Deletion

In step 6 of the GA method given in Section 4.5.2, we quote that each individual that has been obtained by a reproduction instance replaces a less fit individual, chosen for deletion via a noisy selection procedure. This is category of Genetic Algorithms is usually called steady-state GAs and is contrasted to the generational GAs.

The generational GAs keep two distinct populations, the parents population and the children population. In each reproduction instance, two individuals of the parents population produce one or two children that are placed in the children population. Thus, the size of the children population gradually increases. When it becomes equal to that of the parents population, the parents population is deleted, the children population becomes the parents population and a new generation begins, hence the name “generational” GA.

The choice between steady-state and generational genetic algorithms is related to the fundamental issue of selective pressure vs. population diversity. Goldberg and Deb [27] have shown that the steady state GA imposes much higher selective pressure than the generational genetic algorithm. Davis [20], on the other hand, advocates that this is acceptable and in fact more efficient in real world problems. This is nevertheless an empirical observation, dependent on the problem at hand.

There are some more differences between steady-state and generational algo- rithms [23] but, apart from Goldberg’s analysis about selective pressure, arguments backing the one or the other approach are qualitative. We have tested both a generational and a steady-state∗ _{variant of our GA, and our experience confirmed} Goldberg’s analysis. The steady-state GA fairly quickly converged to a sub-optimum solution. The generational variant better found its way to the minimum and was therefore adopted.

4.6

Conclusions

In the above, we have presented the design desicions for a genetic algorithm for the estimation of the tunable parameters of our engineering catalytic converter model. The algorithm was adapted so as to perform successfully using a previously de- fined performance measure (Section 4.3) and engine or driving-cycle tests. The GA evolved after the conjugate gradients method was rendered inadequate because of the multimodality of the problem’s search space.

The genetic algorithm is based in the operating concept of a ‘classic’ GA. It departs from the prototype Genetic Algorithm that is usually the starting point of GA-oriented optimization methods development. It is a generational GA that uses real encoding, tournament selection, simulated binary crossover and random muta- tion. The choices have been made after screening many other common alternatives for GA encoding, selection and operators respectively.

In general, the genetic algorithm provides a flexible paradigm for multi- dimensional optimization in multimodal search spaces that has been in active devel- opment for many years. Apart from the theoretical advances on the subject, genetic algorithms have matured enough so as to be applied in many different scientific dis- ciplines. An early collection of such applications has been presented by Davis [20]; examples of mechanical and chemical engineering applications have also appeared, e.g. [28, 29, 30, 31]. The development of the genetic algorithm described in the present chapter does not aspire to contribute to the field of optimization methods more than just being yet another successful application of the GA paradigm.

Nevertheless, it fills a significant gap in the framework of catalytic converter modeling, in that, together with the developed performance measure, it provides a robust and efficient method for the estimation of tunable parameters of apparent kinetic models without relying on human intuition. The introduction of this ge- netic algorithm is thus a significant step towards an integrated methodology for the catalytic converter modeling.

Specifically, the parameter estimation is accurate and reliable. This fact has removed any uncertainty regarding successful model tuning that was previously in- hibiting the introduction and testing of new model formulations. The advances in catalytic converter modeling that have been presented in Chapter 5 were possible only after the parameter estimation methodology gave acceptable results, because it enabled the reliable and unambiguous assessment of different modeling approaches. Moreover, it paves the way for the reliable use of the model as a tool in the auto- motive industry, because it removes the dependency of the model’s applicability on the experience of the engineer-user.

The evidence for the combined performance of the genetic algorithm and the catalytic converter advanced model is provided in the next chapter in the form of selected case studies.

4.7

Summary

• A computer-aided tuning procedure has been developed that transforms the problem of catalytic converter model tuning into an optimization problem and applies standard optimization techniques to tackle it.

Sec. 4.7 Summary 119 and outlet concentration measurements from routine driving cycle tests.

• Attacking to the problem of model tuning as an optimization problem in- volved the development of two components: A performance measure, which qualitatively assesses the goodness-of-fit of the model for each possible set of parameter values, and an optimization procedure, which finds a set of tunable parameters that gives an optimum value for the performance measure.

• The performance measure is based on the comparison of the measured and computed efficiency of a catalytic converter for the three major pollutants, CO, HCs and NOx. The performance measure has a number of properties

that are advantageous for its use with an optimization routine as well as a stand-alone standard measure for the performance of a model.

• The first optimization algorithm that was implemented was based on the conjugate-gradients method. Preliminary results suggested that the targeted optimization space is multimodal and a global search procedure is appropriate. Nevertheless, this approach revealed that the potential of the tunable param- eters models is higher than initially expected, provided they are appropriately tuned.

• Subsequently, a genetic algorithm was developed and several combinations of encodings, crossover and mutation operators were tested. The best results were obtained with a real encoded generational GA with simulated binary crossover and random mutation which therefore adopted.

• The optimization methodology that combines the genetic algorithm and the performance measure developed herein presents a significant advance in the framework of catalytic converter modeling. It guarantees the reliable tuning of the model without depending on user experience and intuition, and thus it is an invaluable tool for both model development(through reliable testing of further enhancements), and for the model application (by ensuring its efficient use by the exhaust systems designer).

List of symbols

A Pre-exponential factor (tunable parameter), [–] e Error between measurement and computation, [–] E Activation Energy (tunable parameter), [–] E Conversion efficiency, [–]

f Performance function, [–] F Performance measure, [–] ` Binary code string length, [–]

n Number of optimized parameters, [–]

N Number of points for performance measure evaluation (4.11), [–] N_pop Number of individuals in the population, eq. (4.29), [–]

p Selection probability, [–] P computed response

b

P measured response from experiment t Time, [s]

T Time period for performance measure evaluation, [s] y Concentration, [–]

Greek Letters

δ Binary code resolution βq Spread factor, (4.33), [–] η Distribution index (4.33), [–] θ Tunable parameter

µ Average

ϕ Transformed (encoded) tunable parameter

Subscripts exp experiment i response r experimental run res response Abbreviations CG Conjugate Gradients GA Genetic Algorithm SA Simulated Annealing

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Chapter 5

Catalytic Converter Case studies

In this chapter, the model is validated and its predictive ability is demonstrated in three typical case studies, that feature different exhaust line layouts operated under both U.S. and European driving cycle tests conditions.

Different washcoat types are tested. The first case study employes a set of mea- surements of Pd/Rh catalyst, the kinetics of which were tuned with the conjugate gradients optimization method. The results of this case study revealed the poten- tional of optimization methods for the computer aided tuning of the model and the limitations of the conjugate-gradients approach.

The two subsequent case studies, the “Variable Precious Metal Loading” case study and the “Variable cell density/wall thickness” case study were tuned em- ploying the genetic algorithm. The former addresses the effect of precious metal loading variation in the efficiency of the catalytic converter, whereas the latter ad- dresses the effect cell density variation. The model’s predictive ability is very good, demonstrating significant steps towards the prediction of the Precious Metal Loading effect. Moreover, the model’s predictive power suggests further potential regarding emerging technologies in the field of exhaust aftertreatment, such as NOx storage

catalysts.