# A New Method for Constructing the Static-Voltage-Stability Boundary with a Conversion Strategy
## 2 Static Voltage Stability Boundary
### Stable Boundary Construction with Continuous Method  
To achieve the construction of the SVSRB, referring to the continuous parameter tracking algorithm proposed in reference [25], the equation representing all SNB points in a two-dimensional power plane can be characterized as follows:  
\[\boldsymbol{\Psi}(\boldsymbol{S})=\left\{\begin{array}{l}\boldsymbol{f}( \boldsymbol{x})+\eta\lambda_{k}\overset{\rightarrow}{\boldsymbol{b}}_{k}( \boldsymbol{\alpha})=\boldsymbol{0}\\ \boldsymbol{f}^{\mathrm{T}}(\boldsymbol{x})\boldsymbol{L}=\boldsymbol{0}, \quad\boldsymbol{L}^{\mathrm{T}}\boldsymbol{L}=\boldsymbol{1}\end{array}\right., \tag{4}\]  
where \(\boldsymbol{S}=[\boldsymbol{x},\boldsymbol{L},\eta,\boldsymbol{\alpha}]^{ \mathrm{T}}\in\mathbb{R}^{2n+1}\) is the unknown variable, \(\eta\) is the continuous parameter, and \(\lambda_{k}\) is the load margin under the direction of \(\overset{\rightarrow}{\boldsymbol{b}}_{k}(\boldsymbol{\alpha})\) as follows:  
\[\overset{\rightarrow}{\boldsymbol{b}}_{k}(\boldsymbol{\alpha})=[\boldsymbol{0 },\cos(\beta_{k}+\boldsymbol{\alpha}),\boldsymbol{0},\sin(\beta_{k}+ \boldsymbol{\alpha}),\boldsymbol{0}]^{\mathrm{T}}, \tag{5}\]  
where \(\boldsymbol{\alpha}\) is the offset on angle \(\beta_{k}\), and \(\beta_{k}\) is the initial power-growth-direction angle, as shown in Figure 2.  
As is shown in Figure 2, the initial power-growth-direction angle \(\beta_{k}\) can be written as follows:  
\[\beta_{k}=\arctan(\Delta p_{j}/\Delta p_{i}), \tag{6}\]  
where \(\Delta p_{i}\) and \(\Delta p_{j}\) are the power increments along with direction \(\beta_{k}\), as shown in Figure 2. Notice that \(\eta\) is the newly added continuous parameters used to characterize the quantitative impact on the load margin \(\lambda_{k}\) when \(\beta_{k}\) is shifted to \(\alpha\) in Equation (4).  
Figure 1: The SVSR in power injection space. The green lines are P-V cure with CPF method, while the red dashed line is the static voltage boundary.  
Figure 2: The initial direction angle of constructing SVSRB.  
Figure 3 demonstrates the principle diagram of constructing the SVSRB using the continuous parameter tracking method, i.e., Equation (4).  
The CPF simulation can be deployed in the \(\mathbf{b}_{0}\) direction to reach the first boundary point \(\mathbf{S}_{0}\). Thus, the initial values for the boundary construction in Equation (4) can be obtained. Notably, \(\eta_{0}=1\) and \(\alpha=0\) are satisfied at the boundary point \(\mathbf{S}_{0}\). By setting a continuous parameter \(\alpha\) with an angle perturbation of size \(\Delta\alpha\), the direction vector changes from \(\mathbf{b}_{0}\) to \(\mathbf{b}_{1}\). Correspondingly, the small change \(\Delta\eta_{0}\) of continuous parameter \(\eta_{0}\) is calculated, which is added to the previous calculation value \(\eta\) to update \(\eta_{1}\). The product of \(\eta_{1}\) and the initial margin \(\lambda_{0}\) can represent the load margin under the direction vector \(\mathbf{b}_{1}\). By continuously changing the parameter \(\Delta\alpha\) and tracking the parameter \(\Delta\eta_{0}\), all boundary points can be obtained; hence, the SVSRB is formed.