Coordinated restoration of transmission and distribution systems considering frequency security under restoration path establishment

The integration of distributed energy resources calls for coordinated restoration of transmission system (TS) and distribution systems (DSs). However, existing studies face challenges in embedding frequency constraints under TS network energization, during which the topology of the restored part is changing. This paper proposes a two-stage strategy for transmission system and distribution system restoration. Stage 1 identifies optimal network energization paths from black-start generators to transmission system loads, forming expanding restoration partitions. Fixing the sequential transmission system energization paths, stage 2 determines load restoration schemes while explicitly incorporating frequency constraints under both load pickup actions and renewable power fluctuations. To mitigate the impacts of uncertainties from renewable power fluctuations and distribution system component repair time, a rolling optimization paradigm is applied within stage 2. In both stages, a distributed framework is applied to optimize the utilization of energy resources at the system level and preserve operator independence. An enhanced Alternating Direction Method of Multipliers algorithm (ADMM) is developed to resolve potential infeasibility issues in existing algorithms with “Relax-Round-Polish” procedure. Case studies validate the effectiveness of the model in ensuring frequency security and enabling timely decision-making within each time interval.