## 1 Introduction
Reference [14] described the frequency dynamic response of network nodes using swing equations and optimized the inertia distribution to improve the damping ratio of oscillation modes. Based on reference [14], reference [15] determined the spatial distribution of system inertia with the goal of damping optimization. Reference [6] proposed a virtual inertia matching method for the multi-machine parallel operation of virtual synchronous generators. According to this inertia matching principle, the virtual inertia of each VSG is configured to ensure that each VSG can allocate the load in proportion to its capacity in both steady-state and transient processes. Reference [16] configured virtual inertia by considering the stability impact of small disturbances and the frequency stability on voltage-source and current-source virtual synchronous machines. Reference [17] employed state-space small-signal model analysis to improve the transient power distribution performance of virtual generators without compromising steady-state power sharing. Reference [18] proposed an optimization method for virtual inertia allocation based on Voronoi diagram interpolation. Reference [19] established a virtual inertia optimization distribution model with the goal of maximizing the damping ratio of interval oscillation modes in small disturbance analysis, taking virtual inertia control parameters as optimization variables and frequency stability as constraints. Then, the model was solved using a Newton method based on sensitivity analysis to obtain the virtual inertia optimization allocation scheme, but it did not consider the coordinated allocation of virtual inertia among multiple regions. The application environments of the existing virtual inertia allocation methods and the method proposed in this paper are shown in Table 1.  
In summary, existing studies have established fundamental principles and methods for virtual inertia allocation. However, they have not adequately addressed the coordinated allocation of virtual inertia across multiple regions while ensuring system inertial security. Specifically, current approaches lack a mechanism to enhance the uniformity of RoCoF among regions and minimize the overall RoCoF magnitude, preventing scenarios where a region experiences excessive disturbances but possesses insufficient local virtual inertia to maintain its frequency stability. To bridge this gap, this paper proposes a neural network-based coordinated virtual inertia allocation method for multi-region distribution systems, ensuring each region's inertia demand is satisfied. Firstly, utilizing a data-driven approach, RBFNN identifies the boundaries of feasible virtual inertia that distributed resources can offer under various disturbance scenarios. Secondly, an optimization model for virtual inertia allocation is formulated, aiming to minimize both the regional RoCoF and its inter-regional discrepancies, while accounting for the regulatory capabilities of grid-forming PV systems and distributed ESS. Then, a solution method for this optimization model is developed, leveraging a genetic algorithm. Finally, the optimized virtual inertia allocation scheme derived from the model is validated through simulation, demonstrating its effectiveness in enhancing the multi-region frequency stability.