# A New Method for Constructing the Static-Voltage-Stability Boundary with a Conversion Strategy
## 4 Improving Boundary Tracking Methods  
### Alternate Curve Prediction Strategy  
The estimation stage mainly provides iterative initial value in the correction stage. The tangent estimated method, as depicted by Equation (11), can be ineffective during the prediction process mentioned in Section 3. As is mentioned in Equation (4), the essence of the continuous method can be interpreted as all variables being regarded as the function of a small parameter, as follows:  
Figure 5: The schematic diagram of convergence domain.  
\[\phi:\alpha\in\mathbf{\mathcal{E}}\mapsto S=\begin{bmatrix}x(\alpha)\\ L(\alpha)\\ \eta(\alpha)\end{bmatrix}\in\mathbb{R}^{2n+1}, \tag{16}\]  
where \(\phi\) is continuous mapping; \(\mathbf{\mathcal{E}}\) is the set of the continuous parameter \(\alpha\). Consider that for the complicated inverse operation of the gradient matrix, an alternating estimation method can be proposed based on the changing shape of the boundary curve and the known boundary point data, as shown in Figure 6.  
Based on the function relation revealed by Equation (16), the SVSRB can be approximated by a polynomial function locally. As is shown in Figure 6, \(S_{0}\), \(S_{1}\)\(S_{2}\) and \(S_{3}\) can be acquired via Equation (11) and \(T_{0}\), \(T_{1}\), \(T_{2}\) and \(T_{3}\) are the corresponding tangent vectors, respectively. \(\theta_{01}\), \(\theta_{12}\) and \(\theta_{23}\) are the angles between the corresponding tangent vectors. We can observe that \(S_{4}\)\({}^{\text{pre}}\) can be obtained by the curve extrapolation method rather than the tangent method if \(\theta_{01}\), \(\theta_{12}\) and \(\theta_{23}\) are increasing monotonically, because the boundary morphology varies slowly, as shown in Figure 6.  
Consider a series of points (\(\alpha_{0}\), \(x_{0}\)), (\(\alpha_{1}\), \(x_{1}\)), (\(\alpha_{2}\), \(x_{2}\)), \(\ldots\), (\(\alpha_{k}\), \(x_{k}\)) in the mapping of \(\phi\) based on Equation (16), where \(\alpha_{k}\) is the value under the \(k\)-th time step, and \(x_{k}\) is the corresponding component based on Equation (16). The Lagrange interpolation polynomial can be formulated as follows:  
\[P(\alpha)=\sum_{j=0}^{k}x_{j}L_{j}(\alpha), \tag{17}\]  
where \(x_{j}\) is the \(j\)-th of vector \(x_{j}\), i.e., \(x_{j}=[x_{1},\,x_{2},\,\ldots,\,x_{n}]\), \(k\) is the number of fitting points and \(L_{j}(\alpha)\) is the Lagrange coefficient as follows:  
\[\begin{array}{ll}L_{j}(\alpha)=&\prod_{i=0,j\
eq j}^{k}\left( \frac{\alpha-\alpha_{i}}{\alpha_{j}-\alpha_{i}}\right)=\frac{(\alpha-\alpha_ {0})}{(\alpha_{j}-\alpha_{0})}\times\frac{(\alpha-\alpha_{1})}{(\alpha_{j}- \alpha_{1})}\times\cdots\\ &\times\frac{(\alpha-\alpha_{j-1})}{(\alpha_{j}-\alpha_{j-1})}\times\frac{( \alpha-\alpha_{j+1})}{(\alpha_{j}-\alpha_{j+1})}\times\cdots\times\frac{( \alpha-\alpha_{k})}{(\alpha_{j}-\alpha_{k})}\end{array} \tag{18}\]  
As is shown in Figure 6, (\(\alpha_{1}\), \(x_{1}\)), (\(\alpha_{2}\), \(x_{2}\)) and (\(\alpha_{3}\), \(x_{3}\)), i.e., \(S_{1}\), \(S_{2}\) and \(S_{3}\), are known points, and the alternate strategy is satisfied: \(\theta_{01}<\theta_{12}<\theta_{23}\). Then, the next point, \(S_{4}\)\({}^{\text{pre}}\), can be written as follows:  
\[\begin{array}{ll}x_{4}&=\frac{(\alpha_{4}-\alpha_{2})(\alpha_{4}-\alpha_{1} )}{(\alpha_{3}-\alpha_{2})(\alpha_{3}-\alpha_{1})}x_{3}\\ &+\frac{(\alpha_{4}-\alpha_{1})(\alpha_{4}-\alpha_{3})}{(\alpha_{2}-\alpha_{1} )(\alpha_{2}-\alpha_{3})}x_{2}\\ &+\frac{(\alpha_{4}-\alpha_{2})(\alpha_{4}-\alpha_{3})}{(\alpha_{1}-\alpha_{ 2})(\alpha_{3}-\alpha_{3})}x_{1}\end{array} \tag{19}\]  
Figure 6: The schematic diagram of alternate strategy.  
where \(\alpha_{4}=\alpha_{3}+\Delta\alpha\), \(\Delta\alpha\) is an known step size. Therefore, the procedure of applying the tangent and extrapolation prediction method can be summarizes in Table 2 below.