## 4 Discussion  
The simulation results show that the feasible region boundary assessment model of virtual inertia constructed by the RBF neural network achieves optimal mean square error (MSE) values of 0.37 and 0.12 in photovoltaic systems and energy storage systems, respectively, verifying the data-driven model's ability to capture complex nonlinear rela  
Figure 8: Frequency comparison chart of each region: (**a**) Frequency comparison before and after optimization in Region 1; (**b**) Frequency comparison before and after optimization in Region 2. Figure 9: Frequency changes in each region: (**a**) Frequency changes in Region 1; (**b**) Frequency variation in Region 2.  
tionships. Compared with traditional parametric modeling methods [6; 14], this method breaks through the difficulties associated with the precise modeling of distributed resources' dynamic characteristics, especially in non-steady-state scenarios such as sudden changes in light intensity and rapid fluctuations in the state of charge, reducing the boundary assessment error. This advancement overcomes the limitations of the fixed parameter assumptions in [18; 19], providing a more reliable constraint boundary for the dynamic allocation of virtual inertia. The convergence characteristics of the multi-objective optimization model are shown in Figure 10. The genetic algorithm reaches a stable solution within 25 generations, with the rate of change of frequency (RoCoF) in the two regions reduced to 0.209 Hz/s and 0.2205 Hz/s, respectively, and the absolute difference narrowed to 0.0115 Hz/s. Compared with the single-region optimization in [13] and the damping-oriented allocation strategy in [15], this study achieves the first multi-region RoCoF balanced optimization, improving the frequency response synchronization between regions by 41%. This confirms the effectiveness of the \"inertia allocation-frequency coordination\" coupling mechanism and solves the problem of oscillation amplification caused by the isolated configuration of regional inertia in traditional methods [5]. This study incorporated the intelligent SOP as a key control device in the case study. Its primary function serves to establish flexible inter-regional interconnection, thereby enabling coordinated virtual inertia allocation across regions. It should be noted that current research has not yet explored whether SOP devices themselves can provide virtual inertia support or their potential impact on the distribution of system inertia requirements. Significantly, as a grid-connected power electronic device, SOP may introduce new dynamics (e.g., dynamic coupling), imposing additional constraints on both system inertia estimation and regulation. Therefore, subsequent research will systematically analyze the dynamic coupling characteristics and additional constraints introduced by SOP integration. Key focuses include modeling and quantitatively evaluating the interaction between SOP control parameters and the virtual inertia transfer function, ultimately enhancing the theoretical framework for inertia cooperative control in power systems incorporating SOP. Currently, this study is limited to a two-region 114-node distribution system. To enhance the scalability of the proposed method for larger or more complex networks, adaptive improvements are required across three areas: model architecture, algorithm efficiency, and coordination mechanisms. At the model architecture level, the data-driven nature of RBFNN offers inherent scalability. Multi-region modeling can be achieved through a distributed training framework, enabling cross-region feature generalization and reducing data collection costs for new regions. For high-dimensional inputs (e.g., multiple resource types and complex coupled variables), deep learning techniques can hierarchically extract inertia boundary features.