## 5 Discussion
### Compare and Analyze the Results Obtained from Different Methods  
To validate the effectiveness and accuracy of the improved stochastic-batch gradient descent algorithm, different methods before and after the improvement are compared to observe the enhancements in computational efficiency and precision achieved by the improvements to the gradient descent algorithm. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Penetration Rate of New Energy/\%** & **Angle between Each Hyperplane\({}^{\prime}\)** & **Average Fitting Error of Critical Points/\%** \\ \hline S-BGPD & 40 & 2.12 & 1.27 \\ Traditional fitting method & 40 & 2.03 & 1.20 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of errors. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Output of**} & \multicolumn{2}{c}{**Active Power**} & \multicolumn{2}{c}{**Whether It Is**} \\
**Operating Point** & **Generators G66** & **Output of Wind Turbine** & **Located within** & **Time Domain Simulation** \\  & **and G103/MW** & **D27/MW** & **Domain** & \\ \hline \(1\) & 2204.312297 & 1528.687703 & Yes & Stable \\ \(2\) & 2216.342716 & 1516.657284 & No & Unstable \\ \(3\) & 2239.286794 & 1493.713206 & Yes & Stable \\ \(4\) & 2210.594078 & 1522.405922 & Yes & Stable \\ \(5\) & 2206.249615 & 1526.750385 & Yes & Stable \\ \(6\) & 2232.481303 & 1500.518697 & Yes & Stable \\ \(7\) & 2169.626044 & 1563.373956 & Yes & Stable \\ \(8\) & 2225.400152 & 1507.599848 & Yes & Stable \\ \(9\) & 2227.172067 & 1505.827933 & No & Unstable \\ \(10\) & 2229.287542 & 1503.712458 & No & Unstable \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Time domain simulation results with a 40% penetration rate. In Figure 11, it is evident that all four methods converge. BGD converges slower due to the use of a smaller step size to ensure stability. On the other hand, SGD iterates with a single sample at a time, which prevents it from exploring the global optimal solution, leading to oscillations in the convergence curve. Compared to these two methods, S-BGD demonstrates better convergence characteristics. Its convergence is more stable and efficient than BGD and SGD. In Table 4, it can be observed that the S-BGD method has a lower fitting error compared to the SGD and BGD methods, indicating that the S-BGD method, which combines both the SGD and BGD methods, exhibits better optimization capabilities than using the SGD or BGD methods individually. Furthermore, the S-BGPD method outperforms the S-BGD method in terms of fitting accuracy, indicating that the incorporation of the gradient accumulation strategy GP allows the S-BGPD method to obtain relatively superior solutions and thereby improves fitting precision. Regarding training time, the BGD and S-BGD methods have training times of 572.56 s and 312.42 s, respectively. This demonstrates that the S-BGD method achieves a 45% reduction in training time compared to BGD, highlighting its enhanced computational efficiency. On the other hand, the S-BGPD method has a training time of 376.59 s, representing a 20% increase compared to S-BGD. This increase in training time is mainly due to the introduction of the gradient accumulation strategy GP, which elongates the search path to avoid local optima or saddle points and increases the computational load. To validate the effectiveness and accuracy of the choice of hidden layers, the S-BGPD method was employed in the same IEEE-118 bus system with configurations of 5 and 10 hidden layers, respectively, to optimize and observe their error results. In Figure 12, it can be observed that after more than 100 iterations, the cumulative relative errors of the optimization results in five and ten hidden layers that are not significantly different. It can tentatively be concluded that the results obtained using the S-BGPD method with either configuration are similar. The reason for not considering more