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This letter introduces a novel graph convolutional neural network (GCN) architecture for solving the optimal switching problem in distribution networks while integrating the underlying power flow equations in the learning process. The switching problem is formulated as a mixed-integer second-order cone program (MISOCP), recognized for its computational intensity making it impossible to solve in many real-world cases. Transforming the existing literature, the proposed learning algorithm is augmented with mathematical model information representing physical system constraints both during and post training stages to ensure the feasibility of the rendered decisions. The findings highlight the significant potential of applying predictions from a linearized model to the MISOCP form. 

Public safety power shut-offs (PSPSs), implemented by California utilities during high wildfire risk periods, lead to the de-energization of grid sections and leave certain customers out of power for several hours and even days. We propose an agile decision support system (DSS) to mitigate these impacts by harnessing electric vehicles (EVs) as mobile energy sources serving the multiple microgrids (μGs) formed within affected communities. Given that not all μGs possess adequate energy storage and distributed energy resources (DERs), we advocate for the mobilization of vehicle-to-grid (V2G)-enabled EVs for equitable and resilient energy access. Our emergency service relocation (ESR) model incentivizes EV owners to transport stored energy between μGs. However, traditional DSS cannot promptly solve the associated mixed-integer programming (MIP) problem, necessitating a faster solution algorithm for rapid EV deployment under emergency conditions. We introduce a learning framework employing graph convolutional networks (GCNs) that significantly expedites the MIP problem's solution by predicting binary values. Our results demonstrate the effectiveness of the proposed framework in promoting grid resilience and considerably reducing solve time when the problem has 69k binary decision variables.