Optimal linearization of complex buildings envelope

Optimal climate control for building systems is facilitated by linear, low-order models of the building and of its heat, ventilation and air conditioning systems (HVAC). The difficulty of obtaining such linear models greatly hampers the commercial implementation of these optimal controllers. This work focuses on obtaining a linear controller model of the building envelope, consisting of the walls, windows, floors, etc. but also the air present in the building. We present a work-intensive method on how to obtain the best possible linear model and we quantify its Np-steps ahead prediction performance. The obtained models can be used as benchmark for linear models obtained by for example system identification or for investigating the influence of the prediction performance on a model based controller. Furthermore, the method greatly automates the setting up of a model-based control method in a Modelica© simulation environment. The following paragraphs briefly pinpoint the non-linearities presents in the building envelope and outline the proposed method. The non-linearities in a building envelope are mainly due to the Stefan-Boltzmann law for long- and short-wave radiation where the radiative heat transfer of a body is proportional to its absolute temperature to the fourth power. Other non-linearities are the absorption and transmission of radiation through glazing and the convective heat transfer which is a non-linear function of the temperature. Finally, the partial differential equation of heat transport in a solid is three-dimensional and time dependent and its boundary conditions are non-linear. In this work, we built a detailed building envelope model in Modelica© using the IDEAS-library taking these non-linearities into account except for the heat transfer equation (a partial differential equation) which is simplified to ordinary differential equations with a finite number of parameters representing only one-dimensional heat transport. The model is based on the dimensions and the physical properties of a real building. The obtained model is then linearized around a chosen operating point using its equations implemented in Modelica© directly, resulting into a state-space model of hundreds of states, their initial value and their initial derivative. Its inputs are the convective, conductive and radiant heat supply by the HVAC system and its outputs can be any states of the original model. Finally, we apply a model reduction technique to reduce the number of states to 30 and we compute the Np-steps ahead prediction performance of the reduced-order model. Since prediction performance requires an observer for state estimation, state-estimators (Kalman Filter or Luenberger observer) are designed based on the reduced-order model obtained from the large-scale linear model. The obtained model is the best linear approximation of the non-linear model around the chosen working point as it is uniquely derived from the original equations. The model can be used to benchmark controller models obtained by system identification on the original model or it can be used directly into a model-based controller implementation.status: publishe