Reaction mechanism reduction and optimisation for modelling aviation fuel oxidation using standard and hybrid genetic algorithms

Computers and Chemical Engineering 30 (2006) 889–900

Reaction mechanism reduction and optimisation for modelling aviationfuel oxidation using standard and hybrid genetic algorithms

Lionel Elliott a, Derek B. Ingham a, Adrian G. Kyne b,∗, Nicolae S. Mera b,Mohamed Pourkashanian b, Sean Whittaker a

a Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UKb Energy and Resources Research Institute, University of Leeds, Leeds LS2 9JT, UK

Received 15 March 2005; received in revised form 3 January 2006; accepted 17 January 2006Available online 15 March 2006

Abstract

This study describes the development of a new binary encoded genetic algorithm for the combinatorial problem of determining a subset ofspecies and their associated reactions that best represent the full starting point reaction mechanism in modelling aviation fuel oxidation. Thegenetic algorithm has a dual objective in finding a reduced mechanism that best represents aviation fuel oxidation in both a laminar premixed flameagpog9

pponi©

K

1

sccd3td

0d

nd perfectly stirred reactor systems. The number of species in the subset chosen is kept fixed and is specified at the start of the procedure. Theenetic algorithm chooses ever improving mechanisms based on an objective function which measures how well the new reduced mechanismsredict a set of species’ profiles simulated by the full mechanism. In order to verify the validity of our approach, a full enumeration was performedn a reduced problem and it was found that the genetic algorithm was able to find the optimum solution to this reduced problem after a fewenerations. The reduction involved going from 338 reactions involving 67 species to 215 reactions involving 50 species. This corresponded to a0% CPU time saving in each function evaluation.

A second step was to take the reduced reaction mechanism and to use a second real encoded genetic algorithm for the parameter optimisationroblem of determining the optimal reaction rate parameters that best model an experimental set of premixed flame and jet stirred reactor species’rofiles. A significant improvement could be seen in the species profiles obtained using the mechanism with the GA optimised rates over thosebtained from the original reduced mechanism. Further, in order to increase the efficiency of the second reaction rate coefficient optimisation step, aew hybrid method was developed which incorporates a direct optimisation method (Rosenbrock method) into the GA. A significant improvementn both accuracy and efficiency was apparent in using this new hybrid approach.

2006 Elsevier Ltd. All rights reserved.

eywords: Genetic algorithm; Optimisation; Aviation fuel; Rosenbrock; Combustion; Reaction mechanism reduction

. Introduction

The numerical simulation of turbulent reacting flow has madeignificant progress in recent years due to advances in physi-al modelling, development of detailed reaction mechanisms,omputer hardware and numerical algorithms. Nevertheless theetailed modelling of hydrocarbon combustion within complexD geometries is presently out of reach given the tools availableoday, which accounts for the continued interest in methods forescribing chemical kinetics of hydrocarbon combustion more

∗ Corresponding author. Tel.: +44 113 343 2481.E-mail address: [email protected] (A.G. Kyne).

efficiently. Much effort has been invested in developing simpli-fied two- or three-step reaction mechanisms with rate expres-sions in generalised Arrhenius form. The reduction of chemicalreaction mechanisms constitutes an integral part of modelling,and, as such, the reduction strategy and specific techniquesdepend on the objectives of the modelling. The systematic reduc-tion technique pioneered by Peters and Williams (1987), thecomputational singular perturbation method (see Lam, 1993),and the ILDM approach of Maas and Pope (1992) are among themost prominent efforts in this area of research. While replacingdetailed reaction mechanisms with reduced models alleviatesCPU time and memory overheads, the accuracy can often becompromised when applying the reduced schemes to a reactionspace outside their range of validity. Schwer, Lu, and Green

098-1354/$ – see front matter © 2006 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2006.01.003

mailto:[email protected]

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890 L. Elliott et al. / Computers and Chemical Engineering 30 (2006) 889–900

Nomenclature

Ai pre-exponential factor for the ith reactionBi non-Arrhenius parameter for the ith reactionEai activation energy for the ith reaction

foptimisePSR PSR portion of fitness function for real encoded

GA

f ReducedCOMBINED combined fitness function for binary GA

f ReducedPREMIX premix portion of fitness function for binary GA

f ReducedPSR PSR portion of fitness function for binary GA

kfi forward reaction rate for the ith reaction

K total number of speciesKPREMIX the number of species that are compared in the

PREMIX profiles

KPSR the number of species that are compared in thePSR model

‖ ‖L2 L2 norm of a function calculated using the trape-zoidal rule

nchild number of offspringsndef number of definite species (species ever present)ne elitism parameternmut number of mutations that take place along the

child chromosome (binary GA)npop population size

nfullr number of reactions in full starting mechanism

nfulls number of species in full starting mechanism

nreduceds number of species in reduced mechanism

NPREMIX total number of PSR conditions

NPSR total number of PREMIX conditionsNR the number of reactions in the reduced modelp pressure (atm)pc uniform arithmetic crossover probabilitypt tournament selection probabilitypm non-uniform mutation probability (real encoded

GA)pmut probability a child is selected for mutation (binary

GA)Pk starting point for Rosenbrock method at the kth

iterationR universal gas constant (J/mol/K)si ith direction total progress, 1 ≤ i ≤ 3NRT temperature (K)Wk weighting factor applies to the kth species

xki ith direction vector for Rosenbrock method at the

kth iteration, 1 ≤ i ≤ 3NRXjk mole fraction of the kth species in the jth PSR

conditionYjk mole fraction of the kth species in the jth PREMIX

conditionzi chromosome vector (i = 1, 2, . . ., nfull

s )

Greek lettersδi ith step length for Rosenbrock method,

1 ≤ i ≤ 3NRε constant added to PREMIX fitness function to

avoid numerical overflowφ equivalence ratioτ residence time (s)

Superscriptscalc calculated valuesorig original values

(2003) have addressed this problem by replacing the full chem-istry model with an entire library of locally accurate, reducedkinetic models, rather than a single skeletal model. Thus, onereduced model can be used over the large regions of the domainwhere the flow field is constant and the local chemistry is rela-tively simple while another scheme can be used to approximatethe full chemistry model in regions where the flow is changingrapidly such that the reactant mixing time scales are of the sameorder as the chemical reaction time scales.

There have been a number of approaches put forward in theliterature regarding taking a detailed reaction scheme and find-ing the minimum subset of species/reactions that best representa specific combustion regime. Frenklach and Wang (1991) havedeveloped a detailed reduction method whereby reactions fromthe full mechanism that correspond to low reaction rates and lowenthalpy-weighted reaction rates are eliminated. Brown, Li, andKoszykowski (1997), Turanyi, Berces, and Vajda (1989), andVajda, Valko, and Turanyi (1985) developed a similar method(principal component analysis). This time rather than reactionsbeing eliminated if their reaction rates fall below a user definedmaximum threshold, they are eliminated based on a maximumthreshold associated with a sensitivity analysis. The drawbackwith these reduction strategies is that they cannot guarantee thatthe reaction mechanism generated is the most compact mech-anism that can perform to the required accuracy. To addressthis drawback, Bhattacharjee, Schwer, Barton, and Green (2003)hgtrcrtaionrFbto

ave developed a linear integer program which can be solved touaranteed global optimality. Here the smallest reaction subsetaken from a detailed mechanism is selected such that the neweduced model satisfies the user’s specified accuracy/validityriteria. A further mechanism reduction methodology that haseceived considerable attention is the Computer Assisted Reduc-ion Method (CARM) developed by Chen (1988, 1997). Hereskeletal mechanism is developed by identifying and eliminat-

ng reaction steps based upon a combined sensitivity and ratef production analysis at the conditions being investigated. Theext step is to then identify quasi-steady state (QSS) species andemove those species that fall below a specified cut off level.inally, the solution of the coupled and non-linear set of alge-raic equations obtained in the previous steps are used to findhe QSS species concentrations together with the reaction ratesf the non-QSS species.

L. Elliott et al. / Computers and Chemical Engineering 30 (2006) 889–900 891

In this investigation the reduction of chemical reaction mech-anisms is treated, in general, in two steps. First a set of simplifiedreactions is selected that describes the combustion process overa given range of operating conditions. Once a reduced mech-anism has been constructed, the next step is the selection ofsuitable reaction rate coefficients. It is not only necessary toestablish the best-fit values of the unknown parameters but alsoto estimate their uncertainties in rigorous statistical terms, i.e.to determine the joint confidence region, and to examine thecorrelations between the parameter values. Both steps representnon-convex inverse problems, the solution of which is a non-trivial task. Evolutionary algorithms are particularly suitable foroptimising objective functions with complex, highly unstruc-tured landscapes. They are powerful optimisation techniquesthat can be applied for solving optimisation problems for whichthe objective functions have a complex structure and gradientinformation is not available. Computational intelligence meth-ods, such as neural networks, evolution strategies or geneticalgorithms (GA) are highly adaptive methods originated fromthe laws of nature and biology. Unlike the traditional gradientbased methods, one of the most important characteristics ofcomputational intelligence techniques is the effectiveness androbustness in coping with uncertainty, insufficient information,and noise.

Two types of genetic algorithms, a binary coded genetic algo-rithm and a real coded genetic algorithm, were used in ordertnIcrtdnfopvwatttiCRTtabIgtAbns

subset of species and their associated reactions the performancewas based on a “dual objective function” which measured howwell the new reduced mechanisms predicted a set of species’profiles simulated by the full mechanism for both laminar pre-mixed and perfectly stirred reactor (PSR) systems. The secondstep of the reduction process involved taking the optimal skeletalmechanism generated in step one and applying a real encodedgenetic algorithm (Elliott et al., 2002a, 2002b, 2003a, 2003b,2004) in order to find “the best” set of the Arrhenius reactionrate coefficients that provided the best fit to an experimentalset of species’ profiles. For this type of parameter optimisationproblem, either binary or real encoded GAs could have beenimplemented. However, it was found by Elliott et al. (2002a)that real encoded algorithms are more efficient in terms of con-vergence times than those binary encoded for the current type ofproblem. The “best set” of reaction rate coefficients was judgedusing a dual objective function that examined how effectivelythe reaction mechanisms predicted an experimentally observedset of premixed and PSR species’ profiles. The search spaceavailable to the GA was defined according to the reaction rateuncertainty associated with the National Institute of Standardsand Technology (NIST) chemical kinetic database. Defining thesearch space in this way led to reaction mechanisms that couldbe used outside those conditions used in the optimisation pro-cess (see Elliott et al., 2003a). Finally, in order to increase theefficiency of the second reaction rate coefficient optimisationsdHiiip

2

crcpsKtbGihatptmiitu

o deal with the two components of the problem considered,amely combinatorial optimisation and numerical optimisation.t is well known that of all evolutionary algorithms, the binaryoded genetic algorithms are most suitable and the only natu-al option for combinatorial optimisation problems. However,he binary representation traditionally used in GAs has somerawbacks when applied to multidimensional, high-precisionumerical problems. Disadvantages include the inability to per-orm fine local tuning and the inability to operate in the presencef non-trivial constraints. The use of real parameters makes itossible to use large domains (even infinite domains) for theariables, which is difficult to achieve in binary implementationshere increasing the domain would mean sacrificing precision,

ssuming a fixed length for the chromosomes. Another advan-age when using a real coded GA is their capacity to exploithe graduality of the functions with continuous variables, wherehe concept of graduality refers to the fact that slight changesn the variables correspond to slight changes in the function.ontinuing this line of thinking, a highlighted advantage ofCGAs is the capacity for the local tuning of the solutions.here are genetic operators, such as the non-uniform muta-

ion that allows the tuning to be produced in a more suitablend faster way than in BCGAs, where the tuning is difficultecause of the Hamming cliff effect (see Michalewicz, 1996).n the first step of our reduction process, we used a binary codedenetic algorithm to choose “the best” subset of species and allheir associated reactions from a detailed reaction mechanism.

binary encoded GA was necessary as the selection of theest group of species represents a combinatorial problem. Theumber of species in the subset chosen was kept fixed and waspecified at the start of the procedure. In determining “the best”

tep, a new hybrid method was developed which incorporates airect optimisation method (see Rosenbrock, 1960) into the GA.ere a small population GA was used to generate a reasonable

nitial starting point for the Rosenbrock method. A significantmprovement in both accuracy and efficiency could be gainedn using the direct method from this GA generated startingoint.

. Combustion models

Various software packages may be used for the direct cal-ulations to determine the output species concentrations if theeaction rates are known. In this study the perfectly stirred rectoralculations are performed using the PSR FORTRAN computerrogram that predicts the steady-state temperature and compo-ition of the species in a perfectly stirred reactor (see Glarborg,ee, Grcar, & Miller, 1988). Laminar premixed flame struc-

ure calculations were performed using the PREMIX code forurner stabilised flames with a known mass flow rate (see Kee,rcar, Smooke, & Miller, 1985). The code is capable of solv-

ng the energy equation in determining a temperature profile,owever, throughout this investigation an experimental temper-ture profile was imposed in order to rule out the possibility thathe predicted temperature profile leads to an inaccurate species’rofile prediction. For instance, if the energy equation is usedhen there may be discrepancies in the modelled and experi-

ental temperature profiles. These discrepancies may lead tonaccurate species profiles and so there is no way of determin-ng whether the inaccuracies occur as a result of an inaccurateemperature profile or inaccurate reaction rate coefficients. Insing the experimental temperature data we can conclude that


any difference in modelled or experimental species profiles is aresult of the chemical kinetics and not the thermodynamics. Notsolving the energy equation had the additional advantage of asignificant CPU time saving.

Both PSR and PREMIX programs are not stand-alone pro-grams and are designed to be run in conjunction with pre-processors from the CHEMKIN library (see Kee, Miller, &Jefferson 1980), which handle the chemical reaction mecha-nism and the thermodynamic and transport properties. For everyreaction within any given reaction mechanism the forward ratecoefficients are in the three parameter functional Arrhenius form,namely

kfi = AiTβi exp

(−Eai

RT

)(1)

for i = 1, . . ., NR, where R is the universal gas constant and thereare NR competing reactions occurring simultaneously. The reac-tion rates (1) contain the three parameters Ai, βi, and Eai for theith reaction. These parameters are of paramount importance inmodelling the combustion processes since small changes in thereaction rates may produce large deviations when modelling theoutput species’ profiles in PSRs or premixed flames.

3. Optimisation models

3.1. Step 1: mechanism reduction

stotPKpeooaivGumfcpec

itosbci

The number of species, nreduceds selected in each reduced species

model is kept fixed and specified at the start thus,∑nfull

s1 zi =

nreduceds . A further constraint to the model is that a number

of “definite species”, ndef, are defined such that there are ndefpositions in the binary vector, z, where zi must always equal 1.This constraint is used to ensure that fuel and oxidiser speciesinvestigated in the PREMIX/PSR combustion models are alwayspresent in the reduced mechanisms generated by the GA. In addi-tion, major product species or species of interest can be forced tobe ever present thus reducing the search space and subsequentlyincreasing the rate of convergence. The initial starting popu-lation consists of nindiv individuals each containing the samedefinite species with the remaining (nreduced

s − ndef) species ran-domly chosen from the remaining (nfull

s − ndef) species of thefull mechanism, e.g. for an initial population of five individualswhere nfull

s = 20, nreduceds = 10, and ndef = 3 we could have

wrawosiar

siuvcmb

f

f

f

While detailed comprehensive kinetic mechanisms are neces-ary to fully understand the fundamental chemistry of combus-ion they are often impractical for use in industrial codes becausef their size. Using a simplified surrogate aviation fuel-air reac-ion mechanism, AFRMv1.1 (see Elliott et al., 2002c; Kyne,atterson, Pourkashanian, Williams, & Wilson, 2001; Patterson,yne, Pourkashanian, Williams, & Wilson, 2000) as a startingoint, the first step of this investigation was to use a binaryncoded genetic algorithm to establish the best reduced subsetf species and their associated reactions that best represent a setf species’ profiles simulated from the full starting point mech-nism. In a simple GA, the process of evaluation assigns eachndividual solution in a population with an associated fitnessalue indicating the individual’s survivability. Subsequently, theA must evolve this population of individuals into a new pop-lation using the three operations of selection, crossover, andutation. Selection probabilistically chooses better individuals

or a new generation while crossover and mutation manipulateandidate solutions to generate new individuals for the selectionrocedure to process again. After a certain number of gen-rations, when no further improvement is observed, the besthromosome represents an optimal solution.

Starting with the full mechanism, neglecting the NOx chem-stry, the starting point mechanism contains nfull

r = 338 reac-ions involving nfull

s = 67 species. An initial starting populationf individuals is formed where each individual represents a pos-ible reduced species model. Each individual is represented by ainary vector (chromosome), z, of length, nfull

s where zi = 1 indi-ates the ith species is included in the reduced model while zi = 0ndicates that the ith species is not included (i = 1, 2, . . . , nfull

s ).

here the highlighted 1’s represent the definite species and theemaining species of the reduced model are randomly distributedlong the chromosome. Each reduced species set has associatedith it, a set of possible reactions. Here a reaction is chosennly if all reactant and product species exist within the reducedpecies set. Also any third body reactions will only be includedf both the reactant/product and all effective third body speciesppear in the reduced species set. The reaction rate coefficientsemain the same as those of the full starting point mechanism.

Having generated an initial population of individuals the nexttep is to devise a criterion for judging how “fit” an individuals. For this process, CHEMKIN’s PSR and PREMIX codes aresed to generate a set of species’ profiles for each of the indi-iduals considered. This set of profiles is then compared to aorresponding set of profiles generated with the full startingechanism according to the combined objective function given

y Eq. (4).

ReducedPSR =

⎧⎨⎩10−8 +

NPSR∑j=1

KPSR∑k=1

∣∣∣∣∣Xcalc

jk − Xorigjk

Divisor

∣∣∣∣∣⎫⎬⎭

−1

(2)

ReducedPREMIX =

⎧⎨⎩10−8 +

NPREMIX∑j=1

⎛⎝KPREMIX∑

k=1

Wk

∥∥∥Y calcjk − Y

origjk

∥∥∥L2∥∥∥Y

origjk

∥∥∥L2

⎞⎠

⎫⎬⎭

−1

(3)

ReducedCOMBINED = f Reduced

PSR f ReducedPREMIX (4)


• Here Xcalcjk and Y calc

jk represent the mole fraction concentra-tions for PSR and PREMIX systems for the kth species inthe jth set of operating conditions using the set of reactionrate parameters ((Ai, βi, Eai ), i = 1, . . ., NR). NR representsthe number of reactions in the reduced model and NPREMIX

and NPSR represent the number of PREMIX conditions andPSR conditions considered, respectively. KPSR represents thenumber of species that are compared in the PSR model andthis is taken to be the number of definite species chosen atthe start of the reduction process. KPREMIX is the number ofspecies that are compared in the PREMIX profiles and this istaken to be the number of species in the reduced mechanismdefined at the start of the reduction process.

• Xorigjk and Y

origjk are the corresponding simulated species values

generated using the original, full reaction model.• If Xcalc

jk < PSR species threshold (defined in applica-tion parameters) then Divisor = PSR species threshold elseDivisor = Xcalc

jk . This stops the concentrations of very smallspecies leading to large negative contributions to the PSR fit-ness.

• If MAX(Yorigjk ) > PREMIX species threshold (defined in

application parameters) Wk = 1.0 else Wk = 0. This had theeffect of removing the small/less important species profilesfrom contributing to the PREMIX fitness.

• ‖ ‖L2 represents the L2-NORM of a function. The L2-NORM

•

TIsK

odamt(itstatw

fraction concentration is less than 10−7 do not contribute directlyto the fitness but are still important as they can allow importantreactions to be used in the reduced model which may signifi-cantly impact the “more important” species profiles. Similarly,in the PSR contribution to the fitness function of Eq. (4), onlythose “important species” chosen to be definite species at thestart of the reduction process directly impact the fitness. Casescan arise where one reduced model is ranked higher than anotherbut it does not contain a species that the user is interested in orthis species is poorly predicted relative to the other species. Forsuch cases, the fitness function can easily be modified to ensurecertain species are always chosen by adding these species ofinterest to the definite species list and/or to bias the reducedmechanisms’ ability to predict these species by modifying theweighting factors, Wk.

Having calculated the fitness of each of the individ-uals, k-tournament selection (see Harris, Elliott, Ingham,Pourkashanian, & Wilson, 2000) is used to select the fittestindividuals to be parents for the next generation. Genetic oper-ators such as crossover and mutation (see Harris et al., 2000)were then used to exchange information between the parentsin order to form a population of children. In order to constructthe new population the Elitist model was used, in which the ne(the elitism parameter) best individuals from the old popula-tion were retained in the next population along with the fittest(ctoaaisAruo

Fc

is given by

‖f‖L2 =(∫

f 2(x)

)1/2

and is numerically calculated using a trapezoidal rule. Thus,the fitness function is a measure of the difference betweenprofiles in calculated and predicted mole fractions over theircomplete domain.Since the differences in measured and modelled species con-centrations can be very small, 10−8 has been added to thefitness functions in order to avoid numerical overflow.

he calculations of Xcalcjk and Y calc

jk are made via the CHEMKINI software PSR and PREMIX, respectively, and the equationsolved can be found in the studies of Glarborg et al. (1988) andee et al. (1985).

The validity of the method lies in how it decides what makesne reduced mechanism better than another. This is subjective,epending on what the user wants out of their reaction modelnd there are a number of alternative methods of defining theodel’s accuracy which could have been used. The accuracy of

he reduced mechanisms chosen in this study is given by Eq.4), where one individual is defined to be better than anotherndividual if the sum of the percentage errors/100 in predictinghe nreduced

s species profiles simulated by the full mechanism ismaller than that of another reduced reaction model. In ordero avoid comparing “less important” species, a condition wasdded to the PREMIX contribution of Eq. (4) to only considerhose species whose peak simulated mole fraction concentrationas greater than 10−7. The species whose peak simulated mole

npop − ne) child chromosomes. These child chromosomes werehosen without repetition such that no individual would appearwice in the new population. The constraints associated withur problem led us to disregard the more traditional crossovernd mutation operators in favour of a modified approach. Thepproach we adopted does not strictly crossover information butnstead exchanges information between parents. Two parents areelected and those species common to both parents are identified.

child is then formed by keeping these common species andandomly choosing the remaining species from the remainingnused positions of the chromosome such that the total numberf species = nreduced

s (see Fig. 1). In order to avoid convergence

ig. 1. The modified operator used in this investigation to replace the traditionalrossover operator.


to local optima a mutation mechanism has been incorporated.Here a child will be selected for mutation with a probability,pmut. If a child is selected, then those positions not containinga definite species have an equal probability of mutating (i.e.changing from a 0 to a 1 or vice versa). The probability of adefinite species location being mutated is set equal to zero. Thenumber of mutations carried out along the child chromosome,nmut is specified at the start of the algorithm. After mutationthe number of species in the new child chromosome must equalnreduced

s . Thus, for nmut = 1, mutation involves randomly choos-ing a non-definite species location and flipping the value in thatposition. If the value is a 1 another location containing a 0 is ran-domly chosen and flipped so that the number of species remainsequal to nreduced

s .Once the children population has been constructed using

crossover and mutation, Eq. (4) is again used to calculate eachchild’s fitness. A standard filter function (see Harris et al., 2000)is then used to generate a new population by merging the fittestparents and children. From this new population the whole pro-cess is repeated with selection, crossover, and mutation usedto generate a new children population and so on until conver-gence is reached where no improvement is observed to the fittestindividual. The GA reduction parameters used in the study aredescribed in Section 4.1.

3

3

apArttbatdEmTaretcbsuEutEaD

Delfau, Akrich, and Vovelle (1995) of the kth species and jthpremix condition. Also, Wk in both the PSR and PREMIX con-tributions to Eq. (4) are set to 1 for each of the species whereexperimental data exists and 0 for those species not measured.

foptimisePSR ((Ai, βi, Eai )i=1,NR

)

=⎧⎨⎩10−8 +

NPSR∑j=1

KPSR∑k=1

Wk

∣∣∣Xcalcjk − X

origjk

∣∣∣X

origjk

⎫⎬⎭

−1

(5)

3.2.2. The hybrid GA/Rosenbrock approachIt is interesting to note that in the standard GA approach

adopted in the previous section, less than 1% of the CPU timeis associated with the GA. In fact one generation involves call-ing the PSR code 10 (number of individuals) × 10 (number ofPSR conditions) and the PREMIX code 10 (number of indi-viduals) × 1 (number of PREMIX conditions). Thus, with 1000generations needed for a converged solution the whole proceduretook 1 week to run on a 3.2 GHz Pentium 4 processor. Thus, inorder to decrease what some modellers might find as excessiveCPU times, a more direct optimisation method was introduced.

The Rosenbrock method (1960) begins by considering eachcoordinate direction to be a search direction. It then conductssearches along these directions, cycling over all variables in turn,moving to new points that yield success. This continues untiltts

ipx

afnewbfx

Etmmx

eattcifinrs

.2. Step 2: reduced mechanism optimisation

.2.1. The standard GA approachHaving found a reduced set of species and their associ-

ted reactions that best matches a simulated set of species’rofiles generated using the starting point reaction mechanismFRMv1.1, the next step was to optimise this reduced set of

eaction rate coefficients. This represents a non-trivial task ashe large number of parameters in the problem creates a solu-ion landscape that is uneven making it unsuitable for gradientased methods. Thus, a real coded GA was adopted. The GApproach to reaction rate coefficient optimisation is not a newechnique and so only a brief description will be given here. Aetailed description can be found elsewhere in the studies oflliott et al. (2002a, 2002b, 2003a, 2003b, 2004). The overallethodology is similar to that described in the previous section.he differences lie in how the parent and children chromosomesre defined. This time an individual consists of a set of reactionate coefficients (real numbers), i.e. Ai, βi, and Eai (Eq. (1)) forach of the reactions included in the reduced model. The ini-ial population consists of a number of sets of these reaction rateoefficients where each coefficient lies between a predeterminedound. Having constructed the initial population k-tournamentelection is used to select parents which undergo the traditionalniform arithmetic crossover and non-uniform mutation (seelliott et al., 2004). The fitness of the individuals is calculatedsing a similar criterion described by Eq. (4). However, thisime the PSR contribution to the combined fitness is given byq. (5) where X

origjk and Y

origjk represent an experimental PSR

nd PREMIX species concentration measurement taken fromagaut, Reuillon, Boetner, and Cathonnet (1994) and Doute,

here is at least one success followed by a failure. This stageerminates after one success and one failure is obtained for eachearch direction.

In the case of the optimisation of reaction rate coefficients,f we consider the kth iteration of the search at the startingoint Pk, a set of 3NR mutually orthonormal search directionsk1, x

k2, . . . , x

k3NR

and a set of 3NR step lengths δ1, δ2, . . . , δNR

re required. The starting point search directions were obtainedrom running a standard GA with a population size of 8 and theumber of offsprings set to 10. The GA was run for just 100 gen-rations and these partially optimised reaction rate coefficientsere then taken to be the 3NR search directions. The Rosen-rock iteration begins by exploring the given set of directions asollows. Starting from Pk a step δ1 is taken along the directionk1 and if it does not result in an increase in the fitness function ofq. (4) (note this is a minimisation method) then it is considered

o be successful, the new (improved) point is retained, and δ1 isultiplied by α > 1. If the step is a failure it is rejected and δ1 isultiplied with β where −1 < β < 0. The search continues with

k2 in the same way and so on until all 3NR directions have beenxplored and it returns to xk

1 again and continues the cycle untilsuccess is followed by a failure has been recorded at some

ime during the iteration in every direction. The effect of mul-iplying the step sizes by the factors α and β is to expand or toontract and reverse the exploration step in a direction accord-ng to whether the previous step in that direction succeeded orailed. The second part of the iteration is concerned with defin-ng a new set of orthonormal search directions for the use in theext iteration. If Pk+1 is the point arrived at as a result of explo-ation from Pk then the direction of total progress is given by1 = Pk+1 − Pk which can be written s1 = ∑3NR

j=1∆jxkj where ∆j


is the algebraic sum of all the successful steps along the direc-tion xk

j . Since the exploration continues until at least one successhas been achieved in each direction, ∆j �= 0 for all j. The vectorsj where sj = sj−1 − ∆j−1x

kj−1, j = 2, . . ., 3NR represents the

total progress made in all directions other than xk1, . . . , x

kj−1.

The new first direction for the next iteration is chosen to be thenormalised direction of total progress in the present iteration,i.e. xk+1

1 = s1/ ‖s1‖ and the remaining directions are chosento form an orthonormal set with xk+1

1 using the Gram-Schmidt

process. Thus, for j = 2, . . ., 3NR, dj = sj − ∑j−1i=1 (sT

j xk+1i )xk+1

i ,

xk+1j = dj/

∥∥dj

∥∥.Rosenbrock did not include any convergence test in his

method and merely terminated the search after a specified num-ber of function evaluations. However, he did suggest monitoringthe value of ‖s2‖ / ‖s1‖ , which measures the progress madealong all directions other than the first as a fraction of the totalprogress, and that if this remains generally less than 0.3 then itis unlikely that the minimum is located. Also he recommendsthe values of step adjustment parameters α = 3, β = −0.5.

4. Results and discussion

4.1. Mechanism reduction

The starting point mechanism adopted was a surrogate avi-aPcsgtpτ

spmbfarf

frtgrdcsTlsnfy

1 PREMIX case. Thus, for the purpose of full enumeration thetime taken for each individual function evaluation had to bereduced and a much smaller search domain had to be tested. Inorder to achieve this, a test case involving methane–air com-bustion for just one PREMIX condition was considered. TheGRI3.0 reaction mechanism (Smith et al., 2006) was used to gen-erate simulated species profiles (nfull

s = 53) and subsequentlythe search could be reduced by considering a reduced mecha-nism with 16 species (nreduced

s = 16) and the number of definitespecies being ever present increased to 12 (ndef = 12). In doingthis the total number of all possible combinations and thereforefunction evaluations was 53–12C16–12 = 101,270. While 101,270calls to PREMIX may still seem excessive it should be pointedout that many of the reaction mechanisms were small and causedPREMIX to terminate after less than 1 s. The total CPU timetaken for the full enumeration on a 3.2 GHz Pentium 4 was 2days. The 12 ever present species in the reduced mechanismwere: CH4, O2, N2, Ar, CO2, H2O, CO, H2, O, OH, H, and CH3.The fitness of all possible combinations of choosing the final 4species from the remaining 41 species of the full mechanism wascalculated using Eq. (3) and the best fitness and correspondingreduced reaction mechanism was stored. For the same condi-tions the GA was set the task of finding this optimum reducedreaction mechanism. Table 1 demonstrates the success of theGA in arriving at this optimum solution for a number of casesof interest. In addition to increasing confidence in the abilityoinotwlqasapcarscs

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tion fuel-air reaction mechanism, AFRMv1.1, developed byatterson et al. (2000). The full mechanism consists of 67 speciesompeting in 338 reactions and was used to simulate a set ofpecies profiles. The first set of simulated species profiles wereenerated by running CHEMKIN’s PSR code for 10 differentemperatures in the range 820 K < T < 1150 K with the operatingressure, p = 10 atm; equivalence ratio, φ = 1.5; residence time,= 0.5 s; the initial fuel mole fraction of 10−3. Further, a set of

pecies profiles generated by running CHEMKIN’s PREMIXrogram that modelled an atmospheric pressure, fuel rich, pre-ixed aviation fuel/O2/N2 flame (φ = 1.7) was chosen as the

asis for a comparison with the model. The input premixeduel/oxidiser mixture was 2.95% aviation fuel (2.63% n-decanend 0.32% toluene), 28.64% O2 and 68.4% N2 and a mass flowate of 0.0099 g cm−3 s−1 was employed. In both cases aviationuel was modelled as 89% n-decane and 11% toluene.

In order to gain confidence in our reduction technique aull enumeration was conducted to establish the best possibleeduced mechanism that exists based on Eq. (4). This meanshat we perform an exhaustive search, looking at every sin-le combination of the nreduced

s species and their associatedeactions. Then having identified the set that gives the best pre-iction to a set of species’ profiles, for a given set of boundaryonditions, we ensure that the GA can find the same nreduced

species and their associated reactions in a much shorter time.he difficulty in a full enumeration is the vast size of the prob-

em. For instance, in finding the optimum 50 species reducedcheme from the full 67 present in AFRMv1.1, where 12 defi-ite species are ever present, involves 67–12C50–12 = 6.82 × 1013

unction evaluations which is computationally unfeasible whenou consider that each evaluation involves running 10 PSR and

f the GA to find the optimum solution, the cases considerednvestigate the sensitivity of the mutation parameters, pmut andmut, on the rate of convergence. The departure of our mutationperator from more traditional mutation operators has motivatedhe interest in these particular parameters. In all cases the GAas able to arrive at the optimum reduced reaction scheme in

ess than 60 generations. The average number of generationsuoted is calculated from a total of six runs where each run useddifferent starting seed for the random number generator. The

tarting seed determines the initial population. Thus, an aver-ge of six different starting seeds is preferred in case the initialopulation contains a very fit individual, increasing the rate ofonvergence. In taking an average it is a fairer judge as to whethermutation parameter is better than another. While the problem

epresents a relatively simple one for a GA, nevertheless thepeed at which the GA finds the solution is reassuring and breedsonfidence that the method can be extended to successfullyearching a larger parameter space. Following these calculations

able 1nfluence of pmu and nmut on CPU runtime costs in arriving at the best solutionor a simulated problem where there are 101,270 possible function calls

mu nmut Average numberof generations

Average numberof function calls

ApproximateCPU time (s)

.1 1 35 1074 1832

.1 2 21.7 675 1152

.1 3 37.7 1035 1766

.01 2 44.2 1350 2304

.05 2 33.2 1020 1740

.2 2 26.3 813 1387


the genetic operators and GA parameters were taken to be asfollows:

• population size npop = 24;• number of offsprings nchild = 30;• tournament selection, tournament size k = 2, tournament prob-

ability pt = 0.8;• the probability a child is selected for mutation pmut = 0.1;• the number of mutations carried out along the child chromo-

some nmut = 2;• elitism, elitism parameter ne = 2.

It is difficult to comment in terms of the sensitivity that eachof these GA parameters has on the overall accuracy/efficiencyof the model. This is because each parameter can be influ-enced by the others. For instance a particular tournamentsize may work well only when the population size is large.However, it should be pointed out that the speed of conver-gence is very much dependent on the initial population sothat a good starting estimate will enhance the efficiency of themethod.

Having gained confidence in our methodology the nexttask was to return to the problem of interest which was tofind the optimal 50 species and their associated reactions thatbest represent the full AFRMv1.1 in modelling the 10 PSRand 1 PREMIX case described at the start of this section.IctfbCtael2hImnmprtsti5m

spSfti

fers noticeably in the PREMIX comparisons. While going from338 reactions involving 67 species to 215 reactions involving 50species may not seem a dramatic reduction, a 90% CPU savingwas apparent in each function evaluation. Although it has notbeen documented here, it was found that reaction mechanismscontaining less than 50 species did not perform well at predict-ing the simulated data of the full starting point mechanism. Thisis because there becomes a point beyond which a further reduc-tion of the mechanism results in an inadequate description of thecombustion processes.

4.2. Reduced mechanism optimisation

The second step of the investigation uses a number of meth-ods in order to tackle the problem of keeping the same set ofspecies/reactions but finding the best set of reaction rate con-stants that best fit an experimental set of jet stirred reactor (JSR)and premixed species’ profiles (see Dagaut et al., 1994; Douteet al., 1995). The models use the same PSR and PREMIX flameoperating conditions described in the previous section and thebest set of reaction rate coefficients was determined based onthe fitness function of Eq. (4) but this time Eq. (5) represents thePSR contribution to the combined fitness.

Fig. 4 compares the performance in fitting the experimentaldata against function evaluation number for three different opti-misation approaches. Further, the horizontal lines correspondingtfitt

•

•

•

•

n order to enhance conversion 12 definite species werehosen based on their importance both in quantity and inheir popularity with regard to the number of reactions theyeature in. The 12 definite species were defined at the start toe: C10H22, C6H5CH3, O2, H2O, CO, CO2, H2, H, O, OH,6H6, and N2 while the 38 deficit species were chosen from

he remaining 55 species of the full mechanism. In reachingsolution the GA was allowed to run for 500 generations

ven though the fitness evolution had reached a plateau afteress than 200 generations. The final mechanism consisted of15 reactions involving 50 species and is available online via:ttp://www.personal.leeds.ac.uk/∼fuensm/project/mech.html.n order to add further credence to our method, a further test wasade using full enumeration. Taking the first 46 species of our

ew reduced reaction scheme to be definite species, a full enu-eration was performed on the remaining 67–46C50–46 = 5985

ossible combinations of choosing 4 deficit species from theemaining 21 species of the full mechanism in order to confirmhat there were no other combinations involving 4 differentpecies that gave an improved fitness based on Eq. (4). Althoughhis is by no means a guarantee of the method’s success ast assumes the first 46 species will be part of the optimal0 species set, it does provide additional confidence in theethod.The full and dotted lines of Fig. 2 compare species profiles

imulated using AFRMv1.1 with the GA reduced mechanism’sredictions over the 10 PSR conditions that were considered.imilarly Fig. 3 compares the same mechanisms’ predictionsor the laminar premixed flame calculations. It is reassuring thathe new GA reduced mechanism’s PSR predictions are almostndistinguishable from the simulated data while only C2H2 dif-

o the two models that do not undergo any reaction rate coef-cient optimisation, AFRMv1.1 and Reduced have been added

o the figure for comparison. The models compared correspondo:

AFRMv1.1: The full reaction mechanism does not undergoany optimisation in this study and thus its fitness remainsconstant. However, during its construction it has been manu-ally tuned to give the best fit to the same experimental profilesgiven in Figs. 2 and 3 (see Patterson et al., 2000).Reduced: The reduced mechanism obtained using the binaryGA without any optimisation done on its reaction rate coeffi-cients.Rosenbrock: The Rosenbrock method is used to search forthe best set of reaction rate coefficients based on Eqs. (4) and(5). A random initial set of search directions (reaction ratecoefficients) is used to start the method. The reaction ratecoefficients were normalised and a step length = 0.1 with stepadjustment parameters α = 3, β = −0.5 were used. Functionevaluations were only performed if the reaction rate coef-ficients obeyed the constraints associated with the NationalInstitute of Standards and Technology database (see Elliott etal., 2003a).Reduced GA 8-10: Similarly, a standard real coded GA start-ing from a random initial population is used for the optimisa-tion. The genetic operators and GA parameters were taken tobe as follows:◦ population size npop = 8;◦ number of offsprings nchild = 10;◦ uniform arithmetic crossover, crossover probability

pc = 0.65;

http://www.personal.leeds.ac.uk/%7efuensm/project/mech.html


Fig. 2. Concentration profiles of various output species for aviation fuel oxidation in a jet stirred reactor at P = 10 atm, residence time = 0.5 s and equivalence ratio = 1.5as functions of the inlet temperatures. Comparisons are made between the full AFRMv1.1 mechanism (—), the GA reduced mechanism (····) the GA reduced/hybridoptimised mechanism ( ) and the measured experimental data of Dagaut et al. (1994) (�).

◦ tournament selection, tournament size k = 2, tournamentprobability pt = 0.8;

◦ non-uniform mutation, mutation probability pm = 0.5;◦ elitism, elitism parameter ne = 2.

It is very difficult to say that one parameter value works bet-ter than another as they are so dependent on the optimisationproblem. One can only propose to choose values which havebeen successful in previous works (see Elliott et al., 2002a,

2002b, 2003a, 2003b, 2004; Harris et al., 2000) or when anew problem arises perform simulated tests (the exhaustivesearch test was performed in this study).

• Reduced Hybrid GA: A combination of the GA and the Rosen-brock methods are used to again optimise the reaction ratecoefficients to best fit the experimental data.

First a GA is run for 100 generations (1000 function eval-uations are involved) with the starting population, genetic


Fig. 3. Concentration profiles of various species as functions of the distance from the surface of the burner for a rich atmospheric premixed aviation fuel flameobtained by the full AFRMv1.1 mechanism (—), the GA reduced mechanism (····) the GA reduced/hybrid optimised mechanism ( ) and the measured experimentaldata of Doute et al. (1995) (�).

operators and GA parameters identical to those of the previousReduced GA 8-10 model. The output reaction rate coefficientsof the GA are then used as the starting search directions for aRosenbrock method which was run for a further 6500 functionevaluations although it is clear that a plateau is reached afterjust 3000. Again the reaction rate coefficients were normalised

and a step length = 0.1 with step adjustment parameters α = 3,β = −0.5 were used.

It can be seen from Fig. 4 that the GA is set up to maximisethe fitness function given by Eq. (4). It can also be seen fromthe figure that the Reduced Hybrid GA provides a remarkable


Fig. 4. Evolution of fitness as a function of function evaluation number for anumber of optimisation models for reaction rate coefficient optimisation.

improvement over the standard Reduced GA 8-10 model. Notonly does it produce a significant improvement to the correla-tion with the experimental measurements based on Eqs. (4) and(5) but it does so in a fraction of the cost in CPU time. It canalso be seen from the Rosenbrock profile that direct use of theRosenbrock method is unsuitable for our problem and its highlyunstructured solution landscape as it is too dependent on its ini-tial starting point.

The improvement in the fitness produced by Reduced HybridGA over the AFRMv1.1 and Reduced models can be seen in termsof the models’ prediction of the experimental species’ profilesof Dagaut et al. (1994) and Doute et al. (1995) in Figs. 2 and 3,respectively. It can be seen from Fig. 3 that the Reduced HybridGA shows a remarkable agreement with the measured pro-files with what is sometimes (as in the case of CO2 and H2)a significant improvement over the more detailed AFRMv1.1’spredictions. It can be seen from Fig. 4 that the excellent PSR pre-dictions are not at the expense of the PREMIX profiles whichagain show an improvement over the full and reduced models’with the exception of H2 and C6H6. While the precise reasonsfor the poor prediction of the H2 and C6H6 profiles is unknownit is thought that the under predicted benzene profile is partlydue to the under predicted C2H2 profile which is a precursor tobenzene formation through the hydrogen abstraction acetyleneaddition pathway. The confidence in the GA procedure is suchthat if a possible improvement existed then it would have beenfwbi

umauGdaL

Fig. 5. Ignition delay time profiles for aviation fuel-air mixtures, initial pressure1 atm and equivalence ratio 0.5. The models are compared with the measure-ments of Freeman and Lefebvre (1984).

and PREMIX predictions given by Reduce hybrid GA mecha-nism are at the expense of over predicting ignition delay. It isthought that future studies could incorporate these ignition delaymeasurements into the optimisation process to provide a morewidely applicable reaction model. However, Fig. 5 does demon-strate the success of the reduction process as the predictions ofignition delay by Reduced and AFRMv1.1 are indistinguishablefrom each other.

5. Conclusions

A two-step approach to reaction mechanism reduction hasbeen successfully demonstrated. The first step takes a surrogateaviation fuel-air reaction mechanism and using a binary encodedGA finds the optimum subset of species and associated reactionsthat best represents the simulated species’ profiles generated bythe full starting mechanism. The number of species present ineach reduced subset was defined at the start of the process and itwas found that specifying less than 50 species led to unsatisfac-tory reaction mechanisms. Full enumeration was conducted ona reduced problem in order to gain confidence that the GA wasindeed finding the optimum reduced reaction set. The reductioninvolved going from 338 reactions involving 67 species to 215reactions involving 50 species. This corresponded to a 90% CPUtime saving in each function evaluation.

onsdtRass

A

f

ound. Therefore, the authors conclude that within the frame-ork of the present mechanisms, a better correlation can onlye realised once more reaction pathways are added or the errorsn the experimental data are larger than those quoted.

In order to validate the models against experimental datander conditions that lay outside those used during the opti-isation process, the ignition delay measurements of Freeman

nd Lefebvre (1984) for aviation fuel-air combustion weresed as a bench mark. Fig. 5 compares the Reduced HybridA, AFRMv1.1, and Reduced models’ predictions of ignitionelay for aviation fuel-air mixtures of equivalence ratio 0.5 attmospheric pressure with the measurements of Freeman andefebvre (1984). It can be seen that the improvements in the PSR

The second step tested a number of models in finding theptimum set of reaction rate coefficients, for the reduced mecha-ism generated in step 1, which best represented an experimentalet of premix species’ profiles. A standard real coded GA, airect Rosenbrock method, and a hybrid method combining bothhe GA and the Rosenbrock approaches were investigated. Theosenbrock method failed to improve upon the standard GApproach. However, not only did the hybrid scheme produce aignificant improvement in fitness over the standard GA, it dido in a fraction of the CPU time.

cknowledgement

The authors would like to thank the EPSRC for the fundingor this work.


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Documents

Reaction mechanism reduction and optimisation for modelling aviation fuel oxidation using standard and hybrid genetic algorithms