Impact of the controller model complexity on model predictive control performance for buildings

Model predictive control (MPC) for heating, ventilation and air conditioning (HVAC) in buildings requires accurate controller models of the building envelope and its HVAC systems. Controller models are typically obtained by means of black- or grey-box system identification or using a white-box modelling approach. However, the necessary level of model complexity used by each method in order to obtain good MPC performance remains a priori unknown and no systematic method or examples showing the optimal complexity is available. This paper systematically investigates the required controller model complexity necessary to obtain optimal control performance for a given building. First, a 6-room house is modeled in detail using building energy simulation software. The building model is then linearized to obtain a linear time invariant (LTI) state-space model (SSM) and the upper bound of the control performance is computed using an MPC with the SSM both as controller and as plant model. The accuracy of the SSM (containing more than 250 states) is then artificially decreased by reducing its number of states to different orders ranging from 4 to 100 using balanced truncation model order reduction technique. The performances of MPCs using these controller models are then compared with the upper bound for both a standard MPC formulation (S-MPC) and an offset-free formulation (OSF-MPC) and with the performance of a rule-based-controller (RBC). The procedure is repeated for the same house model with a higher level of insulation and for a lighter weight construction. This paper shows that the controller model should contain a minimum of states to model each zone separately, and that the walls and floors separating the zones should also have enough states to act as a low pass filter with correct cut-off frequency. The minimum number of states further increases with the building mass content. In the case of the investigated 6-room house, the thermal comfort achieved by MPC using a controller model with a minimum of 30 states instead of 20 states was improved with a factor 2-6 without significant increase of the energy use, showing that good MPC performances require controller models with a significantly higher number of states than the order used by most of the black- and grey-box system identification techniques. The minimum required number of states might be chosen lower when OSF-MPC is used instead of conventional MPC. However, OSF-MPC might significantly increase the energy use when poor controller models (high model mismatch) are used. Furthermore, if the controller model is an LTI model, this paper shows that the CPU time necessary to solve the MPC optimization problem becomes independent of the number of states of the controller model when a dense approach is used. The controller model can thus be as complex as necessary to produce accurate predictions without increasing the computation time of the optimization.