Efficient energy management is essential for sustainable microgrid operation amid increasing renewable energy integration. In this paper, an imitation learning-based framework to approximate mixed-integer Economic Model Predictive Control (EMPC) is proposed for microgrid energy management, considering fuel generators, renewable energy resources, a unified energy storage unit, and curtailable loads. Within the proposed framework, a neural network is trained to imitate EMPC control actions from offline trajectories, thereby enabling fast real-time decision making without solving online mixed-integer optimization problems, which often exhibit highly variable solution times across instances and do not scale well to large problem sizes; in particular, worst-case solving times can be excessively large and therefore unsuitable for real-time deployment. In contrast, the learned policy provides predictable and consistently low computation times. For robustness and generalization, the learning process incorporates noise injection during training to mitigate distribution shift and explicitly accounts for forecast uncertainty in renewable generation and demand. Furthermore, a constraint-tightening approach combined with a projection layer built on control barrier functions is proposed to ensure recursive feasibility and constraint satisfaction for the learned controller. Simulation results demonstrate that the learned policy achieves economic performance comparable to EMPC, while reducing computation time by approximately one order of magnitude relative to the optimization-based EMPC.

We study feasibility guarantees for safety filters developed using Control Barrier Functions (CBFs) when a safe set is defined using the pointwise minimum of continuously differentiable functions, a construction that is common for the backup CBF (BCBF) method and typically nonsmooth. We replace the minimum by its log-sum-exp (soft-min) smoothing and show that, under a strict safety condition, the smooth function becomes a CBF (or extended CBF) for a range of the smoothing parameter. For compact safe sets, we derive an explicit lower bound on the smoothing parameter that makes the smooth function a CBF and hence renders the corresponding safety constraint feasible. For unbounded sets, we introduce tail conditions under which the smooth function satisfies an extended CBF condition uniformly. Finally, we apply these results to BCBFs. We show that safety of a compact (terminal) backup set under a backup controller, together with a condition ensuring safety of the backup trajectories on the relevant boundary of the safe set, is sufficient for constraint feasibility for BCBFs. These results provide a recipe for a priori feasibility guarantees for smooth inner approximations of nonsmooth safe sets without the need for additional online certification.