Modeling and simulation#
This chapter presents the modeling and simulation of thermal behavior of a multi-layer wall and of a toy model of a house.
The workflow is:
Modelling: obtain the thermal circuit and the state-space representation.
Draw the thermal network based on the description of the building and on modelling assumptions.
Calculate the parameters of the thermal network: conductances and capacities.
Name the temperature and flow-rate sources.
Chose the temperature nodes considered as outputs.
Write the matrices and vectors (\(A, G, C, b, f, y\)) of the differential algebraic equations.
Create the thermal circuit TC data structure and convert it to state-space representation (matrices \(A_s, B_s, C_s, D_s\) and input vector \(u_s\)).
Tests in steady-state and step response: test if the model is wrong (falsifiability). If the model fails these tests, then it is wrong; if the model passes these tests, it does not mean it is correct.
Compare and interpret the steady-state results obtained for thermal circuit and state-space representation.
Perform eigenvalue analysis of the state matrix \(A_s\) of the state-space representation and analyze the maximum time step (for numerical integration with Euler explicit or forward method) and the settling time. If the maximum time step is:
too small: set to zero the capacities that, with the related conductance (or resistance), produce the small time step (e.g., air, windows, doors);
too large: consider the discretization of elements which have thermal capacities and resistances (e.g., walls, ground).
Simulate the time response (by using Euler explicit and implicit method) to different step inputs.
Compare the settling time obtained from simulation of step response with that obtained by eigenvalue analysis.
Compare the steady-state value obtained from thermal circuit and state-space representation.
Inputs: prepare the input data set, i.e., time variation of weather and scheduled sources of temperatures and flow-rates.
Read the weather data (outdoor temperature, direct solar irradiance and diffuse solar irradiance).
Calculate the solar irradiance on the outdoor and indoor surfaces of the walls.
Define the schedules for temperature sources (e.g., indoor temperature set point) and flow rate sources (e.g., internal gains).
Obtain the input data set.
Simulation: integrate in time the state-space model for the given inputs.
Transform the input data set into input vector in time for state-space representation.
Set the initial conditions.
Integrate the states in time (e.g., by using Euler forward and/or backward methods).
Calculate the outputs from the states.
Plot the results.