Physically Informed Data-Driven Identification of District Heating Network under Missing Node Measurements: A Virtual-Node-based Method

Accurate and concise modeling of district heating networks (DHNs) is essential for reliable operation and control, but it remains challenging in practical engineering where measurements at some source or load nodes are completely unavailable. To address this knowledge-intensive modeling problem, we propose a virtual-node-based aggregate model (VN-AGM) and a physically informed data-driven identification method for DHNs under missing node measurements. The central contribution is an explicit and interpretable representation of incompletely observed DHNs, in which unmeasured load or source nodes are equivalently aggregated into a single virtual node while preserving the thermal mapping and hydraulic consistency of the original network. First, we derive the VN-AGM for the DHN with unmeasured load or source nodes, revealing that the load (or source) nodes without measurements can be equivalently aggregated into a virtual node. Second, we formulate the parameter identification model of the VN-AGM, the structural identifiability of which is then analyzed to clarify how physical knowledge supports reliable model construction under limited sensing conditions. Third, the mass flow rate distribution relationship between source and load nodes is embedded as a reasoning constraint, enabling parameter identification without requiring additional measurements beyond the available field data. Finally, we develop a two-stage lasso regression-based parameter estimator for the VN-AGM, which embeds the structural constraints of DHN to improve the robustness under practical measurement errors. Case studies on 51-node and 226-node DHNs demonstrate that the proposed method provides a scalable and practically deployable tool for supporting DHN modeling, operation, and control under incomplete observability.