Grey-Box Modeling of Building Thermal Dynamics

Research Question

How can the thermal dynamics of a real school building be identified from measurement data using a grey-box model that combines physical prior knowledge with data-driven parameter estimation, while quantifying the uncertainty in the identified model?

Method

This project applied grey-box modeling to identify the heat dynamics of a school building from on-site measurement data. Grey-box models combine:

• Physical prior knowledge — a lumped-parameter thermal network model (resistor-capacitor equivalent) encoding building physics: heat capacity of indoor air and building fabric, thermal resistance of walls and windows, heat exchange between the radiator-water circuit, indoor air, and building.
• Stochastic differential equations — the physical model is formulated as a continuous-time stochastic state-space system, where the SDE framework accounts for model uncertainty and unmeasured disturbances.
• Maximum likelihood estimation — parameters of the grey-box model are estimated from measurement data using statistical inference, producing both point estimates and uncertainty intervals.

The approach is distinct from black-box machine learning: the model structure is physically interpretable: its states and parameters retain a relationship to heat storage and transfer in the radiator and building.

Personal Contribution

Frederik Banis was the first author of the conference paper. He contributed to the grey-box model formulation, the statistical estimation framework, and the analysis of measurement data from the school building. Co-authors contributed supervision, building physics expertise, and methodological guidance.

Validation and Key Results

• The grey-box model successfully captured the dominant thermal dynamics of the school building, with physically interpretable parameter estimates for thermal resistance and capacitance.
• Models of increasing order were fitted by maximum likelihood to measurements from an unoccupied school during a holiday period, with ventilation off.
• The 4R3C structure captured the dominant behavior; adding states in the 5R4C and 6R5C variants did not provide a visible improvement in the reported fit.
• Out-of-sample validation and deployment in model-based control remained future work.

The method was published in the SINTEF Proceedings (2020), a peer-reviewed conference proceedings series.

Outputs

• Conference paper: SINTEF Proceedings, Oslo, 2020. Handle: 11250/2684084
• Related software: The SINDYc and MCMC framework (Zenodo: 10.5281/zenodo.3911952) extends the uncertainty-aware system identification methodology developed in this project.

Collaborators

• DTU Compute, Technical University of Denmark — Henrik Madsen, Niels Kjølstad Poulsen, Peder Bacher (supervisors and co-authors)
• Christian Anker Hviid (co-author, building physics)
• Hjörleifur G. Bergsteinsson, Davide Cali (co-authors)

Status and Next Steps

Status: Published (2020). The grey-box modeling methodology is a foundational technique in the research program, demonstrating how physical prior knowledge and statistical inference combine for interpretable, uncertainty-aware system identification.

Transfer to current research: The same grey-box philosophy — encode known physics into the model structure, then estimate remaining parameters and test the resulting model — now informs the radar forward models for physiological monitoring. The shift is from building thermal dynamics to electromagnetic propagation physics, but the inferential architecture of physics-constrained, uncertainty-aware identification is shared.

See the Methods page for the uncertainty-aware inference pillar and the Research page for the broader research program.