A MODIFIED MULTI-POPULATION GENETIC ALGORITHM

FOR PARAMETER IDENTIFICATION OF CULTIVATION

PROCESS MODELS

Olympia N. Roeva, Kalin Kosev

Centre of Biomedical Engineering, Bulgarian Academy of Sciences, 105 Acad. G. Bonchev Str., Sofia 1113, Bulgaria

Tanya V. Trenkova

Institute of Water Problems, Bulgarian Academy of Sciences, 1 Acad. G. Bonchev Str., Sofia 1113, Bulgaria

Keywords: Multi-population Genetic Algorithm, Modification, Mutation, Identification, Cultivation Process.

Abstract: In this work a modified multi-population genetic algorithm (MPGA) without the performance of the

mutation operator is proposed. The idea is to reduce the convergence time and therefore to increase the

identification procedure effectiveness for on-line application of the algorithm. Modified MPGA, classical

multipopulation GA and two other modifications are tested for parameter identification problem of an E.

coli non-linear fed-batch cultivation model. The contribution of each modification measure to the

performance improvement is demonstrated. The obtained results show that the highest accuracy for

parameter identification of the considered model is achieved with the multipopulation GA with

Modification 1. The best calculation time is shown by the multipopulation GA without mutation.

1 INTRODUCTION

The most popular stochastic optimization method is

the evolutionary computation. This is a class of

methods based on the ideas of biological evolution,

which is driven by the mechanisms of reproduction,

mutation, and the principle of survival of the fittest.

Several different types of evolutionary search

methods have been developed independently. One of

them are genetic algorithms (GAs) (Goldberg,

1989), which focuses on optimizing general

combinatorial problems. The GAs are highly

relevant for industrial applications, because they are

capable of handling problems with non-linear

constraints, multiple objectives, and dynamic

components – properties that frequently appear in

real-world problems.

The GAs are widespread optimization techniques

and finding applications in a large scope of

problems. The application of GAs in bioprocess

optimization had been reported in early 1996. Since

then GAs are used widely in the field of

bioprocesses engineering as an alternative

optimization tool to conventional methods (Na,

2002, Roeva, 2009).

Many variations of the standard genetic

algorithm, as presented by Goldberg (Goldberg,

1989), can be found in the literature. Modifications

and hybridizations have been motivated by a desire

to improve the performance of the GA, and to adapt

them to particular problem domains (Alsumait,

2010, Kim, 2007, Roeva, 2006). The purpose of this

work is to propose a modification of GA, improving

the convergence time of the algorithm for a specific

problem – parameter identification of non-linear fed-

batch cultivation process of E. coli.

Better results can be obtained by introducing

many populations, called subpopulations compared

to the standard GA. Each subpopulation evolves for

a few generations isolated before one or more

individuals are exchanged between the

subpopulations. Thus the evolution of a species

resulting in multi-population genetic algorithm

(MPGA), is more similar to nature than the single

population GA.

In this work a modified MPGA (MMPGA) is

proposed and only the operator crossover is

348

N. Roeva O., Kosev K. and V. Trenkova T..

A MODIFIED MULTI-POPULATION GENETIC ALGORITHM FOR PARAMETER IDENTIFICATION OF CULTIVATION PROCESS MODELS.

DOI: 10.5220/0003080603480351

In Proceedings of the International Conference on Evolutionary Computation (ICEC-2010), pages 348-351

ISBN: 978-989-8425-31-7

 2010 SCITEPRESS (Science and Technology Publications, Lda.)

performed. The direct replacement is used where

parents are replaced by their offspring (Rowe, 1995).

When two chromosomes crossover, they are both

replaced by the resulting offspring, then all the

original alleles are preserved. This ensures no loss of

alleles in the subpopulations. Mutation operator is

not performed. The examined GA are tested with a

problem for parameter identification of non-linear

model of fed-batch cultivation process of E. coli.

2 MULTI-POPULATION

GENETIC ALGORITHM

WITHOUT MUTATION

Generally the last operator in the GA is the mutation

algorithm. The importance of its role is still a matter

of debate. Most authors however, consider that

mutation plays a secondary role in the genetic

algorithm. Other authors define mutation as an

opportunity to prevent the solution from entering in

a local maximum.

In connection with the algorithm convergence,

some authors propose increase of mutation rates

(Louis, 1993). A GA converges when most of the

population is identical, i.e. the diversity is minimal.

Therefore the increasing of mutation rates is the

usual way of maintaining the diversity. Although the

high mutation rates may increase the diversity, its

random nature raises problems. Mutation is as likely

to destroy good schemes as bad ones and therefore

elitist selection is needed to preserve the best

individuals in a population (Louis, 1993).

The role of every operator in GA as well as the

role of the mutation operator depends mainly on the

specific problem. For the considered problem here –

model parameter identification, the operator

mutation can be eliminated. The experiments show

that in this way the better convergence time is

achieved without loss of the solution accuracy.

The proposed MMPGA works in a similar way

compared to the SMPGA. The subpopulations

evolve independently from each other for a certain

number of generations (isolation time), like the

single population GA. After the isolation time a

number of individuals is distributed between the

subpopulations (migration). The migration rate, the

selection method of the individuals for migration

and the scheme of migration determines how much

genetic diversity can occur in the subpopulations and

the exchange of information between

subpopulations. The selection of the individuals for

migration can be uniform at random (pick

individuals for migration in a random manner) and

fitness-based (select the best individuals for

migration). There are many variants of the migration

structure of the individuals between subpopulations.

The most general migration strategy is that of

unrestricted migration (complete net topology).

Here, individuals may migrate from any

subpopulation to another. For each subpopulation, a

pool of potential immigrants is constructed from the

other subpopulations. The individual migrants are

then uniformly at random determined from this pool.

3 FED-BATCH CULTIVATION

PROCESS OF E. COLI

As a test problem a fed-batch cultivation process of

E. coli is considered. Cultivation of recombinant

micro-organisms e.g. E. coli, in many cases is the

most economical way to produce pharmaceutical

biochemicals such as interleukins, insulin,

interferons, enzymes and growth factors.

For the parameter identification real

experimental data are used. Detailed description of

the fed-batch cultivation process of E. coli strain

MC4110 is presented in (Arndt, 2004).

The mathematical model of the considered

process has the form (Crueger, 1984):

max

dX S F

dt k S V

−

(1)

()

max in

S/X S

dS S F

X+ S S

dt Y k S V

−

(2)

max

A/ X S

dA S F

dt Y k S V

−

(3)

(4)

where: X is the concentration of biomass, [g/l];

S – concentration of substrate (glucose), [g/l];

A – concentration of acetate, [g/l]; F – feeding rate,

[l/h]; V – bioreactor volume, [l]; S

– substrate

concentration of the feeding solution, [g/l];

max

– maximum growth rate, [h

-1

];

k – saturation

constant, [g/l];

S/X A/X

YY – yield coefficient, [-].

The optimization criterion is presented as a

minimization of a distance measure J between

experimental and model predicted values of state

variables as follows:

XSA

JJJ min

++→

(5)

A MODIFIED MULTI-POPULATION GENETIC ALGORITHM FOR PARAMETER IDENTIFICATION OF

CULTIVATION PROCESS MODELS

349

() ()

()

X exp mod

XiX i

=−

∑

(6)

() ()

()

Sexpmod

SiS i

=−

∑

(7)

() ()

()

Aexpmod

AiA i

=−

∑

(8)

where X

exp

, S

exp

, А

exp

are the vectors of experimental

data for biomass, substrate and acetate, X

mod

, S

mod

– the vectors of simulated data, n – is the

number of data for each variable.

4 RESULTS AND DISCUSSION

For the problem of parameter identification of model

(1) – (4), with an optimization criterion (5) four

genetic algorithms are compared:

• SMPGA: Standard MPGA (Goldberg, 1989);

• MPGA Modification 1: modified MPGA based on

modification proposed in (Roeva, 2006);

• MPGA Modification 2: MPGA without mutation

– here proposed modification;

• MPGA Modification 3: MPGA realized using

both Modifications 1 and 2.

MPGA Modification 1. The reproduction

determines which chromosomes will be chosen as

the basis of the next generation. Generating

populations from only two parents may cause loss

of the best chromosome from the last population.

The obtained good solution may be destroyed by

either the crossover or the mutation or both

operations. Thereby, the best solution in GA pops

up from the new population may be inferior to the

old generations. The aim of the modification

(Roeva, 2006) is to prevent this disadvantage.

The modified GA possesses a structure similar to

the standard GA. However, the modified GA

distinguishes itself from the standard GA in a way

the reproduction is processed after both the

crossover and mutation have been performed. Thus

the deterioration problem never happens since the

best solution from the current generation will be

superior to or at least the same as the past.

Using the modification proposed in (Roeva,

2006), a modified MPGA is realized – (MPGA

Modification 1).

MPGA Modification 2. Modified multi-population

genetic algorithm proposed in this paper is

considered as MPGA Modification 2.

MPGA Modification 3. The MPGA Modification 3

is realized based on application of the two

modifications described above. A MPGA without

mutation operator and reproduction processed after

the crossover operator is developed.

All numerical experiments are done on Windows

Vista platform, with an Intel Core2Duo, 2.16 GHz,

3GB DDRIII RAM.

The considered GAs are realized in Matlab 7.5

environment.

As a suitable genetic operators and parameters

different authors propose different solutions,

depending on the specific problem. The defined in

this work GA operators and parameters are based on

previous studies of the considered problem here –

parameter identification of cultivation process model

(see Roeva, 2007).

The parameter identification problem of the

model (1) – (4) is solved on the basis of real

experimental data for process variables – biomass,

substrate and acetate (Arndt, 2004).

The obtained results (the values of the

optimization criterion (J), as well as the values of

the criteria

J ,

and the convergence time

(T)) from the four GA – SMPGA, MPGA

Modification 1, MPGA Modification 2, MPGA

Modification 3 are presented in Tables 1. For each

MPGA are presented estimates and criterion mean

values of 30 runs (average). The results for minimal

(min time) and maximum (max time) computing

time are also shown. The results show that the

algorithm produces the same estimations with more

than 85% coincidence.

Considering the three indicators (average value,

minimal time and maximum time) it is clearly

noticeable that MPGA Modification 1 finds the

solution for less computing time compared to

SMPGA. Using this modification, the best accuracy

of the solution is also achieved. The obtained results

confirm that the modification (Roeva, 2006)

prevents the loss of “the best chromosome” and

achieves an increase in solution accuracy.

The best computing time for the three indicators

is shown by MPGA Modification 2 (see Table 4).

The elimination of the operator mutation decreases

considerably the computing time. Minimal solution

time of 195.36 s is achieved. The error in this case is

slightly higher. The value of the optimization

criterion for minimal time solution is 4.0749

compared to the one of MPGA Modification 1 –

3.9090. The mutation operator changes the

individual representation by introducing new genetic

material to the gene pool. For this reason, mutation

operator tends to preserve or increase the diversity

ICEC 2010 - International Conference on Evolutionary Computation

350

of the population. As the new material is completely

untested, mutation operator often ends up decreasing

the fitness of an individual and increasing the

convergence time.

Table 1: Results from parameter identification – criteria

values and convergence time.

GA Indicator J

J T, s

SMPGA

average 2.3831 1.2583 0.0004 3.6418 316.0752

min time 2.4876 1.2554 0.0004 3.7434 288.1806

max time 2.4363 1.1442 0.0005 3.5810 359.8007

Modif. 1

average 2.4198 1.3892 0.0012 3.8102 277.8347

min time 2.3131 1.5951 0.0008 3.9090 259.2737

max time 2.2126 1.2794 0.0019 3.4939 297.6343

Modif. 2

average 2.4615 1.3631 0.0021 3.8267 203.0400

min time 2.4631 1.6115 0.0003 4.0749 195.3601

max time 2.3878 1.6648 0.0004 4.0530 214.7198

Modif. 3

average 2.4276 1.3830 0.0023 3.8130 240.6050

min time 2.5629 1.5171 0.0043 4.0844 208.8073

max time 2.4597 1.0401 0.0008 3.5005 264.4685

When the operator mutation is eliminated in the

proposed in (Roeva, 2006) modification of GA –

MPGA Modification 3, the convergence time is

decreased compared to MPGA Modification 1 –

from 259.27 s to 208.80 s. The proposed MPGA

Modifications 2 and 3 considerably decrease the

convergence time of GA, and in the same time the

increase of the error is slightly smaller.

5 CONCLUSIONS

Based on performed numerical experiments the

following conclusions for the performance of the

examined MPGA could be generalized:

1. Applying MPGA Modification 1, the estimates of

the considered model parameters with highest

accuracy are obtained. The value of the optimization

criterion J is 3.4939 obtained for a time of 297.6343

2. By the MPGA Modification 2 the best

convergence time is achieved. The average results

are: J = 3.8267 and T = 203.04 s. The obtained

minimal time for solution finding is 195.36 s with an

optimization criterion value of 4.0749.

As a result from the conducted experiments and

analysis of the received data the multi-population

genetic algorithm without mutation (MPGA

Modification 2) is defined as suitable for on-line

application for optimization and control of

bioprocesses. This is the algorithm with the best

convergence time and in the same time the accuracy

of the model is comparable with the higher accuracy

achieved by MPGA Modification 1.

ACKNOWLEDGEMENTS

This work is partially supported by the National

Science Fund Grants DMU 02/4 and DID-02-29.

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A MODIFIED MULTI-POPULATION GENETIC ALGORITHM FOR PARAMETER IDENTIFICATION OF

CULTIVATION PROCESS MODELS

351