Robust hosting capacity Maximization of Renewable-Rich distribution microgrids via a hierarchical Planning–Scheduling–Real-Time optimization framework

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• A master problem optimizing asset capacities. • Iterative coordination via Benders decomposition to align planning with real-time performance. • A multi-objective function aware of risk, investment costs, operating costs and conditional value at risk. • The results show that it increases the renewable hosting capacity by 15-20% compared to conventional methods. This paper proposes a hierarchical multi-stage optimization framework for maximizing the hosting capacity of distributed renewable energy resources in distribution microgrids while rigorously accounting for operational security and uncertainty. The problem is formulated as a tightly coupled three-level decision structure comprising long-term planning, day-ahead scheduling, and real-time corrective control. The upper-level planning stage determines the siting and sizing of renewable generation, energy storage systems, and controllable grid assets under investment, regulatory, and spatial constraints, with the hosting capacity explicitly embedded in the objective function. Given these design decisions, the intermediate stage performs scenario-based day-ahead operational scheduling over a multi-period horizon, considering AC power flow physics through convex second-order cone relaxations, energy storage dynamics, inverter reactive power capability, demand response, and renewable curtailment. The lower-level stage evaluates real-time feasibility under fast timescale uncertainty realizations and deploys limited corrective actions at a 10‑minute resolution including renewable active power curtailment, smart inverter reactive power regulation, fast storage charging/discharging, and OLTC/capacitor step changes to mitigate voltage and thermal violations. Operational infeasibility is quantified through slack variables and aggregated using expectation or conditional value-at-risk metrics, which are fed back to the planning stage to ensure robustness against extreme events. To solve the resulting large-scale stochastic mixed-integer problem, a decomposition-based algorithm is developed that combines convex relaxation, McCormick linearization, scenario reduction, and Benders-like feasibility cuts derived from real-time assessments. The proposed approach is validated on a modified IEEE 33-bus distribution microgrid with high renewable penetration. Simulation results demonstrate that the method significantly increases achievable hosting capacity while maintaining voltage and line-flow security across representative uncertainty scenarios, offering a scalable and practically implementable tool for robust microgrid planning.