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This work considers the task of finding an ac-feasible operating point of a severely damaged transmission network while ensuring that a maximal amount of active power loads can be delivered. This AC maximal load delivery (AC-MLD) task is a nonconvex nonlinear optimization problem that is incredibly challenging to solve on large-scale transmission system data sets. This work demonstrates that convex relaxations of the AC-MLD problem provide a reliable and scalable method for finding high-quality bounds on the amount of active power that can be delivered in the AC-MLD problem. To demonstrate their effectiveness, the solution methods proposed in this work are rigorously evaluated on 1000 $N$ - $k$ scenarios on seven power networks ranging in size from 70 to 6000 buses. The most effective relaxation of the AC-MLD problem converges in less than 20 seconds on commodity computing hardware for all 7000 of the scenarios considered. [ABSTRACT FROM AUTHOR]

Energy supply in rural and off-grid communities has traditionally relied on diesel-based microgrids, due to limited access. But global environmental concerns are pushing for the transformation of these systems into renewable-based microgrids. This transition to more complex systems with a mix of dispatchable and nondispatchable resources requires new planning tools that ensure the security of supply. This paper presents a novel mixed-integer linear optimization model that determines optimal technology mix, size, placement, and associated dispatch for a multi-energy microgrid. The model satisfies microgrid's electrical and heat transfer network limitations by integrating linear power flow and heat transfer equations. It captures the efficiency gains from waste heat recovery through combined heat and power technologies, by modeling the interplay between electrical and heat sources. To ensure a secure design against generator outages, the optimization maintains sufficient reserve capacity in the system, which is dynamically allocated based on system operating conditions. Several case studies on an isolated microgrid model, developed based on a real microgrid in Alaska, illustrate how the proposed model works. The results show the effectiveness of the model and are used to discuss various aspects of the optimization solution. [ABSTRACT FROM PUBLISHER]