Bulgarian Academy of Sciences A MODIFIED MULTI-POPULATION GENETIC ALGORITHM FOR PARAMETER IDENTIFICATION OF CULTIVATION PROCESS MODELS Olympia N. Roeva, Kalin Kosev, Tanya V. Trenkova Institute of Biophysics and Biomedical Engineering, …… olympia@clbme.bas.bg, … 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. Three genetic algorithms (GAs) are compared to modified MPGA – classical multi-population GA and two other modifications. The algorithms 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 (Modification 2). Table 1. Genetic algorithm operators and parameters Operator Type Parameter Value encoding binary generation gap 0.97 crossover double point crossover rate 0.70 mutation bit inversion mutation rate 0.1 reinsertion fitness-based migration rate 0.2 selection roulette wheel selection precision of binary representation 20 fitness function linear ranking subpopulations 10 isolation time number of individuals 100 number of generations APPLICATION OF GA FOR FEED RATE PROFILES SYNTHESIS Multi-population genetic algorithm without mutation 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. After the isolation time a number of individuals is distributed between the subpopulations. 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. 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. The obtained results from the four GA – SMPGA, MPGA Modification 1, MPGA Modification 2, MPGA Modification 3 are presented in Table 2. 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. Outline of the proposed genetic algorithm could be presented as: 1. [Start] Generate k random subpopulation of n chromosomes 2. [Fitness] Evaluate the fitness f(x) of each chromosome x in the subpopulation 3. [New population] Create a new population by repeating following steps until the new subpopulation is complete 3.1. [Selection] Select two parent chromosomes from the subpopulations 3.2. [Crossover] With a crossover probability cross over the parents to form new offspring 3.3. [Accepting] Place new offspring in the new subpopulations 4. [Replace] Use new generated populations for a further run of the algorithm 5. [Test] If the end condition is satisfied, stop, and return the best solution in current subpopulation 6. [Loop] Go to step 2 Table 2. Results from parameter identification – estimates GA Indicator µmax kS YS/X YA/X SMPGA average 0.5716 0.0422 2.0197 0.0136 min time 0.5760 0.0434 2.0192 0.0152 max time 0.5601 0.0388 2.0200 0.0158 Modif. 1 0.5843 0.0459 2.0201 0.0129 0.6000 0.0514 2.0197 0.0110 0.5752 0.0430 2.0206 0.0193 Modif. 2 0.5780 0.0445 2.0204 0.0170 0.6102 0.0531 2.0199 0.0137 0.6082 0.0535 2.0211 0.0127 Modif. 3 0.5833 0.0458 2.0207 0.0170 0.5958 0.0501 2.0202 0.0223 0.5483 0.0352 2.0211 0.0169 FED-BATCH FERMENTATION PROCESS OF E. COLI MC4110 The mathematical model of fed-batch fermentation of E. coli MC4110 can be represented by the following dynamic mass balance equations: ; (1) (2) (3) The best computing time for the three indicators is shown by MPGA Modification 2. 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 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. (4) The optimization criterion is presented as a minimization of a distance measure J between experimental and model predicted values of state variables as follows: CONCLUSIONS The proposed modification of MPGA is compared to the standard MPGA and two other modifications. Here considered MPGA are tested for problem of non-linear model parameter identification. Real experimental data of E. coli fed-batch cultivation process are used. 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 s. 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. RESULTS AND DISCUSSION For the problem of parameter identification of model (1) – (4), with an optimization criterion (5) four GA are compared: • SMPGA: Standard MPGA; • MPGA Modification 1: Modified MPGA; • MPGA Modification 2: MPGA without mutation – here proposed modification; • MPGA Modification 3: MPGA realized using both Modifications 1 and 2. The basic GA operators and parameters of the proposed here MPGAs are summarized in Table 1. 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. ACKNOWLEDGEMENTS: This work is partially supported by the National Science Fund Grants DMU 02/4 “High quality control of biotechnological processes with application of modified conventional and metaheuristics methods” and DID-02-29 “Modelling Processes with Fixed Development Rules” . 24 – 26 October, 2010 Valencia, Spain ICEC 2010 International Conference on Evolutionary Computation