Parameter estimation and analysis for grey-box models of building thermal behavior

Abstract

Reduction of anthropogenic CO2 emissions is one of the most important scientific endeavours of our time. Space heating of buildings is responsible for a considerable portion of the worlds total energy consumption. The Energy Performance of Buildings Directive, issued by the European Union, estimates that approximately 20% of the energy consumption within the EU is the result of heating, ventilation and air conditioning in buildings. Consequently, the reduction of energy consumption in buildings has received significant scientific attention. Towards this goal, methods for creating models of building thermal behaviour is an important subtask.

The first of two main goals within building thermal behaviour modelling research is to create models that can accurately predict future thermal behaviour of buildings. The second, but equally important, goal is construction of models that can be used as classification tools to evaluate the thermal performance based on data collected from a specific building. The former of these goals aims to reduce energy consumption by improved control of temperature thus reducing the amount of energy required to maintain comfortable living conditions. The latter approach is useful towards understanding energy demands of individual buildings, such that the building occupants and owners can make qualified decisions on what energy conserving measures to implement, and also for the authorities to compose taxation schemes based on energy efficiency.

Modelling building thermal behaviour is challenging due to the complex nature of buildings, i.e., use of a wide variety of materials and different building geometries. Further, the physical buildings often does not match the building specifications and blueprints, due to workmanship issues or continued modification and renovation of existing buildings. Additionally, weather conditions and occupant demands makes experimental design difficult. Because of these inherent uncertainties involved in building thermal modelling it is useful to formulate such models as stochastic differential equations. This type of models, often called grey-box models, allows the combination of prior expert knowledge with parameters that are calibrated to fit a specific building. This approach produces models that tends to provide good prediction accuracy for future behaviour while also being interpretable by humans. The stochastic modelling framework has a strong mathematical foundation which provides a framework that can be used to estimate parameters, analyse estimation uncertainty, and to perform model selection and validation. The grey-box modelling framework also fits naturally with the Bayesian statistics framework and the Markov Chain Monte Carlo methods, which has gained popularity over the recent years.

Grey-box models of building thermal behaviour are typically simplified description of the physics involved. Since the models are constructed using prior system specific knowledge, the parameters are often cognitively connected to the thermal properties of the physical buildings, i.e., the model parameters are used as soft-sensors. However, interpreting model parameters as representative of the physical properties of the building requires a careful analysis of the parameter identifiability to ensure that the calibrated parameters are unambiguous, and to estimate the uncertainty of the obtained parameters.

In this thesis, the stochastic modelling framework is combined with Kalman filter implementations that does not require differentiable models. This allows estimation of parameters for externally simulated models which facilitates experimentation with model structures. Further, the grey-box parameter estimation uncertainty is analysed using several different methods, including the Profile Likelihood framework, and the extended Profile Posterior method. Both profiling methods are extended to create 2D profiles which allows more detailed identifiability analysis of the parameter space. The Profile Posterior method is compared to the results obtained using Markov Chain Monte Carlo methods. The combination of model formulation as stochastic differential equations with Markov Chain Monte Carlo methods offers a particularly powerful and efficient model calibration framework, which can be utilised also for calibration of external software simulations.

The use of stochastic model formulations is applicable to a wide range of modelling challenges. Given that almost every conceivable model is in some way an approximation of the real system, the stochastic differential equation parameter estimation framework has been argued as a natural framework for modelling dynamic system models in general. The benefits of performing model calibration utilising a framework with a solid statistical foundation that provides tools for model validation and parameter identifiability analysis well out-ways the complexities of the methods involved.