Frequency-constrained microgrid-distribution network coordinated load restoration: A distributed carbon-aware optimization approach

In the event of extreme contingencies, microgrids (MGs) have emerged as a reliable solution for enabling rapid load restoration and enhancing power supply resilience in active distribution network (ADN). However, MGs with high penetration of renewable energy sources often exhibit low physical inertia and damping capabilities, posing significant frequency security risks during load restoration. Additionally, the coupled relationship between energy supply and carbon emissions during restoration, critical for achieving low-carbon operation, remains underexplored. To address these challenges, this paper proposes a novel frequency-constrained microgrid-distribution network coordinated load restoration model (FCM-LRM) considering electricity-carbon coupling, ensuring both system stability and environmental sustainability. In the model, the system frequency security constraints, regulation reserve and ramping constraints and carbon allowance trading constraints are embedded to collectively ensure system's frequency safety and low-carbon performance during restoration, and the uncertainty-related constraints across subsystems are modeled as unified joint chance constraints (JCCs) to enhance decision reliability. Finally, a modified sample average approximation (MSAA) method is used to transform the JCCs into linear formulations, while the alternating direction method of multipliers (ADMM) algorithm enables fast distributed cooperative solution. Case studies demonstrate that the proposed framework can maintain system's frequency within safe threshold during restoration processes, improve renewable energy utilization and reduce carbon emissions by 4.63 % through electricity-carbon coupling. The distributed solving method achieves convergence within 80.13 s, representing a 74.40 % reduction compared to conventional approaches, satisfying real-time restoration requirements.