The Development and Application of an Evolutionary Algorithm for the Determination of Kinetic Parameters in Automotive Aftertreatment Models
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Abstract
Accurate mathematical models are an essential tool in the development of aftertreatment systems, as they can provide detailed information on the impact of design changes while simultaneously reducing development costs and the time between the initiation and completion of a catalytic process development. Identifying the set of kinetic parameters that achieves a perceived acceptable level of accuracy may require significant time and effort. Optimisation techniques can be used to speed up the tuning process; however, these techniques can require a large computation time, and may not produce a satisfactory answer. This invariably leads to questioning regarding the accuracy of such automated approaches. In this paper, the performance of a number of optimisation algorithms including a GA, nPSO and a hybrid algorithm has been explored with respect to their applicability to kinetic parameters optimisation in the context of aftertreatment modelling. The focus is therefore on the assessment of the optimisers’ performance, rather than the characterisation of catalytic reactions. The different algorithms were applied to the optimisation of parameters in a number of mathematical functions and theoretical aftertreatment systems. The optimisation algorithms were tested on theoretical aftertreatment systems since these have known absolute solutions thereby allowing the optimisers’ performance to be assessed in the absence of any other external source of inaccuracy such as model structure and experimental error. The results obtained demonstrate that such optimisation approaches facilitate the determination of kinetic parameters with suitable accuracy. The proposed hybrid optimisation algorithm achieved excellent performance in considerably shorter computation time than the GA or nPSO optimisers.
Keywords
Automotive Emissions Catalyst Kinetics OptimisationNomenclature
 [i]
Concentration of gas species i (mol m^{−3})
 A
Preexponential factor (m^{3} mol^{−1} s^{−1})
 A_{n}
Preexponential factor of reaction n (m^{3} mol^{−1} s^{−1})
 c_{1}
Arbitrary constant
 c_{1f}
Final value of c_{1}
 c_{1i}
Initial value of c_{1}
 c_{2}
Arbitrary constant
 c_{2i}
Initial value of c_{2}
 c_{2f}
Final value of c_{2}
 e_{t}
Individual error value at time t
 e_{total}
Overall error value at for the individual
 E_{A}
Activation energy (kJ mol^{−1})
 E_{An}
Activation energy of reaction n (kJ mol^{−1})
 E_{rrCO}
Error between current and objective function CO lightoff curves
 E_{rrNO}
Error between current and objective function NO lightoff curves
 E_{rrTHC}
Error between current and objective function THC lightoff curves
 E_{rrTotal}
Total error between current and objective functions for all toxic gas species
 Ê_{t}
objective function conversion efficiency at time t
 E_{t}
individual conversion efficiency at time t
 G
Inhibition term
 G_{best}
Global best solution
 i ith
dimension
 i_{T}
Total number of data points considered
 i_{G}
Initial generation
 j j^{th}
dimension
 k_{ai}
Inhibition coefficient (m^{3} mol^{−1})^{m}
 n n^{th}
iteration number
 nG
Final generation
 N
Number of gas species considered in error calculation
 P_{best}
Personal best solution
 P_{i}
Penalty function (0 < P_{ i } < 1)
 r
Reaction rate (mol s^{−1} m^{−3})
 R_{i}
Random constant 0 < R_{ i } < 1
 v
Particle velocity
 w
Inertia weighting
 x_{i}
Independent variable in mathematical functions
 X
Particle position
Acronyms
 DOC
diesel oxidation catalyst
 GA
genetic algorithm
 GAPSO
genetic algorithm, practical swarm optimisation hybrid
 GPS
Generalised pattern search
 HC
Hydrocarbons
 LOC
Lightoff curve
 nPSO
Particle swarm optimisation with niching
 NEDC
New European drive cycle
 PM
Particulate matter
 PSO
Particle swarm optimisation
 RDE
Real driving emissions
 SCR
Selective catalytic reduction
 SoTS
Sum of the squares
 THC
Total hydrocarbons
 TWC
Threeway catalyst
 WGS
Watergas shift
Subscript/superscript
 i i^{th}
individual
 j j^{th}
dimension
 n n^{th}
iteration number
 m
Global reaction order
Chemical symbols
 C_{3}H_{6}
Propylene
 C_{3}H_{8}
Propane
 C_{7}H_{8}
Decane
 C_{10}H_{22}
Global reaction order
 CO
Carbon monoxide
 CO_{2}
Carbon dioxide
 H_{2}
Hydrogen
 H_{2}O
Water
 NH_{3}
Ammonia
 NO
Nitric oxide
 NO_{2}
Nitrogen dioxide
 NO_{x}
Oxides of nitrogen
 O_{2}
Oxygen
1 Introduction
Emissions legislation has become increasingly stringent in recent years and the introduction of real driving emission (RDE) regulations places automotive manufacturers under greater pressure to reduce emissions from their vehicles. Emissions are controlled using a number of aftertreatment devices such as threeway catalysts (TWC) and diesel oxidation catalysts (DOC). These aftertreatment devices are subject to constant development in an effort to meet vehicle emissions limits while reducing cost.
Accurate mathematical models are essential tools in the development of aftertreatment systems, as they can provide useful information on the impact of design changes while simultaneously reducing development costs and lead times. These models commonly use global kinetics to calculate the rate of reactions on the surface of the catalyst. In the global kinetic approach, each reaction is represented by an equation that provides a rate of surface reaction for a wide range of concentrations of all gases that inhibit or promote that reaction [1]. Rival techniques such as microkinetics [2] incorporate adsorption and desorption steps associated with the surface reactions and therefore do not usually make assumptions regarding the ratelimiting step. While this has the potential to improve the accuracy of the model [3], the number of parameters that require validation increases dramatically [4]. For all approaches, the determination of the model’s kinetic parameters is crucial since it determines its predictive capabilities with respect to the reaction conditions experienced by the catalyst.
Ais the preexponential factor (m^{3} mol^{−1} s^{−1})
E_{ A }is the activation energy (kJ mol^{−1})
[i]is the concentration of gas species i
Gis the inhibition term
k_{ ai }is the inhibition coefficient (m^{3} mol^{−1})^{m}
Assuming the relevant reactions and structure of the LangmuirHinshelwood rate expression have been determined, the preexponential factor and activation energy terms may be tuned to match experimental data. However, the determination (or “tuning”) of these kinetic parameters is an extensive and complex task. In a typical automotive TWC, there are approximately 20 reactions occurring simultaneously, which equates to up to 40 such parameters that require tuning. This does not include the crossinhibition or selfinhibition terms. Thus, models may require considerable manual effort on the part of the model developer to achieve an acceptable capture of the catalyst’s behaviour by the simulation.
In this respect, optimisation techniques based on computer algorithms become vital to speed up the tuning process. However, the application of these techniques to aftertreatment models can require a large computation time and may not produce a solution deemed satisfactory. This invariably leads to questioning regarding the accuracy of such an automated approach. i.e. whether the absence of fit is due to the lack of accuracy of the optimiser, suitability of the model in capturing the chemical and/or physical phenomena, the quality of the experimental data or a combination of these factors.
Several optimisation techniques have been applied to the tuning of automotive catalysis models, and various authors have used third party programs for optimisation of aftertreatment systems. Pandya et al. [7] used the DVODE solver in their work to optimise the kinetic parameters of five reactions in a DOC system. The DVODE solver uses a generalised pattern search (GPS) optimisation algorithm which has been shown to be outperformed by a standard genetic algorithm (GA) in later work [8]. Ramanathan and Sharma used the iSIGHT optimisation package to tune kinetic parameters in a threeway catalyst model [9]. The package, in conjunction with the model, achieved good accuracy. However, the optimisation process required a large number of iterations, between 5000 and 10,000, to optimise the kinetic parameters [9]. This was acceptable due to the rapid computation time of the kinetic model, which required seconds to run, but this number of iterations would be prohibitive on a model with a larger computation time. iSight uses a variety of optimisation algorithms, and uses various forms of a genetic algorithm approach for multiobjective functions [10]. While the use of a thirdparty optimisation program is a convenient method to solve engineering problems, the user is provided with a black box optimisation package, with a limited ability to alter search variables within the optimiser.
Sola et al. [11] used a GPS algorithm within the MATLAB optimisation toolbox to optimise kinetic parameters of CO and C_{3}H_{6} oxidation on a platinum DOC model. Pontikakis and Stamelos [12] programmed a genetic algorithm approach to optimise a set of ten reactions in a threeway catalyst model, showing a high level of accuracy when applied to the new European drive cycle (NEDC).
Recent development of metaheuristic optimisation techniques applied to scientific and engineering modelling has demonstrated the superiority of approaches such as genetic algorithms and particle swarm optimisation (PSO) over the more traditional local search optimisation approaches collectively referred to as “hill climbing”. Beheshti notes that as optimisation design problems become more complex, the design space for these variables grows exponentially [13].
GA and PSO techniques have been applied to a number of mathematical problems in recent years due to their versatility and optimisation power. Liu et al. [14] applied a GA technique to a support vector machine model of a SCR catalyst. The GA was tasked with maximising the NO_{ x } conversion and minimising NH_{3} under certain conditions. This multiobjective optimisation approach produced accurate results when compared with experimental data, procuring error values of the order of 10^{−6}, with an absolute error of 5%.
PSO algorithms are a useful technique in model development and, in recent years, have been used to assist in predictive models and control strategies. Mozaffari et al. [15] have developed a dynamic PSO algorithm which was applied to a nonlinear model predictive control strategy. This dynamic PSO technique was applied to a theoretical cold start model and it was found that the PSO technique could guarantee convergence on a solution for their application. Bertram et al. [16] applied a basic PSO technique and a hybridised GAPSO technique to engine control parameters in order to reduce NO_{ x } emissions from a diesel engine. The hybrid approach outperformed the standard PSO technique, resulting in a 27% reduction in NO_{ x } emissions and 60% reduction in PM emissions.
While generic algorithms have been explored for the optimisation of kinetic parameters, the particle swarm approach does not seem to have been reported. More interestingly, when such optimisation approaches are employed, their performance assessment seems to solely rely on the level of the fit between the simulated and experimental data which, by definition, will also be affected by the “quality” of the model used. Reporting of an assessment of the intrinsic performance of the different algorithms when applied to such problems is rare.
In this present work, a comparative investigation of the intrinsic performance of several optimisation algorithms including a GA, particle swarm optimisation with niching (nPSO) and a hybrid algorithm, has been performed with respect to their applicability in aftertreatment modelling optimisation. The GA approach has been selected as it has demonstrated a high level of accuracy in previous studies [12, 16]. The GA algorithm has been shown to be accurate across multiple test scenarios, including automotive modelling problems. nPSO was explored since it has been reported as superior to GA for different types of optimisation problems [17], but has not yet been applied to the optimisation of kinetic parameters. Finally, a hybrid optimiser aimed at capturing the strengths of both GA and nPSO was created and tested with the aim of reducing computational cost without impacting accuracy.
The optimisation algorithms were tested on theoretical aftertreatment data which was generated using the Axisuite® catalyst simulation package. This approach is advantageous since it isolates the possible sources of error in the system and, consequently, allows an assessment of the intrinsic performance of the algorithms. Errors may be generated in different areas of an aftertreatment model, including inaccurate values for the convective and conductive heat transfer parameters, mass transfer parameters, errors generated by the inability of global kinetics to accurately represent a complex chemical system, and finally errors generated by an inability of the chosen optimiser to find an accurate solution. In the present case, using the AxiSuite software to generate both the mathematical model and the objective function negates any error due to inaccurate modelling of the system, thereby isolating the optimiser as the only possible source of error.
Using such conditions, it is possible to analyse and compare the intrinsic performance of different optimisation techniques within the mathematical framework of an aftertreatment model. This approach does not seek to validate aftertreatment models using experimental data, but rather attempts to validate the suitability of using different metaheuristic optimisation techniques in multiple established aftertreatment conditions.
2 Optimisation Techniques
Details about the theoretical and mathematical background of the different algorithms are extensively reported in the literature, and only a short description of their main features and context in which they are employed in the present work is provided in the following section.
3 Genetic Algorithm
Genetic algorithms have previously been used to optimise complex mathematical systems, such as reaction kinetic problems [12, 17, 18, 19, 20, 21, 22], grouping problems [23], and the mathematical travelling salesman problem [24]. GAs are inspired by nature and mimic the biological process of natural selection [23]. In a GA each optimisation variable is known as a bit, and these bits are grouped into a single bit array known as an “individual”. For example, when a GA is applied to the optimisation of catalyst kinetic parameters, each bit array contains a unique set of preexponential values and activation energies.
In the study presented in this paper, a floating point (real number) GA was used instead of binary coding. The floating point GA uses real numbers to form bits in the array which is preferential due to the large magnitude of the kinetic parameters, which would be cumbersome for a binarycoded algorithm to handle. Consequently, the values for the preexponential and activation energies can be directly used by the GA code. The GA search methodology has been fully described by Pedlow et al. [25]. The algorithm uses a doublepoint crossover mechanism and a nonuniform mutation rate to assist its search process.
4 Particle Swarm Optimisation
 j

is the jth dimension
 i

is the ith individual
 w

is the inertia weighting
 n

nth is the iteration number
 c _{1}

is the arbitrary constant
 c _{2}

is the arbitrary constant
 R _{1}

is the random constant 0 < R_{1} < 1
 R _{2}

is the random constant 0 < R_{2} < 1
 P _{best}

is the personal best solution
 G _{best}

is the global best solution
 c _{1i}

is the initial value of
c _{1}
 c _{1f}

is the final value of
c _{1}
 iG

is the initial generation
 nG

is the final generation
The c_{1} constant is linearly altered from a value of 2.5 to 0.5, while the c_{2} variable is altered from 0.5 to 2.5. The c_{1} constant is a multiplier for the local search functionality of PSO and c_{2} for the global search functionality. Altering the values of these constants enables the PSO to identify multiple optimum values in a search space, which helps its search functionality [7]. The PSO algorithm incorporated a nonuniform mutation rate, as used in the GA optimiser. Previous studies have shown that the addition of a mutation technique can help the search process of the PSO algorithm [29, 30, 31], since the mutation function prevents premature convergence, a problem to which PSO is particularly susceptible.
Particle swarm optimisation with niching or nPSO is a PSO with an integrated niching technique. A niche is a subpopulation, based on geographical location, of the main population. Each niche has its own variable limits and will produce its own best solution. Initially, the nPSO algorithm operates as a standard PSO algorithm using an increased local search (c_{1}) parameter. After a set number of iterations, the algorithm detects if individuals in the population have stagnated, which occurs when an individual’s best solution has remained unchanged over a number of iterations [32, 33]. The best individual is identified and removed from the population to form a niche seed. Individuals geographically close to the niche seed are also removed from the population and added to the niche. This process is repeated for the allowable number of niches. Individuals which do not fall into a niche are kept as part of the main population. Next, all the niches are processed simultaneously, with each niche producing its own best solution. Finally the best results from the niches are compared and the best of all solutions is chosen.
5 GAPSO Hybrid Model (Hybrid)
This algorithm uses different aspects of both the GA and PSO algorithms, utilising the ability of PSO to search large spaces associated with the GA’s better converging ability, in addition to the stochastic nature of its mutation capability. In this algorithm the initial population, which must be a multiple of four, is ranked in terms of fitness. The population is separated into two parts, with the fittest individuals sent to the GA algorithm. This subpopulation is run through the GA as previously described, including roulette selection, doublepoint crossover and a nonuniform mutation rate [25]. The ranked subpopulation is then returned to the main population.
The less fit subpopulation is fed to a PSO search algorithm. This algorithm uses the solution from the GA subpopulation as each of its individuals’ best solutions and the overall best solution from the GA subpopulation is kept as the global best solution. The PSO subpopulation is then returned to the main population, the entire population is ranked and the process iterates until a convergence criterion or iteration limit is met.
This Hybrid approach was expanded upon that reported by Kao et al. by adding the operators from the GA section of the optimiser i.e. a doublepoint crossover mechanism, a nonuniform mutation rate and roulette selection. These operators increase the search capability of this section of the optimiser by increasing its stochastic nature. A doublepoint crossover can exchange smaller sections of the genetic information in each individual, which is a considerable advantage in an aftertreatment scenario. A roulette selection mechanism decreases the chance of the system stagnating by choosing random individuals within the population for the crossover mechanism. Finally, a nonuniform mutation rate increases the chance of mutation as a function of number of iterations. This is an important consideration as the mutation search function becomes the main search function for GA as the system starts to converge [25, 34].
The PSO section of the optimiser has been adapted to include a number of operations, such as linearly decreasing inertia and acceleration constants which have been shown by Chaturvedi et al. to improve the PSO search performance [35]. As previously stated, these operators promote the ability of the PSO algorithm to search wide areas of the design space. This is particularly important in the format of the Hybrid algorithm. The PSO section of the code also utilises an adaptive mutation function, based on the work by Rajakamur et al. [30], which increases the chance of mutation as the solution gets closer to the objective function.
These operators have been added to specific sections of the Hybrid optimiser, improving the search functionality of each section and thereby improving its overall search capability. In addition to these operators, additional functions were added to the Hybrid optimiser, such as a userdefined temperature range for optimisation. This function enables the user to select one or more sets of temperature ranges across a lightoff curve to consider for optimisation. This increases the sensitivity of the optimiser since small deviations from this region of the objective function will have a greater effect on the fitness value of the optimiser. Full details on this operator have been published by Pedlow et al. [25]. A convergence function was also added which identifies when the Hybrid optimiser has stagnated by monitoring the best results at each iteration. Once the population has stagnated, this function greatly reduces the limits around the best solution. Hereby, the population is reinitialised within this smaller design space and will continue searching until the population has stagnated or the iteration limit has been reached.
6 Testing Method
The algorithms were tested for two separate scenarios. The first scenario applied the algorithms to a set of mathematical functions, and the second a set of AxiSuite® generated lightoff curves corresponding to different aftertreatment systems.
7 Mathematical Functions
Comparison of optimiser accuracies for the mathematical functions
Optimiser  GA  nPSO  Hybrid  

Shubert function (−)  Best error  3.2 × 10^{−5}  4.0 × 10^{−9}  3.2 × 10^{−6} 
Average error  2.1 × 10^{−1}  6.1 × 10^{−5}  1.4 × 10^{−1}  
Standard deviation  3.3 × 10^{−1}  1.5 × 10^{−4}  2.1 × 10^{−1}  
Griewank function (−)  Best error  0  1.0 × 10^{−12}  0 
Average error  3.4 × 10^{−4}  1.5 × 10^{−3}  0  
Standard deviation  2.7 × 10^{−3}  2.7 × 10^{−3}  0  
Twodimensional problem (−)  Best error  7.8 × 10^{−8}  2.9 × 10^{−7}  1.9 × 10^{−8} 
Average error  3.2 × 10^{−4}  2.7 × 10^{−3}  9.9 × 10^{−5}  
Standard deviation  4.5 × 10^{−4}  3.1 × 10^{−3}  1.2 × 10^{−4} 
As expected, the standard GA algorithm is outperformed by the other two algorithms, which is consistent with what has been reported previously [35]. However, it is important to note that the performance of the optimiser is partly dependent on the optimisation problem. The Hybrid algorithm consistently outperforms the standard GA algorithm, and outperforms the nPSO algorithm on two of the tests. In the Griewank function and the twodimensional problem, the Hybrid algorithm produces a smaller value for the average and best errors, indicating that the hybrid algorithm can reproducibly achieve a more accurate answer than the nPSO algorithm. These functions are slightly more complex than the Shubert function, which may indicate that the Hybrid model is more appropriate for complex systems. This is due to the Hybrid algorithm’s incorporation of both PSO and GA characteristics, enhancing the ability of the Hybrid to search large parts of the design space and converge towards an optimum point.
The analysis of the mathematical systems indicates that the hypotheses underlying the anticipated enhanced performance of the Hybrid algorithm, and which guided its code development, appear validated since it outperformed the standard GA algorithm and the nPSO algorithm for complex tasks. However, all the functions listed have small design spaces when compared against an aftertreatment system model, and can be considered as comparatively simple systems.
8 Mathematical Aftertreatment System
The second testing scenario for the optimisation algorithms was a set of theoretical aftertreatment systems. This scenario introduces a more complex system while still containing a global optimum and perfect solution to the problem (i.e. the default set of kinetic parameters used to generate to lightoff curves are later used as objective functions). The mathematical models were created using Axisuite®, a commercially available aftertreatment modelling software package. The software offers a Simulink® connection [36], which was used to run the aftertreatment system within an optimisation loop coded using MatLab®. Axisuite is able to simulate a number of aftertreatment devices, provides a good degree of flexibility and has been used in the development of previous aftertreatment models [25, 37, 38]. Using Axisuite to generate a set of lightoff curves removes experimental and modelling errors and facilitates assessment of the optimisers’ intrinsic performance in isolation.
 e _{ t }

is the error value at time t
 E _{ t }

is the individual conversion efficiency at time t
 Ê _{ t }

is the objective function conversion efficiency at time t
 e_{total}

is the overall error value for the individual
 N

is the number of gas species considered
 iT

is the total number of data points considered
In each of the theoretical test systems, the algorithms operated with 100 starting individuals and were run for 100 iterations, or generations. Each algorithm was allowed to run to completion, and its accuracy was recorded at each iteration. Wide limits were used in the search process, with each preexponential term allowed a deviation of three orders of magnitude from the set value, and each activation energy term permitted to deviate by 20% from the set value.
The rates of each of the chemical reactions considered in this study are affected by the concentrations of other components in the gas mix. For example, the oxidation rate of CO, described in Eqs. 2 and 3, is inhibited by CO (selfinhibition), and also C_{3}H_{6}, and NO (crossinhibition). The parameters which characterise these inhibition effects are contained in the denominator of the reaction rate equation. The Axisuite model used in this study allows the user to introduce inhibiting effects for numerous other gas species. The determination of the kinetic coefficients for these effects is normally undertaken as a separate phase of the overall process of determining the kinetic parameters, in which the reaction rate is measured in the presence of varying concentrations of each inhibiting gas component. In this regard, an automated optimisation technique could be used to assist in the determination of the inhibition coefficients. However, for the study presented in this paper, the inhibition coefficients were held at their default values, and only the promoting coefficients were considered. By increasing the number of reactions included in the optimisation process to 12, and thereby considering 24 individual kinetic parameters, it was felt that this provided sufficient challenge to the simultaneous optimisation process and allowed the relative performance of each algorithm to be determined.
9 DOC Aftertreatment System
Input gas concentrations for the DOC system
Gas species  Concentration (%)  Gas species  Concentration (%) 

CO  0.015  NO_{2}  0.01 
O_{2}  10  C_{3}H_{8}  0.05 
CO_{2}  7.113  C_{3}H_{6}  0.05 
H_{2}O  6.58  C_{7}H_{8}  0.015 
H_{2}  0.005  C_{10}H_{22}  0.025 
NO  0.09  –  – 
List of reactions optimised in second theoretical DOC system
1)  NO + 0.5 O_{2} ⇌ NO_{2} 
2)  CO + 0.5 O_{2} → CO_{2} 
3)  C_{3}H_{8} + 5 O_{2} → 3 CO_{2} + 4 H_{2}O 
4)  C_{3}H_{6} + 4.5 O_{2} → 3 CO_{2} + 3 H_{2}O 
5)  C_{7}H_{8} + 9 O_{2} → 7 CO_{2} + 4 H_{2}O 
6)  C_{10}H_{22} + 15.5 O_{2} → 10 CO_{2} + 11 H_{2}O 
7)  C_{7}H_{8} + 18 NO_{2} → 7 CO_{2} + 4 H_{2}O + 18 NO 
8)  C_{10}H_{22} + 31 NO_{2} → 10 CO_{2} + 11 H_{2}O + 31 NO 
During implementation of the optimisation algorithms to aftertreatment systems, a test was performed to determine the effect of using a logarithmic search space for the activation energy values. Using the logarithmic search space improved the search function for these variables and thus was chosen for both parameters. During the postoptimisation analysis, these values were transformed back from their logarithmic values.
10 DOC Aftertreatment System—Results
Optimiser results for DOC system
Optimisation algorithm  GA  nPSO  Hybrid 

CO error  0.164 × 10^{−4}  0.880 × 10^{−4}  0.797 × 10^{−4} 
NO error  2.143 × 10^{−4}  0.569 × 10^{−4}  0.113 × 10^{−4} 
THC error  2.075 × 10^{−4}  0.815 × 10^{−4}  0.633 × 10^{−4} 
Total error  4.382 × 10^{−4}  2.264 × 10^{−4}  1.542 × 10^{−4} 
Kinetic parameters that produced the optimisers’ best results and the objective parameter values for the DOC system
Reaction  Objective function  GA  nPSO  Hybrid  

A  E_{A}  A  E_{A}  A  E_{A}  A  E_{A}  
R_{1}  3.00 × 10^{8}  3.00 × 10^{4}  7.30 × 10^{8}  3.18 × 10^{4}  3.00 × 10^{8}  3.00 × 10^{4}  1.97 × 10^{8}  2.79 × 10^{4} 
R_{2}  1.00 × 10^{21}  1.00 × 10^{5}  5.38 × 10^{20}  9.85 × 10^{4}  1.00 × 10^{20}  1.05 × 10^{5}  1.78 × 10^{21}  1.02 × 10^{5} 
R_{3}  1.00 × 10^{19}  1.50 × 10^{5}  1.14 × 10^{17}  1.28 × 10^{5}  1.00 × 10^{19}  1.55 × 10^{5}  4.12 × 10^{20}  1.70 × 10^{5} 
R_{4}  1.00 × 10^{20}  1.10 × 10^{5}  1.30 × 10^{21}  9.96 × 10^{4}  1.00 × 10^{19}  1.12 × 10^{5}  4.29 × 10^{20}  1.15 × 10^{5} 
R_{5}  1.00 × 10^{21}  1.00 × 10^{5}  1.00 × 10^{18}  8.64 × 10^{4}  1.00 × 10^{21}  1.08 × 10^{5}  3.01 × 10^{21}  1.06 × 10^{5} 
R_{6}  5.00 × 10^{20}  1.00 × 10^{5}  2.93 × 10^{17}  1.26 × 10^{5}  5.00 × 10^{19}  1.05 × 10^{5}  3.35 × 10^{21}  1.04 × 10^{5} 
R_{7}  1.00 × 10^{18}  4.00 × 10^{4}  2.14 × 10^{16}  3.20 × 10^{4}  1.00 × 10^{18}  4.00 × 10^{4}  1.71 × 10^{17}  3.89 × 10^{4} 
R_{8}  1.00 × 10^{18}  4.00 × 10^{4}  1.51 × 10^{16}  3.20 × 10^{4}  1.00 × 10^{18}  4.00 × 10^{4}  1.10 × 10^{19}  4.60 × 10^{4} 
In this scenario, the shape of the NO curve is challenging to match, as is the distinctive THC data. Figure 6 and Table 4 demonstrate that the GA algorithm had difficulties matching these complex lightoff curves; however, both the nPSO and Hybrid optimisers were able to better fit those curves. There is a minimal difference in accuracy between the nPSO and Hybrid optimisers; however, the Hybrid optimiser needed a five times shorter computation time to provide its best solution. Examination of Figs. 7 and 8 and Table 4 demonstrates that the Hybrid optimiser is slightly more accurate in the THC lightoff curve, whereas the nPSO algorithm matches the NO peak with a slightly greater accuracy. The best input variables reported by both these optimisers, Table 5, are similar to the input objective variables, indicating that the best solutions found are within the area of the global optima, any difference being attributed to the known convergence issues of these optimisers [12, 34].
11 Sensitivity Analysis
Sensitivity analysis results for DOC system
Reaction  Kinetic parameter  Percentage deviation from objective function (%)  Kinetic parameter  Percentage deviation from objective function (%) 

R_{1}  A_{1}  40.2  E_{A1}  9.09 
R_{2}  A_{2}  28.05  E_{A2}  19.7 
R_{3}  A_{3}  16.06  E_{A3}  12.55 
R_{4}  A_{4}  3.88  E_{A4}  1.47 
R_{5}  A_{5}  2.42  E_{A5}  6.74 
R_{6}  A_{6}  11.51  E_{A6}  13.26 
R_{7}  A_{7}  4.58  E_{A7}  0.3 
R_{8}  A_{8}  7.33  E_{A8}  2.26 
Average algorithm error and standard deviation
Algorithm  SoTS mean (−)  SoTS standard deviation (−) 

GA  3.64 × 10^{−2}  1.02 × 10^{−1} 
nPSO  3.07 × 10^{−4}  1.14 × 10^{−4} 
Hybrid  1.58 × 10^{−4}  3.36 × 10^{−5} 
12 ThreeWay Catalyst—Lean Conditions
Input gas species for TWC under lean conditions
Gas species  Concentration (%) 

CO  0.28 
O_{2}  1.03 
CO_{2}  14.08 
H_{2}O  13 
NO  0.215 
C_{3}H_{8}  0.055 
C_{3}H_{6}  0.017 
List of reactions optimised in theoretical Lean TWC system
1)  CO + 0.5 O_{2} → CO_{2} 
2)  H_{2} + 0.5 O_{2} → H_{2}O 
3)  C_{3}H_{8} + 5 O_{2} → 3 CO_{2} + 4 H_{2}O 
4)  CO + NO → CO_{2} + 0.5 N_{2} 
5)  H_{2} + NO → H_{2}O + 0.5 N_{2} 
6)  C_{3}H_{8} + 3 H_{2}O → 3 CO + 7 H_{2} 
7)  C_{3}H_{6} + 3 H_{2}O → 3 CO + 6 H_{2} 
8)  CO + H_{2}O → CO_{2} + H_{2} 
13 TWC—Lean Condition Results
Optimiser results for mathematical TWC system under lean conditions
Optimisation algorithm  GA  nPSO  Hybrid 

CO error  12.59 × 10^{−5}  0.363 × 10^{−5}  2.055 × 10^{−5} 
NO error  20.55 × 10^{−5}  1.433 × 10^{−5}  2.131 × 10^{−5} 
THC error  24.85 × 10^{−5}  0.251 × 10^{−5}  2.090 × 10^{−5} 
Average error  57.99 × 10^{−5}  2.048 × 10^{−5}  6.281 × 10^{−5} 
Table of the kinetic parameters that produced the optimisers’ best results and the objective parameter values for the lean TWC system
Reaction  Objective function  GA  nPSO  Hybrid  

A  E_{A}  A  E_{A}  A  E_{A}  A  E_{A}  
R_{1}  1.00 × 10^{19}  1.00 × 10^{5}  4.83 × 10^{17}  1.10 × 10^{5}  3.85 × 10^{16}  1.00 × 10^{5}  5.76 × 10^{18}  1.00 × 10^{5} 
R_{2}  1.00 × 10^{19}  1.00 × 10^{5}  3.93 × 10^{17}  7.50 × 10^{4}  1.70 × 10^{16}  1.00 × 10^{5}  2.07 × 10^{19}  1.00 × 10^{5} 
R_{3}  5.00 × 10^{21}  1.20 × 10^{5}  4.95 × 10^{19}  1.07 × 10^{5}  1.41 × 10^{21}  1.20 × 10^{5}  1.04 × 10^{22}  1.20 × 10^{5} 
R_{4}  2.00 × 10^{16}  8.00 × 10^{4}  2.00 × 10^{14}  6.00 × 10^{4}  2.71 × 10^{14}  8.00 × 10^{4}  1.45 × 10^{16}  8.00 × 10^{4} 
R_{5}  5.00 × 10^{16}  8.50 × 10^{4}  3.28 × 10^{15}  9.76 × 10^{4}  1.75 × 10^{15}  8.50 × 10^{4}  5.39 × 10^{16}  8.50 × 10^{4} 
R_{6}  6.00 × 10^{14}  7.50 × 10^{4}  5.16 × 10^{13}  5.63 × 10^{4}  1.87 × 10^{13}  7.50 × 10^{4}  5.78 × 10^{15}  7.50 × 10^{4} 
R_{7}  6.00 × 10^{14}  6.50 × 10^{4}  6.12 × 10^{13}  5.51 × 10^{4}  3.50 × 10^{16}  6.50 × 10^{4}  3.98 × 10^{14}  6.50 × 10^{4} 
R_{8}  5.00 × 10^{10}  5.00 × 10^{4}  5.00 × 10^{9}  3.85 × 10^{4}  4.91 × 10^{11}  5.00 × 10^{4}  5.63 × 10^{10}  5.00 × 10^{4} 
Sensitivity analysis results for TWC lean system
Reaction  Kinetic parameter  Percentage deviation from objective function (%)  Kinetic parameter  Percentage deviation from objective function (%) 

R_{1}  A_{1}  43.08  E_{A1}  3.76 
R_{2}  A_{2}  6.3  E_{A2}  18.78 
R_{3}  A_{3}  9.64  E_{A3}  4.58 
R_{4}  A_{4}  51.31  E_{A4}  2.34 
R_{5}  A_{5}  28.29  E_{A5}  4.63 
R_{6}  A_{6}  4.18  E_{A6}  0.11 
R_{7}  A_{7}  12.26  E_{A7}  3.62 
R_{8}  A_{8}  4.64  E_{A8}  0.01 
Average accuracy and standard deviation for the TWC lean system
Algorithm  SoTS mean  SoTS standard deviation 

GA  1.95 × 10^{−3}  2.38 × 10^{−3} 
nPSO  1.28 × 10^{−4}  1.86 × 10^{−4} 
Hybrid  2.40 × 10^{−4}  1.61 × 10^{−4} 
It is important to remember that while the kinetic parameters deviated from the objective kinetic parameters, the actual simulated lightoff curves achieved were very similar to the objective function. As in the case of the DOC system, this highlights that the solution does not correspond to a single set of parameters but larger areas of the design space, even when considering the sensitive parameters only. This is a common problem and has been identified and discussed by other researchers such as Stewart et al. [2] who detailed ten separate sets of published reaction kinetic values for the simple CO oxidation reaction. The sensitivity analysis confirmed that both the nPSO and Hybrid optimisers were able to closely match the objective function reaction parameters.
14 ThreeWay CatalystRich Conditions
Input gas conditions for a TWC under rich conditions
Gas species  Concentration (%) 

CO  0.59 
O_{2}  0.28 
CO_{2}  13.94 
H_{2}O  13 
H_{2}  0.2 
NO  0.22 
C_{3}H_{8}  0.056 
C_{3}H_{6}  0.0168 
List of reactions optimised in theoretical TWC system
1)  CO + 0.5 O_{2} → CO_{2} 
2)  H_{2} + 0.5 O_{2} → H_{2}O 
3)  C_{3}H_{8} + 5 O_{2} → 3 CO_{2} + 4 H_{2}O 
4)  C_{3}H_{6} + 4.5 O_{2} → 3 CO_{2} + 3 H_{2}O 
5)  CO + NO → CO_{2} + 0.5 N_{2} 
6)  H_{2} + NO → H_{2}O + 0.5 N_{2} 
7)  C_{3}H_{8} + 3 H_{2}O → 3 CO + 7 H_{2} 
8)  C_{3}H_{6} + 3 H_{2}O → 3 CO + 6 H_{2} 
9)  CO + H_{2}O → CO_{2} + H_{2} 
10)  CO_{2} + H_{2} → CO + H_{2}O 
Algorithm results for a TWCrich system
Optimisation algorithm  GA  nPSO  Hybrid 

CO error  27.566 × 10^{−4}  2.671 × 10^{−4}  0.0126 × 10^{−4} 
NO error  3.228 × 10^{−4}  0.216 × 10^{−4}  0.120 × 10^{−4} 
THC error  44.381 × 10^{−4}  2.404 × 10^{−4}  0 
Average error  75.175 × 10^{−4}  5.291 × 10^{−4}  0.133 × 10^{−4} 
The nPSO and Hybrid optimisers achieved an accuracy of 5.2 × 10^{−4} and 1.3 × 10^{−5} respectively. Both the optimisers provide parameters that allow satisfactory simulation of the system, generating conversion profiles that are similar to the objective functions. The GA optimiser produced parameters leading to simulated curves that only partially match the objective function as highlighted by Fig. 13. The difference between the PSObased best results and the objective function can be attributed to the known difficulty of this algorithm to finely converge towards an optimum, a finding which has been previously reported [30].
Sensitivity analysis results for TWCrich system
Reaction  Kinetic parameter  Percentage deviation from objective function (%)  Kinetic parameter  Percentage deviation from objective function (%) 

R_{1}  A_{1}  61.09  E_{A1}  17.09 
R_{2}  A_{2}  17.09  E_{A2}  10 
R_{3}  A_{3}  0  E_{A3}  0 
R_{4}  A_{4}  35.23  E_{A4}  15.02 
R_{5}  A_{5}  23.41  E_{A5}  38.63 
R_{6}  A_{6}  20.45  E_{A6}  0.22 
R_{7}  A_{7}  14.29  E_{A7}  5.23 
R_{8}  A_{8}  21.22  E_{A8}  16.48 
R_{9}  A_{9}  12.49  E_{A9}  0.37 
R_{10}  A_{10}  0  E_{A10}  0 
Table 17 also highlights that two reactions that had no perceptible effect on the lightoff curves. These were Reactions 3 and 10 from Table 15, i.e. the oxidation of propane and the reaction between CO_{2} and H_{2}. While these single parameters had no effect on the lightoffcurves using the chosen gas concentrations, they might have an effect using alternative gas mixes and/or temperatures. As is the case with a manual tuning process, it is therefore important that the calibration of the kinetic coefficients be undertaken using a suitable range of gas compositions.
Table of the kinetic parameters that produced the optimisers’ best results and the objective parameter values for the TWCrich system
Reaction  Objective function  GA  nPSO  Hybrid  

A  E_{A}  A  E_{A}  A  E_{A}  A  E_{A}  
R_{1}  2.01 × 10^{19}  1.03 × 10^{5}  7.02 × 10^{18}  1.01 × 10^{5}  2.93 × 10^{18}  1.05 × 10^{5}  1.99 × 10^{19}  1.03 × 10^{5} 
R_{2}  2.00 × 10^{19}  1.00 × 10^{5}  1.75 × 10^{18}  8.32 × 10^{4}  5.17 × 10^{18}  8.36 × 10^{4}  3.43 × 10^{19}  1.05 × 10^{5} 
R_{3}  8.00 × 10^{5}  1.20 × 10^{5}  1.42 × 10^{5}  1.46 × 10^{5}  1.36 × 10^{5}  1.41 × 10^{5}  1.24 × 10^{5}  9.76 × 10^{4} 
R_{4}  5.00 × 10^{19}  1.10 × 10^{5}  3.96 × 10^{19}  8.25 × 10^{4}  2.11 × 10^{18}  9.65 × 10^{4}  6.44 × 10^{18}  9.96 × 10^{4} 
R_{5}  8.56 × 10^{17}  7.90 × 10^{4}  7.46 × 10^{17}  8.18 × 10^{4}  3.27 × 10^{16}  6.92 × 10^{4}  5.87 × 10^{17}  7.75 × 10^{4} 
R_{6}  9.00 × 10^{16}  8.70 × 10^{4}  3.32 × 10^{16}  7.56 × 10^{4}  3.73 × 10^{15}  9.91 × 10^{4}  7.35 × 10^{16}  7.76 × 10^{4} 
R_{7}  1.25 × 10^{13}  7.50 × 10^{4}  2.89 × 10^{13}  8.06 × 10^{4}  1.97 × 10^{13}  7.66 × 10^{4}  6.27 × 10^{12}  7.61 × 10^{4} 
R_{8}  2.00 × 10^{18}  1.40 × 10^{5}  2.62 × 10^{18}  1.72 × 10^{5}  3.37 × 10^{18}  1.66 × 10^{5}  1.16 × 10^{17}  1.19 × 10^{5} 
R_{9}  2.00 × 10^{10}  5.00 × 10^{4}  2.46 × 10^{9}  5.77 × 10^{4}  3.12 × 10^{10}  5.31 × 10^{4}  1.22 × 10^{9}  4.75 × 10^{4} 
R_{10}  2.00 × 10^{9}  5.00 × 10^{4}  2.48 × 10^{8}  5.22 × 10^{4}  3.02 × 10^{8}  4.76 × 10^{4}  7.51 × 10^{9}  4.97 × 10^{4} 
Average accuracy and standard deviation for the TWCrich system
Algorithm  SoTS mean  SoTS standard deviation 

GA  1.27 × 10^{−2}  7.13 × 10^{−3} 
nPSO  3.34 × 10^{−3}  2.95 × 10^{−3} 
Hybrid  4.25 × 10^{−4}  3.58 × 10^{−4} 
15 ThreeWay CatalystRich Conditions with Oxygen Storage
Input gas conditions for TWC under Rich conditions with oxygen storage
Gas species  Concentration (%) 

CO  0.59 
O_{2}  0.28 
CO_{2}  13.94 
H_{2}O  13 
H_{2}  0.2 
NO  0.22 
C_{3}H_{8}  0.056 
C_{3}H_{6}  0.0168 
List of reactions optimised in theoretical rich TWC system including oxygen storage
1)  CO + 0.5 O_{2} → CO_{2} 
2)  H_{2} + 0.5 O_{2} → H_{2}O 
3)  C_{3}H_{8} + 5 O_{2} → 3 CO_{2} + 4 H_{2}O 
4)  CO + NO → CO_{2} + 0.5 N_{2} 
5)  H_{2} + NO → H_{2}O + 0.5 N_{2} 
6)  C_{3}H_{8} + 3 H_{2}O → 3 CO + 7 H_{2} 
7)  C_{3}H_{6} + 3 H_{2}O → 3 CO + 6 H_{2} 
8)  CO + H_{2}O → CO_{2} + H_{2} 
9)  Ce_{2}O_{3} + 0.5 O_{2} → 2 CeO_{2} 
10)  Ce_{2}O_{3} + NO → 2 CeO_{2} + 0.5 N_{2} 
11)  2 CeO_{2} + CO → Ce_{2}O_{3} + CO_{2} 
12)  2 CeO_{2} + 1/6 C_{3}H_{6} → Ce_{2}O_{3} + 0.5 CO + 0.5 H_{2}O 
Algorithm results for a TWCrich system including oxygen storage reactions
Optimisation algorithm  GA  nPSO  Hybrid 

CO error  344.79 × 10^{−5}  2.66 × 10^{−5}  3.58 × 10^{−5} 
NO error  6.97 × 10^{−5}  1.54 × 10^{−5}  4.69 × 10^{−5} 
THC error  8.50 × 10^{−5}  5.60 × 10^{−5}  2.57 × 10^{−5} 
Average error  360.26 × 10^{−5}  9.80 × 10^{−5}  10.84 × 10^{−5} 
In this scenario the nPSO optimiser and the Hybrid optimiser achieved similar performances. It is, however, important to note that the Hybrid optimiser is significantly less computationally intensive than the nPSO algorithm, so while the nPSO algorithm produced a similar result, the increase in computation time required (by a factor of 5) reiterates that it is more beneficial to use the Hybrid optimiser for this type of system.
Sensitivity analysis results for TWCrich including OSC system
Reaction  Kinetic parameter  Percentage deviation from objective function (%)  Kinetic parameter  Percentage deviation from objective function (%) 

R_{1}  A_{1}  13.98  E_{A1}  0.17 
R_{2}  A_{2}  25.03  E_{A2}  1.89 
R_{3}  A_{3}  7.24  E_{A3}  1.89 
R_{4}  A_{4}  37.42  E_{A4}  1.12 
R_{5}  A_{5}  22.67  E_{A5}  1.7 
R_{6}  A_{6}  2.02  E_{A6}  1.59 
R_{7}  A_{7}  8.53  E_{A7}  4.65 
R_{8}  A_{8}  11.61  E_{A8}  3.31 
R_{9}  A_{9}  17.34  E_{A9}  24.27 
R_{10}  A_{10}  15.07  E_{A10}  0.29 
R_{11}  A_{11}  3.19  E_{A11}  15.32 
R_{12}  A_{12}  17.73  E_{A12}  0.6 
Table of the kinetic parameters that produced the optimisers’ best results and the objective parameter values for the Rich TWC system including optimised oxygen storage reactions
Reaction  Objective function  GA  nPSO  Hybrid  

A  E_{A}  A  E_{A}  A  E_{A}  A  E_{A}  
R_{1}  1.00 × 10^{19}  1.00 × 10^{5}  4.24 × 10^{19}  9.03 × 10^{4}  5.78 × 10^{16}  7.77 × 10^{4}  3.18 × 10^{18}  8.48 × 10^{4} 
R_{2}  1.00 × 10^{19}  1.00 × 10^{5}  3.51 × 10^{21}  1.15 × 10^{5}  8.53 × 10^{17}  8.83 × 10^{4}  2.25 × 10^{19}  1.00 × 10^{5} 
R_{3}  5.00 × 10^{21}  1.20 × 10^{5}  9.13 × 10^{21}  1.36 × 10^{5}  2.19 × 10^{24}  1.18 × 10^{5}  1.34 × 10^{22}  1.20 × 10^{5} 
R_{4}  2.00 × 10^{16}  8.00 × 10^{4}  2.63 × 10^{15}  6.97 × 10^{4}  2.55 × 10^{18}  8.38 × 10^{4}  2.74 × 10^{15}  9.07 × 10^{4} 
R_{5}  5.00 × 10^{16}  8.50 × 10^{4}  2.89 × 10^{16}  7.71 × 10^{4}  1.89 × 10^{15}  1.04 × 10^{5}  2.31 × 10^{15}  7.58 × 10^{4} 
R_{6}  6.00 × 10^{14}  7.50 × 10^{4}  2.21 × 10^{14}  6.56 × 10^{4}  6.16 × 10^{16}  6.93 × 10^{4}  4.61 × 10^{13}  6.88 × 10^{4} 
R_{7}  6.00 × 10^{14}  6.50 × 10^{4}  1.03 × 10^{15}  6.38 × 10^{4}  7.97 × 10^{16}  7.41 × 10^{4}  5.02 × 10^{14}  6.50 × 10^{4} 
R_{8}  5.00 × 10^{10}  5.00 × 10^{4}  5.00 × 10^{10}  4.53 × 10^{4}  1.37 × 10^{11}  5.63 × 10^{4}  6.75 × 10^{10}  5.00 × 10^{4} 
R_{9}  6.00 × 10^{9}  8.00 × 10^{4}  6.00 × 10^{7}  6.40 × 10^{4}  1.33 × 10^{12}  1.00 × 10^{5}  4.18 × 10^{9}  8.00 × 10^{4} 
R_{10}  3.00 × 10^{13}  8.50 × 10^{4}  3.00 × 10^{11}  6.80 × 10^{4}  2.31 × 10^{15}  1.06 × 10^{5}  2.65 × 10^{14}  8.50 × 10^{4} 
R_{11}  6.00 × 10^{9}  8.00 × 10^{4}  6.00 × 10^{7}  6.40 × 10^{4}  7.73 × 10^{8}  6.00 × 10^{4}  3.17 × 10^{11}  8.00 × 10^{4} 
R_{12}  6.00 × 10^{9}  8.00 × 10^{4}  6.00 × 10^{7}  6.40 × 10^{4}  1.71 × 10^{7}  9.74 × 10^{4}  2.98 × 10^{11}  8.00 × 10^{4} 
Average accuracy and standard deviation for the TWCrich system including oxygen storage reactions
Algorithm  SoTS mean  SoTS standard deviation 

GA  4.56 × 10^{−3}  8.30 × 10^{−4} 
nPSO  3.39 × 10^{−3}  2.85 × 10^{−3} 
Hybrid  3.43 × 10^{−4}  1.66 × 10^{−4} 
16 Discussion
The various scenarios considered in this research cover a range of aftertreatment systems and reaction conditions and offer a strenuous test to the optimisers. The results from the mathematical aftertreatment systems show that the Hybrid algorithm and the nPSO algorithm consistently achieve a high degree of accuracy for each of the systems. The Hybrid algorithm slightly outperforms the nPSO algorithm for three out of the five testing scenarios and has a significantly shorter computation time than the nPSO algorithm. It is important to note that there is only a minimal difference between the two optimisers throughout two of the four tests, specifically the DOC and Rich TWC system scenarios. This set of testing scenarios provides confidence in the ability of the nPSO and Hybrid optimisers to optimise aftertreatment systems. However, it must be remembered that the scenarios considered in this paper are theoretical systems that contain a unique, global solution, which can be perfectly defined using a global kinetic approach. These were created using the assumption that other areas of a mathematical aftertreatment model, such as convective and conductive heat transfer, and mass transfer, effectively model the experimental environment. Therefore when these optimisers are applied to experimental data any discrepancy may be caused by experimental error, inaccurate values within the heat and mass transfer models, the kinetics model or an inability of global kinetics to accurately represent the conditions within the catalyst over the range of conditions explored.
The results show that both the nPSO and Hybrid algorithms are suitable optimisers to apply to kinetic parameters within a mathematical aftertreatment model. This is demonstrated by their ability to provide parameters that allow satisfactory matches between the simulated and objective function lightoff curves, in addition to their ability to arrive at kinetic values that are close to the objective variables which were used to create the objective curves.
In terms of computational time, the Hybrid optimiser achieves its solution in a factor of five times quicker than the nPSO algorithm. This is highly advantageous as the results from the presented scenarios have shown that the Hybrid optimiser is able to achieve a similar accuracy to that of the nPSO. Therefore, the Hybrid algorithm may be run for a greater number of iterations to generate a more accurate solution while requiring a shorter computation time than the nPSO algorithm.
The results indicate that a good performance in more simple systems, such as the mathematical functions, is not representative for more complex systems. This is shown through the standard GA performance in the scenarios tested. While it achieved a comparable accuracy for the more simple systems proposed in the mathematical functions, it struggled to obtain an accurate solution in the more complex systems of the aftertreatment tests.
It is important to note that, while the presented work was targeted at discriminating the relative performances of different optimisation algorithms and therefore assumed a situation where the reaction mechanism is known, in most real life situations, the mechanism itself will be in question. In such situations, the difficulty will lie with deciding the relevance or not of various reactions, species, and inhibition terms in the mechanism. In the case of the global mechanism approach, the goal will be to find the simplest mechanism capable of fairly representing the experimental data. Such a mechanism will use the smallest number of possible competing reaction pathways and inhibition terms to reduce the number of degrees of freedom. If such an approach is not employed, many parameters will have low sensitivity towards the resulting calculated reaction profiles and therefore produce large standard deviations. Ultimately, such an approach will result in too many parameters which will produce large regions of the design space that can adequately fit the reaction profile, and also low accuracy for the resulting parameters which is obviously not desirable.
This situation illustrates one of the limitations of the “full reaction network” optimisation approach and highlights the need to bring the maximum constraint to the system to limit as much as possible the degrees of freedom. This is usually brought about using a combination of approaches aimed at gathering independent information regarding the relative importance of different reaction pathways involving common molecules. In the present work, this is illustrated by the case of propane that can be converted through either steam reforming reactions or oxidation with oxygen. There is a possible risk of having the optimiser favouring the former while the oxidation reactions have been shown to be dominant. This can be disentangled using targeted independent experiments under simpler conditions. Another aspect to consider is the fact that while a complete mechanism is necessary to capture the catalyst behaviour under the whole set of possible conditions experienced under reallife operations, the lightoff approach to experimental data may cover only a limited set of reaction conditions. This usually means that not all reactions listed in the reaction mechanism will be “active”. It is therefore necessary to select a range of conditions that will ensure, collectively, that all reactions are “activated” and consequently that the set of experimental data used for the parameters optimisation is fully relevant.
In all cases, the results obtained for the optimised kinetic parameters in the present study highlights the difficulty in obtaining perfect matches with the objective function parameters even though the resulting curves appeared to match. This discrepancy is easy to interpret for the parameters with low sensitivities since large deviations in their values will lead to limited impact on the shape of the curves. This comes from the presence of either “nonactivated” reactions under the conditions used or the presence of redundant reaction steps.
Redundant reactions can occur when a complex reaction network is present. In such situations, duplicate reactions are present in the network and identical behaviour can de facto be obtained by either reaction. If such duplicates are not identified and removed, the optimiser will, by definition, generate a family of poor accuracy optimal parameters for these reactions. For example, in Table 9, the watergas shift (WGS), i.e. reaction 8, is stoichiometrically equivalent to CO and H_{2} oxidation reactions (Reaction 1 minus Reaction 2). Anything accomplished with Reaction 8 can also be achieved by incrementally increasing Reaction 1 and simultaneously decreasing Reaction 2 by the same amount. In this case, it will be possible to study the WGS independently of the oxidation reactions by running tests in the absence of oxygen. However, the oxidation reactions cannot be analysed without interference from the WGS reaction. The difficulties associated with redundant, or nearly redundant, reaction pathways in global reaction kinetics are a recognised issue in the aftertreatment community. This is precisely why the analysis and simulation of kinetics requires a measure of “manual” tuning in order to account for the chemical information obtained during the studies which is aimed at disentangling complex pathways. In the case of the WGS reaction example, this will mean performing a priori fits of the WGS reaction in the absence of oxygen, and then holding the fitted parameters fixed during subsequent optimisation for more complex reactions conditions.
In such cases, and in the absence of manual intervention, an effect of compensation between parameters will therefore be at play which highlights that the possible solutions do not correspond to a single set of parameters but encompass larger areas of the design space. This interdependence (i.e. compensation between parameters) is often referred as “correlation” and its identification is achieved through correlation analyses. Such correlations are linked to the system studied and the mathematical structure of the model adopted. It is a difficulty but, as illustrated with the WGS, it is usually possible to find conditions which will break such correlations. When that is not the case, this leads to an intrinsic reduction of the possible accuracy on the determination of such correlated parameters. Additional possible ways to improve confidence in the kinetic variables obtained usually involves conditions leading to increased constraints on the model. These can involve simultaneous optimisation of parameters for the same catalyst for different environmental conditions. Another possible approach will make use of recent advances in spatial resolution of catalytic reactors by using the spatially resolved profiles rather than end pipe data.
17 Conclusions
In the present comparative study, the nPSO and Hybrid optimisation algorithms were found to perform best for an aftertreatment system. The Hybrid optimiser outperformed the nPSO in three out of the five aftertreatment scenarios and the difference was negligible in the remaining scenarios. Due to the significantly shorter computation time required, the Hybrid optimiser performed best out of the tested optimisation algorithms. The results from the mathematical functions test demonstrate that, in this application, it is not necessarily possible to use these simple systems with small design spaces as a predictor of how optimisation algorithms will perform on more complex systems, such as in an aftertreatment scenario. In the context of mechanistic investigations, it is expected that the algorithms reported here will facilitate and accelerate the identification of the optimal kinetic parameters of different proposed scenarios which will then be discriminated via appropriate statistical analysis.
Notes
Acknowledgements
This work was supported by the Department of Education and Learning, and the authors would like to thank Jaguar Land Rover for financial support.
Compliance with Ethical Standards
Conflict of Interest
The authors declare that they have no competing interests
Supplementary material
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