Virtual Smart Metering in District Heating Networks via Heterogeneous Spatial-Temporal Graph Neural Networks

The gist: A heterogeneous spatial-temporal graph neural network (HSTGNN) is proposed to construct virtual smart heat meters by incorporating functional relationships and dedicated branches for flow, temperature, and pressure measurements to achieve joint modeling of cross-variable and spatial correlations.

Motivation

Intelligent operation of thermal energy networks aims to improve energy efficiency, reliability, and operational flexibility through data-driven control, predictive optimization, and early fault detection Achieving these goals relies on sufficient observability, requiring continuous and well-distributed monitoring of thermal and hydraulic states District heating systems are typically sparsely instrumented and frequently affected by sensor faults, limiting monitoring Virtual sensing offers a cost-effective means to enhance observability without extensive hardware deployment Existing data-driven methods generally assume dense synchronized data, while analytical models rely on simplified hydraulic and thermal assumptions that may not adequately capture the behavior of heterogeneous network topologies The lack of publicly available benchmark datasets hinders systematic comparison of virtual sensing approaches.

Model Architecture

The proposed HSTGNN is a three-branch heterogeneous spatial-temporal graph neural network that learns dedicated graph structures and temporal dependencies for each sensor modality. The model incorporates functional relationships inherent in district heating networks and employs dedicated branches to learn graph structures and temporal dynamics for flow, temperature, and pressure measurements. Each branch follows the same processing pipeline: (i) input encoding with sensor-specific information, (ii) temporal modeling at the sensor level, and (iii) spatial modeling using a learned graph. Input encoding involves projecting measurements into a higher-dimensional hidden space using a linear transformation and adding learnable node embeddings associated with each sensor Temporal modeling uses Gated Recurrent Units (GRUs), where sensors of the same type share the same GRU parameters, resulting in one GRU per sensor type. Spatial modeling is achieved using a diffusion-based graph convolution (DGC) that propagates information across the learned graph and allows each sensor to incorporate information from its neighbors.

Cross-Type Interaction

After intra-type processing, all sensor nodes are combined to form a single, unified graph in which every node can interact with nodes of other sensor types. To capture cross-type dependencies, the model applies a self-attention mechanism across this unified graph. This attention layer allows each sensor to selectively attend to other sensors, learning interactions such as how pressure and flow sensors influence one another. The final representations are aggregated into a single feature vector and passed through a linear decoder to produce the predictions.

Dataset and Evaluation

The study introduces a controlled laboratory dataset collected at the Aalborg Smart Water Infrastructure Laboratory, providing synchronized high-resolution measurements of flow, temperature, and pressure under representative operating conditions. The datasets consist of four distinct operating periods defined by different combinations of fan speeds and a control strategy to introduce further variability in the thermal conditions. Performance is evaluated using Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE) across multiple datasets, demonstrating that the proposed HSTGNN consistently outperforms existing baselines. The ablation study confirms that the joint modeling of flow and temperature is essential for accurate inference, while pressure provides additional but comparatively moderate improvements.

Conclusion

The proposed HSTGNN framework effectively leverages complementary information from different sensor types by explicitly accounting for heterogeneous sensor behavior in district heating networks. The extensive evaluation results demonstrate that the proposed method consistently outperforms a range of baseline models. This work is significant because it addresses the challenges posed by sparse instrumentation and provides a reproducible benchmark for future studies through the controlled laboratory dataset. The proposed architecture has some limitations, suggesting that future work should focus on introducing more flexible branch designs and validating the approach in larger operational networks.

How it works

The HSTGNN model is designed to infer unobserved sensor measurements by leveraging the spatial and temporal dependencies inherent in the system. The node set V is partitioned into three disjoint subsets according to sensor type: temperature sensors denoted by Vtemp, pressure sensors by Vpress, and flow sensors by Vflow. Temporal dependencies are captured using sliding windows of length T for each sensor type, with the objective being to learn a mapping from the windowed inputs to the target vector yˆt.

Key Components

The model architecture explicitly accounts for heterogeneous sensor types through three dedicated branches, one for each physical measurement type. Each branch processes its respective sensor type using a GRU to capture local temporal behavior before spatial aggregation. The graph structure within each branch is learned directly from data by associating edges with learnable scores ϕij, allowing for a sparse and learned graph structure.

Performance Insights

Empirical evaluation across four datasets demonstrates that methodologies combining temporal modeling with graph-based spatial modeling (GRU-GCN and HSTGNN) outperform other baselines. The proposed HSTGNN exhibits superior performance for thermodynamic variables, achieving the lowest errors for both inlet and outlet temperatures while GRU-GCN remains competitive for flow rate prediction on certain smart meters. The ablation study confirms that the joint modeling of flow and temperature is essential for accurate inference, while pressure provides additional but comparatively moderate improvements.

Contribution

The main contributions of this work are:

• A three-branch heterogeneous spatial-temporal graph neural network that learns dedicated graph structures and temporal dependencies for each sensor modality.

• A controlled laboratory dataset with synchronized flow, temperature, and pressure measurements, specifically designed for virtual sensing in district heating networks.

• A comprehensive evaluation demonstrating that the proposed framework consistently outperforms conventional machine learning methods, and homogeneous STGNN baselines.

References

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1. The HSTGNN can perform joint reconstruction of unobserved sensor readings by exploiting both cross-variable dependencies and network-level spatial correlations, enabling it to infer missing smart meter values with high accuracy across heterogeneous modalities.

2. The system can achieve superior prediction performance compared to baselines, as demonstrated by the claim that the proposed method consistently outperforms conventional machine learning methods, and homogeneous STGNN baselines.

3. The model can capture modality-specific dynamics by using a three-branch heterogeneous spatial-temporal graph neural network that learns dedicated graph structures and temporal dependencies for each sensor modality, addressing the challenge of distinct physical dynamics and coupling patterns between flow, temperature, and pressure measurements.

4. The HSTGNN can learn complex inter-sensor relationships through a mechanism that allows it to selectively attend to other sensors, regardless of type, specifically learning how different physical quantities influence one another.

5. The system can be robust against data scarcity by being trained on a controlled laboratory dataset collected at the Aalborg Smart Water Infrastructure Laboratory, which provides synchronized high-resolution measurements representative of real operating conditions.

6. The framework can adapt to unknown network structures by modeling spatial dependencies directly from data, as it associate[s] each possible edge within a sensor type with a learnable score ϕij, moving beyond assumptions about fixed topologies.

Intelligent operation of thermal energy networks aims to improve energy efficiency, reliability, and operational flexibility through data-driven control, predictive optimization, and early fault detection. Achieving these goals relies on sufficient observability, requiring continuous and well-distributed monitoring of thermal and hydraulic states. However, district heating systems are typically sparsely instrumented and frequently affected by sensor faults, limiting monitoring. Virtual sensing offers a cost-effective means to enhance observability, yet its development and validation remain limited in practice. Existing data-driven methods generally assume dense synchronized data, while analytical models rely on simplified hydraulic and thermal assumptions that may not adequately capture the behavior of heterogeneous network topologies. Consequently, modeling the coupled nonlinear dependencies between pressure, flow, and temperature under realistic operating conditions remains challenging. In addition, the lack of publicly available benchmark datasets hinders systematic comparison of virtual sensing approaches. To address these challenges, we propose a heterogeneous spatial-temporal graph neural network (HSTGNN) for constructing virtual smart heat meters. The model incorporates the functional relationships inherent in district heating networks and employs dedicated branches to learn graph structures and temporal dynamics for flow, temperature, and pressure measurements, thereby enabling the joint modeling of cross-variable and spatial correlations. To support further research, we introduce a controlled laboratory dataset collected at the Aalborg Smart Water Infrastructure Laboratory, providing synchronized high-resolution measurements representative of real operating conditions. Extensive experiments demonstrate that the proposed approach significantly outperforms existing baselines.