## 5 Discussion  
### S-BGPD Is Used to Construct the SSR (Static Security Region)  
In this example, the IEEE-118 system is used to construct the boundary of the hyperplane. Firstly, time domain simulation results are obtained using MATLAB software (9.10.0.1602886 R2021a) to gather the power flow data. Then, based on the constraints of the stochastic-batch gradient descent method for the hyperplane, the safety domain's hyperplane is constructed. Finally, the optimal coefficients for the hyperplane are obtained.  
In the example of the IEEE-118 node system, the structure is illustrated in Figure 7.The system operates at a frequency of 50 Hz. The improved IEEE-118 node system consists of 186 transmission lines, 91 loads, and 54 generator units. The peak load of the system is 3733 MW. In this study, the penetration rate is set to 10 gradients with an increase of 5%, starting from an initial penetration rate of 5%.  
Figure 6: Flowchart for constructing security domains based on improved stochastic–batch gradient descent.  
The formula for calculating the new energy penetration rate, denoted as \(f_{NEW}\), is given as the ratio of the total new energy generation capacity to the overall power demand of the system, PL. The formula can be expressed as follows:  
\[f_{NEW}=\frac{\sum P_{N}}{\sum P_{L}}\times 100\% \tag{43}\]  
In the case study, ten nodes, namely, Node 27, Node 31, Node 46, Node 54, Node 59, Node 61, Node 73, Node 87, Node 111, and Node 107, are connected to wind power generation. These nodes are equipped with DFIG (Doubly Fed Induction Generator) wind turbines. Additionally, three critical nodes, G66, G103, and D27, are considered for the construction of the safety domain. Their active power output is used to create a three-dimensional vector to represent the safety domain.  
As an example, the boundary surface of the security domain fitted for a new energy penetration rate of 40% is shown in Figures 8 and 9.  
The nodes Bus 66 and Bus 27 are chosen as key nodes. In the 2D space with P66 and P27 as coordinate axes, the approximate boundary of the SSR is searched using the method proposed in this chapter. Analytical expression for the approximate 2D SSR hyperplane of the IEEE-118 system is shown in Table 1. The analytical expression for the approximate boundary of the SSR in the 2D active power injection space is shown in Figure 8.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Hyperplane** & **Analytical Expression** & **Range of P27** \\ \hline H1 & \(-0.1700\)P27-P66 + 2.8423 = 0 & [0.1600, 1.8960] \\ H2 & \(-0.4980\)P27-P66 + 3.4641 = 0 & [1.8960, 2.6850] \\ H3 & \(-1.1148\)P27-P66 + 5.1203 = 0 & [2.6850, 3.5350] \\ H4 & \(-2.2942\)P27-P66 + 9.2897 = 0 & [3.5350, 3.9794] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Analytical expression for the approximate 2D SSR hyperplane of the IEEE-118 system.  
Figure 7: Improved IEEE 118-node test system.  
Further, taking Bus 66, Bus 27, and Bus 103 as voltage stability critical nodes, with \(\theta\)max = 13\({}^{\circ}\) and hmax = 0.05, the approximate SSR boundary in three-dimensional space with P66, P27, and P103 as axes is depicted in Figure 9. The analytical expression for this boundary is provided in Appendix A.  
In Figure 9, the surface edges are irregular with jagged protrusions, indicating the influence of new energy uncertainties at a 40% wind power penetration rate. Overall, the surface transitions smoothly from the lowest penetration point (blue area), representing a relatively stable state of the system, to the highest penetration point (red area), which may approach the limits of system stability. The transition is relatively smooth, and the area distribution of various states on the surface is balanced, indicating that the system can accept static safety and stability at a 40% wind power penetration rate.  
Figure 8: Two-dimensional SSR of the IEEE-118 system.  
Figure 9: The three-dimensional approximate SSR boundary of the IEEE-118 system.