Published in Energy Informatics, the study introduces a task-driven approach that identifies specific groups of substations suitable for replacement by machine learning surrogate models. This hybrid simulation technique allows engineers to reduce the computational burden of complex thermo-hydraulic models while maintaining the fidelity required for operational planning and optimization.
District heating networks are increasingly complex as they evolve toward fourth and fifth-generation systems. These modern networks feature decentralized, ultra-low-temperature, and bi-directional topologies that integrate renewable energy sources and distributed thermal storage. While these configurations offer significant potential for decarbonizing the heat supply sector, they introduce intricate physical inter-dependencies. High-fidelity dynamic simulations of such networks are computationally intensive, with costs growing non-linearly as the number of network components increases. For instance, a pseudo-dynamic simulation of a meshed network with more than 162 nodes can require over four hours of computation for a two-week period.
The core challenge in reducing this computational cost lies in how the network is simplified. Previous approaches often relied on model order reduction or empirical aggregation methods, which can compromise the preservation of thermo-hydraulic dynamics or lack generalizability under changing boundary conditions. The authors propose a novel solution by focusing on the definition of substation clusters. They argue that the efficacy of replacing these clusters with machine learning surrogates is highly sensitive to the clusters’ definition. Poorly selected clusters degrade the accuracy of the surrogate models, thereby undermining the overall performance of the hybrid simulation.
The proposed methodology employs physics-informed distance metrics to guide the clustering process. These metrics encode the primary drivers of machine learning surrogate model errors, specifically the number of water temperatures to predict and the temporal evolutions of these temperatures. The team defines three task-driven distances: topology distance, which estimates the cut size of a cluster; temperature distance, which measures total heat transport losses; and transport distance, which quantifies the delay time of heat transport between adjacent nodes. These distances are combined into a single clustering metric using weighting factors that allow for the integration of specific operational requirements.
The clustering process utilizes a hierarchical agglomerative framework. It begins by treating each node as a single-node cluster and iteratively merges the closest clusters based on the defined distance metrics. The process continues until the minimum clustering distance between any two clusters exceeds a defined threshold. This threshold, along with the weighting factors for topology, temperature, and transport, serves as a hyperparameter that directly influences the resulting cluster complexity. By adjusting these parameters, users can calibrate the balance between physical fidelity and computational speed according to their specific application needs.
To evaluate the methodology, the researchers analyzed 16 distinct district heating networks with diverse topological and thermal characteristics. These networks varied in size, connectivity, and thermal loss coefficients, ranging from well-insulated modern pipes to poorly insulated configurations. The team systematically varied the clustering hyperparameters across their full range to assess their impact on cluster quality. They used Pareto front analysis to identify hyperparameter combinations that offered the best trade-offs between spatial reduction and physical accuracy, measured by mean absolute error, energy error, and energy error relative to losses.
The analysis revealed a clear, generalizable trade-off between simulation accuracy and computational time reduction. The results indicated that the degradation in surrogate model accuracy is not a direct function of cluster size but rather a consequence of the increased complexity within the formed clusters. As the distance threshold increases, the clustering process allows for the aggregation of more nodes, creating clusters with higher cut sizes, larger thermal loss profiles, and longer transport delay times. This inherent complexity impacts the learning capacity of the surrogate models, leading to increased errors in the hybrid simulations.
The methodology was further validated on an independent district heating network containing 123 nodes and three heat producers. This validation phase tested the generalizability of the approach on a previously unseen network. The team compared the performance of the reduced network against the full-scale physical simulation over a 24-hour period. The original network exhibited a total thermal production of 2.0626 GWh, with thermal losses of 171.54 MWh, representing approximately 8% of the total heat produced. The full physical simulation of this network required 636 seconds to complete.
Two distinct user preferences were tested during validation: a preference for minimum energy error relative to losses and a preference for minimum remaining nodes. The high physical accuracy preference yielded a deviation of 6.45 MWh in total thermal energy production estimation, which corresponds to 3.76% of the total thermal losses within the network. However, this high accuracy came with modest computational gains, reducing the simulation time by only 7% and the network size by 16%. In contrast, the high spatial reduction preference reduced the network from 123 to 5 remaining physical nodes and achieved a 91% decrease in computational time.
While the high spatial reduction preference offered significant speed improvements, it resulted in a 45.24 MWh deviation in total thermal energy production estimation. This error represents 26.37% of the network’s thermal losses. Such high relative errors may render downstream tasks, such as operational optimization, unreliable due to the substantial misestimation of heat production relative to the network’s inherent thermal loss. The study highlights that expressing errors relative to thermal losses, rather than total production, provides a more meaningful measure of the spatial reduction’s impact on key operational objectives.
The findings demonstrate that the proposed clustering framework provides a robust and flexible tool for calibrating the balance between physical fidelity and computational speed. The 91% reduction in simulation time achieved under the high spatial reduction preference brings the simulation time down to 56 seconds, making iterative optimization computationally feasible for strategic applications such as network planning or extension studies. In these contexts, the ability to rapidly simulate a significant number of scenarios is often prioritized over the high-fidelity precision required for real-time control.
A current limitation of the methodology is that the distance metrics guiding the clustering are static and derived from physical states assumed known a priori. Future work will target the development of online adaptive distance metrics capable of adjusting selected clusters in real-time to fluctuating operating conditions. The approach is most beneficial for networks where the baseline simulation time significantly constrains the target application, such as in Model Predictive Control or operation optimization. For small-scale networks or limited simulation requirements, the overhead of surrogate model development may not be justified, but for large-scale, complex systems, the proposed method offers a viable path to efficient digital twin implementation.
The study, published in Energy Informatics, was conducted by a team including Dubon Rodrigue, Mohamed Tahar Mabrouk, Bastien Pasdeloup, Patrick Meyer, and Bruno Lacarrière. The research addresses a specific gap in the application of machine learning to district heating networks. While previous studies demonstrated that replacing substation clusters with surrogate models could reduce simulation time, they did not fully resolve how to optimally identify the most meaningful clusters for substitution. Existing graph clustering methods for these networks typically focused on geographical proximity or demand-profile similarity, which are not designed to support the objective of training accurate cluster-level surrogate models. This work introduces a methodology that explicitly targets the requirements of ML-based spatial reduction by incorporating thermo-hydraulic dynamics into the clustering process.
The physical data used to train and validate the models were generated using a pseudo-dynamic simulation model. This model combines steady-state hydraulic simulation to compute mass flow rates and pressure losses with dynamic thermal simulation to simulate the temporal evolutions of water temperatures. Each of the 17 distinct district heating networks considered in the study was simulated for 3,945 hours, spanning from January to June. This time window captures seasonal variability in both network dynamics and heating demands. Heat producer supply temperatures, ranging between 90 and 100 degrees Celsius, were set to follow a heating law that varies proportionally with outdoor conditions. For networks with multiple producers, dispatch was managed by ranking producers according to assigned production costs.
The surrogate models employed in the hybrid simulation utilize a many-to-one Recurrent Neural Network architecture based on Gated Recurrent Units. This architecture was selected because it performs best in capturing the temporal dependencies required to model thermal inertia and transport delays within the replaced clusters. Standard feed-forward architectures cannot retain memory of past inputs, while Transformers, although capable of retaining memory, introduce greater complexity that complicates training and slows inference. The specific architecture includes two GRU layers with 20 memory units and hidden dense layers with 60 neurons. Training uses the Adam optimizer with a step-decreasing learning rate, up to 100 epochs, and a batch size of 128. This architecture was previously evaluated against alternatives, including Convolutional Neural Networks and Multi-Layer Perceptrons, and was found to consistently yield the best prediction accuracy across varying cluster topologies and demand levels.
The evaluation of the clustering methodology was conducted through two successive phases: analysis and validation. In the analysis phase, the researchers examined the clustering outcomes for 16 of the 17 studied networks. They systematically varied the hyperparameters across their full range to provide insight into clustering performances and how the outcomes depend on these parameters. This exhaustive process helped identify general trends and select optimal hyperparameters supporting different user preferences. The validation phase was performed on the left-out network, which contains 123 nodes and three heat producers. This phase tested the generalizability of the approach on a previously unseen network, ensuring that the identified hyperparameters remained relevant outside the training set.
From a practical standpoint, the ML-based spatial reduction approach involves an upfront implementation effort, including training data generation, surrogate model training, and hyperparameter selection. This overhead is amortized over repeated use of the reduced network in operational contexts. The approach is most beneficial when the network must be simulated numerous times under varying boundary conditions and different operating strategies, as is the case in Model Predictive Control or operation optimization. For small-scale networks where a single simulation already completes in seconds, or when only a limited number of simulations are required, the overhead of surrogate model development may not be justified. The Pareto fronts presented in the study can serve as a decision-support tool for practitioners to evaluate this trade-off for their specific network and application.
Subject of Research: Graph clustering methodology for machine learning-based spatial reduction in district heating network simulation
Article Title: Flexible clustering of substations for accurate and rapid hybrid simulation of district heating
Article References: Rodrigue, D., Mabrouk, M. T., Pasdeloup, B., Meyer, P., & Lacarrière, B. (2026). Flexible clustering of substations for accurate and rapid hybrid simulation of district heating. Energy Informatics, 9(1), Article 60. https://doi.org/10.1186/s42162-026-00664-3
Image Credits: AI Generated
DOI: 10.1186/s42162-026-00664-3
Keywords: district heating, graph clustering, machine learning, surrogate models, hybrid simulation, energy informatics, network optimization, thermal dynamics, Flexible, clustering, substations, accurate
News Source: Denise Maddox. (November 10, 2026). New Clustering Method Balances Speed and Accuracy in District Heating Simulation. Scienmag.
