Reduced Order Model Design for Three Way Catalytic Converters

This research topic aims to design and simulate a reduced order model of a physics based three way catalyst (TWC) in automotive systems. A catalytic converter carries out a chemical redox reactions of pollutant gases in order to convert them to less harmful gases before emission through the vehicle exhaust. The thermal and oxygen storage dynamics of a TWC are described using a set of partial differential equations (PDE) which are coupled and non-linear in nature along the spatial axis of the catalyst. Computing the solution for a PDE based system of equations is arduous since the state of the system is distributed in an infinite domain along the spatial axis. Hence a model order reduction needs to be carried out in order to observe the behavior of PDE based systems using only a finite number of states. In order to solve for such systems, the PDEs need to be transformed into ordinary differential equations (ODE) which result in a finite, but still a large, number of states. This transformation from a set of PDEs with infinite states to an ODE system with a finite number of states is carried out using the Galerkin\u27s projection method. Since we are dealing with a non-linear PDE, the resulting ODEs are time dependent non-linear system of equations. The model order reduction of the ODE based system is achieved using Proper Orthogonal Decomposition (POD). The POD method uses a basis representation of the system variable, and these basis functions are derived from a matrix containing the values of the variable at different instants over time and space obtained through real time measurement or simulation. The basis functions are extracted from the data matrix by performing Singular Value Decomposition (SVD). SVD computes the eigen values of the system and stores them in a matrix arranging them from the most dominant modes to the least. The number of eigen values selected represents the number of basis functions for the system. An energy estimation criterion is selected to limit the number of basis functions that will capture the entire system dynamics. Therefore we end up with a reduced number of modes that describe the order of the ODE based system. The accuracy of the reduced order model will be calculated for different number of basis functions and a selection of an optimal model order will be determined by computing the root-mean-square error (RMSE) for each of these modes. The reduced model will be developed over the Federal Test Protocol (FTP) driving cycle and the results will be validated alongside the measurement values over the US06 driving cycle