Model reduction of a lean NOx trap catalyst model

The desire to increase fuel efficiency and reduce carbon dioxide emissions of vehicles has led to an increased use of vehicles equipped with lean-burn engines, such as diesel and lean-burn gasoline engines. This type of engine uses excess oxygen when compared to the amount required to stoichiometrically combust fuel. A serious disadvantage of this type of engine, however, is the increased emission levels of nitrogen oxides. These airborne pollutants are toxic for hu- mans when inhaled and are harmful to the environment. They are therefore subject to increasingly strict environmental regulations, which motivate the au- tomotive industry to innovate in order to decrease the nitrogen oxide emission levels of new engines and vehicle designs. An important method to accomplish this is the after-treatment of exhaust gas with the help of catalytic converters. Owing to the composition of the exhaust gas of lean-burn engines, however, the reduction of nitrogen oxide emission obtained with standard three-way catalysts (TWC) is nevertheless insufficient. One type of catalytic converter that might be used to reduce nitrogen oxide emissions of vehicles with lean-burn engines below the regulated maximum is the Lean NOx Trap (LNT). In addition to the platinum used in TWC’s, LNT catalytic converters use barium as active catalytic material. This gives the LNT the ability to store a certain amount of nitrogen oxides on its catalytic surface. The operating principle of a LNT catalyst is the temporary storage of the nitrogen oxides emitted during a time period of lean operation and a brief switchover of the engine to rich operation which produces exhaust gas containing excess reductants the reduce the stored nitrogen oxides to nitrogen, thereby regenerating the trap. This periodic operation requires active control of the engine to determine how and at which time instant to switch to rich operation. Model-based design methods can be used to derive estimation and control algorithms for LNT in a systematic manner. These methods require a dynamic model of the LNT of limited complexity, which describes the dynamics of the LNT that are essential to estimation and control and which is referred to as a control-oriented LNT model. Such a model can be obtained by two routes. The first route is to empirically determine the structure of the control-oriented LNT model from the observed behavior and tune the parameters of the model using vehicle test data. This approach has two disadvantages: it requires knowledge of the LNT dynamics to determine the right model structure and the vehicle tests required to tune the parameters are expensive and time-consuming. There is an alternative route provided we have a LNT model at our dis- posal based on physical laws, physical constants and the reaction constants of the elementary reaction steps which determine the kinetics of the chemicals reactions in the LNT. This type of model is referred to as a first-principles mo- del. Such a model can be obtained by combining existing physical knowledge with laboratory tests designed to determine the kinetics of elementary reacti- on steps. This alternative route means systematically deriving control-oriented models from the first-principle model using model reduction methods. These methods determine which parts of the first-principle model are relevant for esti- mation and control and help to arrive at a control-oriented model by reducing the first-principle model in such a way that it retains only these parts. This thesis examines the reduction of the complexity of LNT models using model reduction methods. First, the relation between detailed first-principle models and existing control-oriented models is explored. The essential simplifi- cation steps that link both model types are 1) the simplification of the kinetics and 2) the compression of the information on the axial spatial concentration profiles for gas-phase species and species on the catalyst surface into a limited number of states. A link can be established between parts of both models using a re-modeling approach in which assumptions with a physical interpretation are used to mathematically derive the control-oriented model from the first-principle model. The existing control-oriented model also contains relations that are em- pirical, however, and these cannot directly be related to the first-principle model using a re-modeling approach. The compression of the axial spatial concentration profiles was approached using existing model reduction methods known as projection methods. Two types of projection methods were used. The Balanced Truncation method uses an energy norm to quantify the energy transfer from model inputs to model states and from model states to model outputs. States which are associated with limited energy transfer from the input to the output are then truncated, removing them from the model. The Proper Orthogonal Decomposition (POD) method exploits correlations between the states of a dynamic model that occur if the model is simulated to redefine the state-space so that only states with limited correlation are retained in the model. It was found, however, that these methods perform poorly for models that are similar to the LNT model. The reason for this poor performance is that the dynamics of the LNT model are characterized by nonlinearity of the kinetics, the occurrence of traveling wave phenomena and large changes in time in the magnitude of species concentrations. One contribution of this thesis is the adaptation of POD algorithms to impro- ve their performance for systems which exhibit wave phenomena on a bounded domain, such as the LNT catalyst model. This was accomplished by inclu- ding phase information in the basis-functions used in the POD method. Data- correlation methods and extrapolation were used to gather information on the dominant features of the traveling wave from simulation data. This information was then used to construct the basis-functions. Next, we have considered simplification of the kinetic part of the LNT mo- del. A large number of different schemes exist to accomplish simplification of complex kinetic models. In this thesis lumping methods are examined, in parti- cular an approach whereby Quasi-Steady-State (QSS) and Partial Equilibrium (PE) assumptions are used to lump species. The effect of lumping on mass conservation and on the positive invariance of kinetic models is discussed. A requirement for the derivation of reduced order models using lumping methods the definition of an expansion relation. We determine under which conditions such a relation exists. Furthermore, in this thesis we take two steps towards a practical implementation of lumping methods. First, a method is developed to retain positive invariance of species concentrations in lumped models, when the expansion relation is approximated using polynomials. This method is based on existing results of function approximation using Sum-of-Squares polynomials. Next, a greedy-type algorithm is proposed to select QSS and PE relations so that the expansion relation can be obtained as the symbolic solution to a set of algebraic equations. A possible application of LNT models is the goal-oriented optimization of LNT operation. The application of model reduction as a part of goal-oriented optimization procedure is investigated. The techniques used for simplification of kinetic mechanisms are extended to spatially distributed models and applied to a test-case model similar to LNT models. It was found that application of model reduction methods during optimization is possible although it requires careful tuning of the model reduction accuracy versus the accuracy of the optimization algorithm. This thesis finally illustrates the difficulties with a straightforward applica- tion of existing model reduction methods to LNT models. The essential part of LNT models are convection-reaction equations required to model changes in species concentrations along the catalyst axis. A number of steps have been taken towards making these methods more suitable in practice for reduction of convection-reaction equations. However, a number of fundamental theore- tical problems remain, such as error control, and closely related to this, the preservation of stability and positive invariance of concentrations.