Hierarchical Physics-Informed Heterogeneous Graph Network-Based Optimal Energy Flow for Integrated Electricity–Heat Virtual Power Plants

In Virtual Power Plants (VPPs), computing the Optimal Energy Flow (OEF) for integrated electricity–heat systems is essential but challenged by highly asymmetric topologies and multi-timescale characteristics. Traditional data-driven models often fail to extract cross-domain coupling features and violate physical laws, yielding infeasible solutions. To address these challenges, this paper proposes an efficient OEF framework based on a Hierarchical Physics-Informed Heterogeneous Graph Neural Network (HPI-HGNN). First, a lossless mapping mechanism transforms physical networks into a heterogeneous graph, utilizing generalized edge attributes and feature projection to resolve dimensional discrepancies. Second, an attention-driven feature fusion algorithm is developed to adaptively evaluate path sensitivities and deeply extract cross-domain features. Finally, a hierarchical constraint mechanism embeds polar-coordinate power flow and thermo-hydraulic balance equations directly into hidden layers. This approach enables mechanism-guided optimization through layer-wise cross-gradient feedback. Simulations on a coupled IEEE 33-bus power and 32-node thermal system show that HPI-HGNN improves prediction accuracy for key variables (voltage magnitude, phase angle, and temperature) by 42.58–78.51% compared to a baseline PINN. Furthermore, it effectively suppresses voltage and temperature limit violations to a near-zero level, ensuring a highly accurate and physically reliable solver for secure online VPP scheduling.