Stochastic Black Start Resource Allocation to Enable Dynamic Formation of Networked Microgrids and DER-Aided Restoration

Extended outages in distribution systems (DSs) dominated by distributed energy resources (DERs) require innovative black start (BS) strategies to ensure efficient and secure restoration. This paper proposes a two-stage stochastic resource allocation method within synchronizing dynamic microgrids for black start (SDMG-BS), enabling risk-averse and adaptive restoration across diverse scenarios while maintaining frequency security. Virtual synchronous generator (VSG)-controlled grid-forming inverters (GFMIs) with primary frequency governors are modeled as BS sources, with frequency response constrained by three transient indices to ensure stability during load pick-up. SDMG-BS enables location-independent synchronization among restored microgrids and the transmission grid (TG) via smart switches (SSWs). A scenario-based stochastic programming model addresses multi-source uncertainties, including seasonal variations in renewable output, load demand, and TG outage duration. The linear frequency model used in planning is validated to exceed 90 $\%$ accuracy. Case studies on a modified IEEE 123-node feeder demonstrate that the proposed approach restores up to 20 $\%$ more critical load than deterministic baselines, executes two MG synchronizations prior to TG reconnection, and maintains all transient frequency indices within safe bounds across all scenarios. The results highlight the effectiveness of SDMG-BS in enhancing resilience and operational flexibility under uncertainty.