A two-stage framework for optimal electricity‑hydrogen energy flow based on explainable graph learning and GPU-accelerated optimization

The deep integration of hydrogen systems into power systems poses a significant computational challenge for optimal energy flow (OEF) analysis of electricity‑hydrogen energy system (EHS), particularly under numerous renewable scenarios. While pure numerical solvers guarantee feasibility and optimality at high computational cost, data-driven methods often sacrifice solution explainability and feasibility. To address this problem, this paper proposes a two-stage framework for OEF to balance efficiency, feasibility and explainability. In the upper stage, an explainable distance-based graph neural network (EX-DGNN) rapidly generates initial solutions. The EX-DGNN uses the distance measurements between different graph nodes to capture the dependencies from distant nodes. Critically, a sample-based explanation mechanism is integrated. Different from explanation method based on importance values or decision reasons, the sample-based explanation method provides human operators with analogous evidences to justify the prediction solutions. In the lower stage, a restoration module, guided by the initial solutions and explanations, fixes discrete variables and rectifies constraint violation. The resulting continuous non-linear OEF model is then efficiently solved using a GPU-accelerated interior point method (IPM) solver. Case studies show that the proposed two-stage method achieves high feasibility and solving efficiency. On the modified IEEE-30 system and 20-node hydrogen system, compared to commercial solvers based on numerical method, the proposed method reduces the solving time by over two orders of magnitude, while maintaining 99.97% satisfied constraints. Traditional graph neural network can only satisfy 3% constraints. The proposed framework necessitates trade-offs among the computational time, feasibility and optimality of solutions. The sample-based explanations offer actionable insights, effectively bridging the gap between data-driven speed and operational transparency. The proposed method is also tested on the modified IEEE 118-bus system and 90-node hydrogen system, verifying the performance on the larger cases.