# A New Method for Constructing the Static-Voltage-Stability Boundary with a Conversion Strategy
## 5 Case Studies
### WECC 3-Machine, 9-Bus Testing System  
The WECC 3-machine, 9-bus testing system was deployed to verify the accuracy and effectiveness of the proposed method for constructing the SVSRB, as shown in Figure 15. The detailed parameters of the WECC 3-machine, 9-bus testing system can be found in reference [26].  
Bus 9 and bus 7 were selected as the key nodes to construct the corresponding SVSRB with the active power variations. The SVSRBs constructed by the conventional tracking method and the proposed method are shown in Figure 16. It can be observed that both methods can effectively track the boundary and highly overlap at the boundary points.  
Figure 14: The number of iterations with different methods.  
Figure 15: The diagram of WECC 3-machine, 9-bus test system.  
The accuracy of different methods in constructing the SVSRB were compared and analyzed. In theory, the SVSRB is composed of a series of SNB points. Hence, the determinant of the Jacobian matrix at each SNB boundary point system should be zero, and the minimum eigenvalue of the system is zero. Due to the inevitable numerical errors brought about by iterative computations, the Jacobian matrix at each boundary point cannot be strictly zero ideally. Thus, the accuracy of different methods can be measured using the minimum eigenvalue of the Jacobian matrix corresponding to each boundary point, as shown in Figure 17.  
From Figure 17, it can be observed that in the WECC 3-machine, 9-bus power system, the conventional method and the proposed method maintain the same minimum eigenvalue of the Jacobian matrix at the initial tracking point \(S_{1}\) (because they have the same initial boundary point). The minimum eigenvalue of the Jacobian matrix at each boundary point, except \(S_{1}\), is smaller than that of the conventional tracking method, indicating that the accuracy of the proposed method in tracking boundary points is better than that of the conventional method.  
Figure 16: The SVSRB of WECC 3-machine, 9-bus test system.  
Figure 17: The minimum eigenvalue at different boundary points.  
To compare the timeliness of different methods in constructing the SVSRB, the timeliness was measured by the consumed computation time, as shown in Figure 18.  
From Figure 18, the time difference for constructing the SVSRB in a single-machine, single-load system is relatively minor in general, but the proposed method still has advantages in timeliness. In complex systems, i.e., a 3-machine, 9-bus power system, as the size of the computational scale increases, the differences in time consumption become increasingly clear, and the proposed method has the advantage of saving computation time, indicating that the proposed method can construct boundaries quickly.  
Compared with the traditional CPF method, the proposed method reduces the time complexity because it avoids the point-by-point calculation in the boundary construction process. To be specific, supposing that the boundary shape is known in advance as a standard circle, the number of two-dimensional boundary points can be set to \(n\) for discretization processing, as shown in Figure 19 below.  
The time complexity of the proposed method is \(O(n)=n\), while for the CPF method is \(O(n)=m\)-\(n\approx n^{2}\), where \(m\) is the computation time performed in a CPF calculation along a particular direction, as shown in Figure 19, and \(m>>n\) is satisfied in most cases. And, this time gap will be further widened in the construction of a three-dimensional SVSRB.