## 1 Introduction  
With the integration of distributed renewable energy sources into the new distribution system, the reserve capacity of the conventional power grid is insufficient, and the anti-disturbance capability of some power supply units in the distribution network has decreased, leading to the low inertia characteristic of the new distribution system being prone to causing large-scale frequency fluctuations. This poses significant challenges to the safe and reliable power supply of the distribution system and the large-scale consumption of distributed PV power. The RoCoF is one of the important factors for measuring the stability of a low-inertia distribution system after being disturbed. A high RoCoF can cause the large-scale disconnection of distributed PV inverters in the distribution network due to frequency change rate protection [1; 2], exacerbating power shortages in the distribution network and leading to dual instability in both the active power and voltage. In the new distribution system, limiting RoCoF under disturbances has become a critical research objective for maintaining system stability. The magnitude of RoCoF is mainly determined by the inertia of the power supply units in the distribution system. In the new distribution system, with a high penetration of renewable energy, maintaining frequency stability across multiple autonomous regions following disturbances in the off-grid state has become a major research focus. Virtual inertia, implemented through control strategies applied to grid-forming PV systems and distributed ESS, can counteract system frequency changes. Consequently, frequency stability across multiple regions can be achieved through the coordinated allocation and distribution of virtual inertia. Researchers worldwide have explored areas such as inertia demand assessment [3; 4] and virtual inertia allocation [5; 6]. Regarding inertia demand assessment, reference [7] defines the minimum inertia demand for a microgrid based on an in-depth analysis of their inertia and frequency characteristics and also analyzes the evolution of these characteristics under the coordinated effects of virtual inertia control and demand response. Reference [8] proposes a probabilistic algorithm for assessing available wind farm inertia to account for discrepancies between the nominal value and the actual value influenced by wind speed distribution and turbine operating conditions. Reference [9] introduces a multi-dimensional evaluation framework for the grid inertia level. This framework utilizes three quantitative indicators: the rotational kinetic energy of operational units, the inertia change rate, and the inertia distribution index. This allows us to effectively address the diverse evaluation requirements of DC systems and power grids with high renewable energy penetration. Based on the dynamic behavior of the nodal frequency during disturbances, reference [5] analyzes the spatio-temporal characteristics of power system inertia and constructs an inertia index derived from the energy of nodal frequency signals under small disturbances. Reference [10] proposes an optimal operation strategy for power systems with high renewable penetration. This strategy, leveraging minimum inertia assessment technology, employs a three-stage framework: preprocessing, minimum inertia assessment, and optimal operation. Reference [11] introduces the concept of critical inertia, defining the minimum system inertia required to maintain acceptable frequency conditions and adhere to the RoCoF limits for a given load level. Regarding virtual inertia allocation, reference [12] calculated the regional inertia demand using a time-dependent Gaussian mixture model encoded with a Markov chain, illustrated by the British power system, and provided a corresponding virtual inertia allocation strategy. Reference [13] proposed the concept of optimal inertia allocation and allocated inertia from an optimal control perspective; however, the overly simplistic models preclude its application in new distribution system analysis.