figure 11, the voltage of the load bus can be monitored by the SVSR. Once the SVSRB is constructed, the voltage stability can be determined by the load consumption, i.e., the active and reactive power of the load. Moreover, it can be seen that the proposed method can effectively achieve accurate boundary fitting, and the boundary constructed by the traditional method is consistent in general, which further demonstrates the effectiveness of the proposed method. The detailed numerical result of some boundary points in Figure 11 are shown in Table 4.  
Based on Equation (16), all variables can be regarded as the function of the continuous parameter. Figure 12 demonstrates the variation in \(x_{1}\), \(x_{2}\), \(L_{1}\), \(L_{2}\) and \(\eta\) with respect with continuous parameter \(\alpha\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**No.** & \(S\) & \(\theta\) & \(V\) & \(l_{0}\) & \(l_{1}\) & \(\eta\) & \(\alpha\) \\ \hline
0 & \(S_{0}\) & \(-0.464\) & \(0.559\) & \(0.667\) & \(0.744\) & \(1\) & \(0\) \\ \hline \multirow{2}{*}{1} & \(S_{0}^{\text{pre}}\) & \(-0.381\) & \(0.541\) & \(0.383\) & \(0.928\) & \(0.852\) & \(0.300\) \\  & \(S_{1}^{\text{lor}}\) & \(-0.382\) & \(0.538\) & \(0.373\) & \(0.927\) & \(0.855\) & \(0.302\) \\ \hline \multirow{2}{*}{2} & \(S_{2}^{\text{pre}}\) & \(-0.314\) & \(0.527\) & \(0.301\) & \(0.947\) & \(0.792\) & \(0.578\) \\  & \(S_{2}^{\text{cor}}\) & \(-0.336\) & \(0.525\) & \(0.309\) & \(0.951\) & \(0.795\) & \(0.578\) \\ \hline \multirow{2}{*}{3} & \(S_{3}^{\text{pre}}\) & \(-0.090\) & \(0.501\) & \(0.090\) & \(0.996\) & \(0.903\) & \(1.404\) \\  & \(S_{3}^{\text{cor}}\) & \(-0.089\) & \(0.502\) & \(0.089\) & \(0.995\) & \(0.906\) & \(1.403\) \\ \hline \multirow{2}{*}{4} & \(S_{4}^{\text{pre}}\) & \(0.001\) & \(0.499\) & \(0.001\) & \(1.00\) & \(1.035\) & \(1.641\) \\  & \(S_{4}^{\text{cor}}\) & \(0.002\) & \(0.500\) & \(-0.002\) & \(0.999\) & \(1.038\) & \(1.639\) \\ \hline \end{tabular}
\end{table} 
Table 4: Detailed numerical results of boundary points. Figure 11: The diagram of SVSRB with different method. Figure 12: The trend in each variable with respect to \(\alpha\). It can be seen from Figure 12 that the continuous tracking method itself can reflect the characteristics of continuous changes in other variables when the parameters continue to change, and it has good fitting characteristics. Hence, combined with the curve extrapolation technique, not only does it avoid the complex matrix inversion operation caused by traditional tracking methods in the predictor process, but the intermediate calculation results can also be effectively saved, providing an approximate expression for boundary stability in a linear estimation manner. The approximate linear expressions for some boundary curves are shown in Table 5.  
As is illustrated in Table 5, the SVSRB can be fitted by linear expression. The system stability can be determined and quantified by the distance between the current operation point and the analytical linear expression.