# A New Method for Constructing the Static-Voltage-Stability Boundary with a Conversion Strategy
## 4 Improving Boundary Tracking Methods
### Orthogonal-Arc-Correction Conversion Strategy  
As is mentioned in Section 3, the hyperplane correction method may fail, which is caused by the fact that the normal hyperplane cannot cross the SVSRB. The series curve clusters in Figure 6 can be projected into the power injection space, as shown in Figure 7.  
It can be observed that different correction methods can affect convergence. If Equation (13) is adopted, it is difficult to converge when the boundary curve cannot be intersected geometrically, as shown in Figure 7. Conversely, the arc-correction method can guarantee convergence because of the intersection with the SVSRB. Thus, the arc supplementary equation can be formulated as follows:  
\[\begin{array}{ll}\sigma^{2}&=\sum\limits_{j=1}^{n}{(x_{j}^{cor}-x_{j})^{2}+ \sum\limits_{j=1}^{n}{(L_{j}^{cor}-L_{j})^{2}}}\\ &\quad+{(\alpha^{cor}-\alpha_{j})^{2}+(\eta^{cor}-\eta_{j})^{2}}\\ &\triangleq<S^{cor},\mathbf{S}_{j}>\end{array}, \tag{20}\]  
where \(x_{j}\) is the \(j\)-th component of \(\mathbf{x}_{j}\), and \(L_{j}\) is the \(j\)-th component of \(\mathbf{L}_{j}\), and \(\mathbf{S}_{j}=[\mathbf{x}_{j},\mathbf{L}_{j},\alpha_{j},\eta_{j}]\) in Equation (4). \(\kappa\) is the arc radius; the symbol \(<\cdot,\cdot>\) represents the inner product of two vectors, and the corrected variable can be written as \(\mathbf{S}_{j}^{cor}=[\mathbf{x}_{j}^{cor},\mathbf{L}_{j}^{cor},\alpha^{cor},\eta^{cor}]\).  
\begin{table}
\begin{tabular}{l} \hline
**Tangent and Extrapolation Prediction** \\ \hline
**Pre-requirements**: current time step \(k\), a series of the boundary point, i.e., \(\mathbf{S}_{K}=[\mathbf{x}_{K},\mathbf{L}_{K},\)\(\eta,\alpha]\), and the corresponding tangent vector \(\mathbf{T}_{K}\), where K = \([\)k, k \(-\) 1, k \(-\) 2, k \(-\) 3\(]\), step size \(\Delta\alpha\). \\
**STEP 1**: Calculate the angle between the tangent vector \(\mathbf{T}_{K}\), marked as \(\mathbf{\theta}_{(\text{k}-3)-(\text{k}-2)}\), \(\mathbf{\theta}_{(\text{k}-2)-(\text{k}-1)}\), \(\mathbf{\theta}_{(\text{k}-1)-(\text{k})}\). \\
**STEP 2**: Judge whether \(\mathbf{\theta}_{(\text{k}-3)-(\text{k}-2)}<\mathbf{\theta}_{(\text{k}-2)-(\text{k}-1 )}<\mathbf{\theta}_{(\text{k}-1)-(\text{k})}\). If satisfied, turn to STEP 3. If not satisfied, turn to STEP 4. \\
**STEP 3**: Calculate the \(S_{k+1}^{\text{pre}}\) based on Equation (19). Calculate the angle \(\mathbf{\theta}_{(\text{k})-(\text{k}+1)}\) based on the newly predicted point \(S_{k+1}^{\text{pre}}\). \\
**STEP 4**: Judge whether \(\mathbf{\theta}_{(\text{k})-(\text{k}+1)}>0\). If satisfied, turn to STEP 6. If not satisfied, turn to STEP 5. \\
**STEP 5**: Calculate \(S_{k+1}^{\text{pre}}\) based on the tangent method. \\
**STEP 6**: Output the predicted value \(S_{k+1}^{\text{pre}}\). \\ \hline \end{tabular}
\end{table}
Table 2: Tangent and extrapolation prediction method.  
Figure 7: The schematic diagram of orthogonal-arc correction.  
Simultaneously using Equations (4) and (20), we can obtain the following:  
\[\mathbf{\varTheta}(S^{cor})=\left\{\begin{array}{c}\mathbf{\varPsi}(S^{cor})=\mathbf{0}\\ \left\langle S^{cor},S_{j}\right\rangle-\kappa^{2}=0\end{array}\right., \tag{21}\]  
where the distance between \(S_{j}\) and \(S^{\text{pre}}\) can be set to \(\kappa\), i.e., \(\left|S_{j}S^{pre}\right|=\kappa\). And, Equation (21) can be illustrated with a graph as shown in Figure 8 below:  
Notice that Equation (21) is a multivariable-coupling nonlinear system, which can be solved by the Newton-Raphson method with \(S^{\text{pre}}\) as the iterative initial value. Combining this with hyperplane correction in Equation (13), the arc-correction Equation (21) can be used as a backup scheme to ensure the correction process has a solution eventually. Thus, the orthogonal-arc-correction strategy proposed in this section is summarized in Table 3 below: