System Approach to Control-Oriented Mathematical Modeling of Thermal Processes of Buildings

Abstract

A heated building is presented as a complex system consisting of elements separated from each other and from the environment by heat-conducting surfaces. The thermal process of a building consists of heat and mass transfer processes in the walls of the building elements and associated changes in the parameters of the internal air. The given mathematical model is a system of ordinary differential equations for internal air and partial differential equations that model the processes of heat transfer through internal and external walls. It is proposed to consider internal and external walls in the form of a discrete system of interconnected heat-transferring flat elements. This allows us to represent a mathematical model of the thermal process of a building in the form of a finite-dimensional system of ordinary linear differential equations, focused on the application of methods of modern control theory. To assess the adequacy of a finite-dimensional model the analytical solution of the problem of two-sided symmetric heat supply is compared with the numerical solution of the corresponding system of ordinary differential equations at various degrees of discreteness. The task of reducing a multilayer wall with a uniform wall with a minimum number of layers has been set and solved for multilayer walls. Computational experiments have shown high accuracy of the two-dimensional reduced model under harmonic perturbations of the ambient temperature.