Causality issue in the heat balance method for calculating the design heating and cooling load

Thermal load calculation based on dynamic models is widely used in simulation software and it is the method recommended by ASHRAE (American Society of Heating, Refrigeration and Air-Conditioning Engineers) and CEN (European Committee for Standardization, in French: Comité Européen de Normalisation). The principle is to make the heat balance for the air volume of a room space considered at uniform temperature and to calculate from this equation the load, i.e. the power needed to obtain the required indoor temperature. The problem is that, by doing so, the physical causality is not respected. If the model is approximated by a piece-wise linear dynamical system, this procedure results in an improper transfer function in the case of non-negligible thermal capacity of the indoor air. In order to point out this problem, a method to obtain state-space and transfer function models from thermal networks is introduced. Then, the transfer function representation is employed to show that changing the physical causality results in an improper transfer function. The practical consequence is that when the space temperature has a step variation, the calculated load tends to infinity if the simulation time step approaches zero. The issue of causality may be a problem in equation-based simulation software, such as Modelica, in which the equations do not represent causal relations: a wrong choice of the causality in a balance equation may result in an improper transfer functions.