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

Keivan Faghih Niresi; Christian Møller Jensen; Carsten Skovmose Kallesøe; Rafael Wisniewski; Olga Fink

cs.LGApr 11, 2026

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

Authors: Keivan Faghih Niresi, Christian Møller Jensen, Carsten Skovmose Kallesøe, Rafael Wisniewski, Olga Fink

Organizations: Intelligent Maintenance and Operations Systems Lab., EPFL, Lausanne, Switzerland · Department of Electronic Systems, Aalborg University, Aalborg, Denmark · Grundfos Holding A/S, Denmark

Abstract

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.

Figures & tables

Figure 1: Schematic layout of the district heating network under study. The tree-structured graph illustrates the symmetrical relationship between the supply and return networks, where edge directions are inverted to reflect flow orientation.

Figure 2: Schematic block diagram of the experimental topology. The system is connected in a tree structure, linking the heat station to consumers.

Figure 3: Hardware implementation of the experimental setup at Aalborg SWIL.

Figure 4: Modules at Aalborg Smart Water Infrastructures Laboratory. For the pipe units, the following input–output pairs are considered: (I/O1_1, I/O1_2), (I/O2_1, I/O2_3), (I/O3_1, I/O3_2), and (I/O4_1, I/O4_3).

Component	Temperature	Pressure	Flow
Heat Station	Boiler_T1	Boiler_dP2	Boiler_Q2_1
	Boiler_T2		Boiler_Q4_6
	Boiler_T3
	Boiler_T4
	Boiler_T5
	Boiler_T6

Table 1: Overview of the available system input sensors used as standard SCADA measurements

Variable	Range (5th – 95th)	Unit
Temperature	16.89 – 49.93	°C
Flow	0.09 – 12.59	lit/min
Pressure	3.27 – 10.71	bar

Table 2: System Operating Ranges (5th – 95th Quantile)

Figure 5: The Proposed HSTGNN Architecture for Virtual Sensing

Hyperparameter	Value
Window Size	$\{4,8,\textbf{16},32,64\}$
Stride	$1$
Number of GNN Layers	$\{\textbf{1},2,3,4\}$
Output Channels for GNN Layers	$\{4,8,\textbf{16},32,64\}$
GNN Kernel Size	$\{1,\textbf{2},3,4\}$
Node Embedding Size	$\{4,8,\textbf{16},32,64\}$

Table 3: Summary of the hyperparameter search space and optimal configurations. Values selected via grid search on the primary dataset and utilized across all experiments are indicated in bold.

	Sensor	Metric	Methods
	Sensor	Metric	GRU-GCN	LSTM	1D-CNN	DGC	GCN	HSTGNN
SM1	Flow Rate ( $\mathrm{l/min}$ )	RMSE	0.1969 $\pm$ 0.0017	0.1749 $\pm$ 0.0011	0.2416 $\pm$ 0.0022	0.1934 $\pm$ 0.0007	0.1940 $\pm$ 0.0013	0.1856 $\pm$ 0.0009
	Flow Rate ( $\mathrm{l/min}$ )	MAE	0.1157 $\pm$ 0.0035	0.1062 $\pm$ 0.0017	0.1561 $\pm$ 0.0006	0.1170 $\pm$ 0.0009	0.1180 $\pm$ 0.0026	0.1017 $\pm$ 0.0015
	Inlet Temperature ( ${}^{\circ}\mathrm{C}$ )	RMSE	0.1868 $\pm$ 0.0114	0.1723 $\pm$ 0.0093	0.1682 $\pm$ 0.0074	0.5080 $\pm$ 0.0124	0.5025 $\pm$ 0.0202	0.1243 $\pm$ 0.0045
	Inlet Temperature ( ${}^{\circ}\mathrm{C}$ )	MAE	0.1398 $\pm$ 0.0073	0.1203 $\pm$ 0.0051	0.1270 $\pm$ 0.0031	0.3978 $\pm$ 0.0080	0.3986 $\pm$ 0.0089	0.0941 $\pm$ 0.0022
	Outlet Temperature ( ${}^{\circ}\mathrm{C}$ )	RMSE	0.1976 $\pm$ 0.0051	0.2692 $\pm$ 0.0186	0.3301 $\pm$ 0.0281	0.6480 $\pm$ 0.0009	0.6478 $\pm$ 0.0111	0.1852 $\pm$ 0.0169

Table 4: Dataset 1: RMSE and MAE comparison across methods (mean

\pm

std over 3 runs)

	Sensor	Metric	Methods
	Sensor	Metric	GRU-GCN	LSTM	1D-CNN	DGC	GCN	HSTGNN
SM1	Flow Rate ( $\mathrm{l/min}$ )	RMSE	0.1561 $\pm$ 0.0030	0.1689 $\pm$ 0.0030	0.2487 $\pm$ 0.0014	0.1668 $\pm$ 0.0024	0.1646 $\pm$ 0.0020	0.1680 $\pm$ 0.0048
	Flow Rate ( $\mathrm{l/min}$ )	MAE	0.0806 $\pm$ 0.0046	0.0952 $\pm$ 0.0018	0.1493 $\pm$ 0.0012	0.0910 $\pm$ 0.0045	0.0869 $\pm$ 0.0027	0.0856 $\pm$ 0.0035
	Inlet Temperature ( ${}^{\circ}\mathrm{C}$ )	RMSE	0.3384 $\pm$ 0.0059	0.2024 $\pm$ 0.0256	0.2261 $\pm$ 0.0169	0.5713 $\pm$ 0.0011	0.5746 $\pm$ 0.0008	0.2003 $\pm$ 0.0135
	Inlet Temperature ( ${}^{\circ}\mathrm{C}$ )	MAE	0.2415 $\pm$ 0.0018	0.1512 $\pm$ 0.0201	0.1623 $\pm$ 0.0108	0.4463 $\pm$ 0.0071	0.4499 $\pm$ 0.0088	0.1471 $\pm$ 0.0132
	Outlet Temperature ( ${}^{\circ}\mathrm{C}$ )	RMSE	0.2351 $\pm$ 0.0125	0.2849 $\pm$ 0.0121	0.2848 $\pm$ 0.0053	0.5952 $\pm$ 0.0149	0.5957 $\pm$ 0.0050	0.1987 $\pm$ 0.0134

Table 5: Dataset 2: RMSE and MAE comparison across methods (mean

\pm

std over 3 runs)

	Sensor	Metric	Methods
	Sensor	Metric	GRU-GCN	LSTM	1D-CNN	DGC	GCN	HSTGNN
SM1	Flow Rate ( $\mathrm{l/min}$ )	RMSE	0.1447 $\pm$ 0.0010	0.2244 $\pm$ 0.0215	0.2450 $\pm$ 0.0090	0.1586 $\pm$ 0.0018	0.1572 $\pm$ 0.0068	0.1439 $\pm$ 0.0031
	Flow Rate ( $\mathrm{l/min}$ )	MAE	0.0762 $\pm$ 0.0005	0.1516 $\pm$ 0.0216	0.1641 $\pm$ 0.0112	0.0930 $\pm$ 0.0015	0.0903 $\pm$ 0.0095	0.0781 $\pm$ 0.0059
	Inlet Temperature ( ${}^{\circ}\mathrm{C}$ )	RMSE	0.2931 $\pm$ 0.0211	0.2465 $\pm$ 0.0112	0.2544 $\pm$ 0.0144	0.6284 $\pm$ 0.0258	0.6302 $\pm$ 0.0069	0.2247 $\pm$ 0.0132
	Inlet Temperature ( ${}^{\circ}\mathrm{C}$ )	MAE	0.2240 $\pm$ 0.0165	0.1834 $\pm$ 0.0146	0.1852 $\pm$ 0.0096	0.4372 $\pm$ 0.0213	0.4391 $\pm$ 0.0011	0.1565 $\pm$ 0.0021
	Outlet Temperature ( ${}^{\circ}\mathrm{C}$ )	RMSE	0.2369 $\pm$ 0.0118	0.2983 $\pm$ 0.0132	0.2661 $\pm$ 0.0139	0.4738 $\pm$ 0.0107	0.4543 $\pm$ 0.0254	0.2320 $\pm$ 0.0136

Table 6: Dataset 3: RMSE and MAE comparison across methods (mean

\pm

std over 3 runs)

	Sensor	Metric	GRU-GCN	LSTM	1D-CNN	DGC	GCN	HSTGNN
	Sensor	Metric	Methods
SM1	Flow Rate ( $\mathrm{l/min}$ )	RMSE	0.1934 $\pm$ 0.0006	0.1908 $\pm$ 0.0016	0.2424 $\pm$ 0.0009	0.2137 $\pm$ 0.0012	0.2127 $\pm$ 0.0005	0.1803 $\pm$ 0.0054
	Flow Rate ( $\mathrm{l/min}$ )	MAE	0.1018 $\pm$ 0.0002	0.1208 $\pm$ 0.0050	0.1492 $\pm$ 0.0013	0.1192 $\pm$ 0.0024	0.1187 $\pm$ 0.0004	0.0974 $\pm$ 0.0010
	Inlet Temperature ( ${}^{\circ}\mathrm{C}$ )	RMSE	0.2355 $\pm$ 0.0006	0.2478 $\pm$ 0.0020	0.2365 $\pm$ 0.0086	0.5202 $\pm$ 0.0235	0.5119 $\pm$ 0.0093	0.1722 $\pm$ 0.0126
	Inlet Temperature ( ${}^{\circ}\mathrm{C}$ )	MAE	0.1838 $\pm$ 0.0004	0.1976 $\pm$ 0.0035	0.1884 $\pm$ 0.0060	0.3986 $\pm$ 0.0251	0.3909 $\pm$ 0.0080	0.1292 $\pm$ 0.0102
	Outlet Temperature ( ${}^{\circ}\mathrm{C}$ )	RMSE	0.1846 $\pm$ 0.0029	0.2519 $\pm$ 0.0233	0.2407 $\pm$ 0.0298	0.4392 $\pm$ 0.0044	0.4400 $\pm$ 0.0044	0.1656 $\pm$ 0.0170

Table 7: Dataset 4: RMSE and MAE comparison across methods (mean

\pm

std over 3 runs)

Figure 6: Outlet temperature prediction for SM 1, Dataset 1. HSTGNN closely follows the true signal; GRU-GCN is second best; graph-only models show larger deviations.

Figure 7: Inlet temperature for SM 1, Dataset 2. HSTGNN is most accurate; temporal-only and graph-only models deviate during transients.

Figure 8: Inlet temperature for SM 2, Dataset 3. HSTGNN tracks the measured signal well; GRU-GCN captures general trends.

Figure 9: Flow rate for SM 1, Dataset 4. HSTGNN and GRU-GCN perform best; graph-only models outperform temporal-only baselines.

	Sensor	Metric	Ablation Variants
	Sensor	Metric	Full Model	w/o Pressure	w/o Flow	w/o Temperature
SM1	Flow Rate ( $\mathrm{l/min}$ )	RMSE	0.1856	0.1863	0.3439	0.1859
	Flow Rate ( $\mathrm{l/min}$ )	MAE	0.1017	0.1021	0.2162	0.1090
	Inlet Temperature ( ${}^{\circ}\mathrm{C}$ )	RMSE	0.1243	0.1598	0.2031	0.7013
	Inlet Temperature ( ${}^{\circ}\mathrm{C}$ )	MAE	0.0941	0.1247	0.1522	0.5139
	Outlet Temperature ( ${}^{\circ}\mathrm{C}$ )	RMSE	0.1852	0.2040	0.2760	0.7307

Table 8: Ablation study of HSTGNN by removing sensor groups (Dataset 1)

	Sensor	Metric	HSTGNN	Simplified
	Sensor	Metric	Models
SM1	Flow Rate ( $\mathrm{l/min}$ )	RMSE	0.1803	0.2124
	Flow Rate ( $\mathrm{l/min}$ )	MAE	0.0974	0.1259
	Inlet Temperature ( ${}^{\circ}\mathrm{C}$ )	RMSE	0.1722	0.3734
	Inlet Temperature ( ${}^{\circ}\mathrm{C}$ )	MAE	0.1292	0.2809
	Outlet Temperature ( ${}^{\circ}\mathrm{C}$ )	RMSE	0.1656	0.3658

Table 9: Ablation study on Dataset 4: HSTGNN compared to a minimal variant without heterogeneous branching, sensor-type spatial modeling, or sensor-specific embeddings.

Figure 10: Learned top- $K$ functional adjacency matrices for the temperature and pressure branches. Rows denote receiving sensors and columns denote source sensors. Non-zero entries correspond to top- $4$ dependencies selected from the learned score matrices. The learned graphs are interpreted as data-driven functional dependency structures rather than direct reconstructions of the pipe topology.

Query / Key	Temperature	Pressure	Flow
Temperature	0.326	0.225	0.449
Pressure	0.270	0.454	0.276
Flow	0.306	0.299	0.395

Table 10: Global row-normalized type-level block-mean attention. Rows denote query sensor types and columns denote key sensor types. Values are computed after type-specific temporal and graph encoding and are interpreted as information-routing diagnostics rather than causal physical dependencies.

Virtual Sensing

Virtual sensing estimates unavailable or costly-to-measure quantities from other sensor data, physical models, and learned relationships.

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Apr 22, 2026J/K move · Enter open · S save

Robustness of Spatio-temporal Graph Neural Networks for Fault Location in Partially Observable Distribution Grids

Burak Karabulut, Carlo Manna, Chris Develder

Ghent University – imec · VITO

Anomaly Localization Graph Neural Networks Graph Construction Graph Attention Networks Power Systems Spatiotemporal GNNs+1

Open paper AskPDF arXiv

Fault location in distribution grids is critical for reliability and minimizing outage durations. Yet, it remains challenging due to partial observability, given sparse measurement infrastructure. Recent works show promising results by combining Recurrent Neural Networks (RNNs) and Graph Neural Networks (GNNs) for spatio-temporal learning. Still, many modern GNN architectures remain untested for this grid application, while existing GNN solutions have not explored GNN topology definitions beyond simply adopting the full grid topology to construct the GNN graph. We address these gaps by (i) systematically comparing a newly proposed graph-forming strategy (measured-only) to the traditional full-topology approach, and (ii) introducing STGNN (Spatio-temporal GNN) models based on GraphSAGE and an improved Graph Attention (GATv2), for distribution grid fault location; (iii) benchmarking them against state-of-the-art STGNN and RNN baselines on the IEEE 123-bus feeder. In our experiments, all evaluated STGNN variants achieve high performance and consistently outperform a pure RNN baseline, with improvements up to 11 percentage points F1. Among STGNN models, the newly explored RGATv2 and RGSAGE achieve only marginally higher F1 scores. Still, STGNNs demonstrate superior stability, with tight confidence intervals (within +/- 1.4%) compared to the RNN baseline (up to +/- 7.5%) across different experiment runs. Finally, our proposed reduced GNN topology (measured-only) shows clear benefits in both (i) model training time (6-fold reduction) and (ii) model performance (up to 11 points F1). This suggests that measured-only graphs offer a more practical, efficient, and robust framework for partially observable distribution grids.

Abstract

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