Computer Information System multi-criteria optimization conditions of the complex chemical processes

An information-computing system for multi-criteria optimization of complex chemical processes has been developed. Considered sensing algorithm for solving the problem of multicriteria optimization. Information and computing system combines subsystems for solving individual computing problems. Each subsystem is designed for separate calculations-direct kinetic task, calculation of the inverse kinetic problem, multi-criteria optimization of the reaction conditions. For the catalytic reaction of the benzylalkyl ethers synthesis, using the developed information-computational system, the problem of optimizing the conditions has been solved. Based on the kinetic model, the optimal values of temperature and reaction time are determined. The values of the parameters for assessing the quality of the Pareto approximation are calculated.


Introduction
An information-computational system is a system for scientific research: systematic data storage, storage, and selection of mathematical models, implementation of numerical algorithms for solving problems, complex and voluminous calculations.
Unified methodology for the analysis of complex catalytic reactions based on kinetic models allows the study of a wide class of catalytic reactions. Methods of multicriteria optimization in the form of an information-computational system will allow analyzing catalytic reactions with a view to their subsequent implementation in production.

Information system structure
The information-computational system for multicriteria optimization of catalytic reactions based on the kinetic model consists of the following groups and modules: Group I -Modules for the development of a catalytic reaction kinetic model: 1) The module for calculating kinetic curves with the choice of a mathematical description [1,2] and a solution algorithm is a direct kinetic problem [3,4].
2) The module for calculating kinetic parameters with the choice of the type of residual functional and the solution algorithm (inverse kinetic problem [5][6][7]).
II group. Database of kinetic models and optimality criteria: 3) Database module of kinetic models and optimality criteria. III group. Modules of optimization and optimal control of the catalytic reaction conditions: 4) A single-criteria optimization module with a choice of variable parameters, an optimality criterion [8], a solution algorithm [9]. 5) Multicriteria optimization module with a choice of variable parameters, optimality criteria, solution algorithm [10].
6) The module for optimal control of the reaction conditions with the choice of control parameters, optimality criterion, solution algorithm [11]. When the processing system, each module interacts with others. Such a decomposition of the original problem flexibly develops and modifies each of the modules. The whole system continues to work stably.

Optimization algorithms in the information-computational system
It is necessary to use single-criterion optimization algorithms to solve the inverse kinetic problem and the problem of determining the optimal conditions for a catalytic reaction with one optimality criterion. The inverse kinetic problem is incorrectly posed. To solve this problem, it is necessary to apply global and local optimization methods. In these methods, a global optimum and a local refinement are sought. For single-criterion optimization, the following algorithms are implemented in the module of the information-computational system: genetic algorithm and Hook Jeeves algorithm.  The genetic algorithm solves the optimization problem by imitating the principles of biological evolution, repeatedly changing the population of individual points using the rules on the model of combinations of genes in biological reproduction. Due to its random nature, the genetic algorithm increases the chances of finding a global solution. It allows solving problems without restrictions, with restrictions on variables, and general optimization problems. The solver does not require differentiability or continuity of the objective function [12].
In fig. 1 shows the scheme of the genetic algorithm. At the first stage, the initial population is formed (introduced or determined) -the first approximation of the desired parameter value. For the inverse kinetic problem, the parameters are the rate constants of the stages and activation energy, for the task of optimizing the conditions -temperature, reaction time, etc. Then, at the Selection level, the population develops with the exception of solutions with a lower fitness function. The sample contains the values of the population with the best values of the fitness function.
For the inverse kinetic problem, the fitness function is the residual functional (1). min; where Z Y is a concentration component of a residual functional; Z T is a temperature component of a residual functional; y ij e и y ij c are experimental and calculated values of component concentrations, N is a number of substances, M is a number of measuring points over time of the observed substances during the reaction.
For the task of optimizing the conditions for carrying out a catalytic reaction, the fitness function is an optimality criterion (2).
where f is a optimality criterion function; t* is a reaction time, min; T is a temperature, К; y 0 is a vector of initial component concentrations, mol / l; X is a vector of variable parameters, D X is a range of permissible values of the vector X.
At the Crossing stage, the population sparse at the previous stage is restored due to the grouping (crossing) of the genes (values) of existing solutions. At this stage, a local refinement of the extremum occurs. At the Mutation stage, random changes occur in the genes of existing solutions. This allows us to diversify the population and determines the global nature of optimization.
To refine the global optimum found by the genetic algorithm in the information-computing system, the Hook-Jeeves method is implemented.
The Hook-Jeeves method is a direct search method or a zero-order method. It is divided into two phases: exploratory search and pattern matching [13]. The general algorithm for the Hook-Jeeves method is presented in Fig. 2.
In the exploratory search is selected a certain initial vector of values of the required parameters U={u 1 ,u 2 ,..., u |U| }. The step value h is set. Then, the value of the optimization optimization function is calculated at three points: U={u 1 , u 2 ,…, u |U| }, U={u 1 +h, u 2 ,…, u |U| }, U={u 1 -h, u 2 ,…, u |U| }, and the transition to the point with the smallest value of the objective function is performed. For the inverse kinetic problem, the optimization target is the residual functional (1), and for the optimization of the conditions, is the optimality criterion (2). The exploratory search ends after iterating over all |U| coordinates. The resulting point is called the base.
In pattern matching search, a step is performed from the obtained base point (the coordinates of the base point are the desired parameters: for the inverse kinetic problem -the rate constants of the stages and activation energy, for the task of optimizing the conditions -temperature, reaction time, etc.) along the straight line connecting this point from the previous base. The step is equal to the distance between the base point and the previous base point. The result is a point for which an exploratory search is conducted. When solving the problem of multicriteria optimization by a grid algorithm (sensing algorithm), the entire D X region is divided into subregions (cells). In each cell, a calculation is made and the Pareto front D f *, the non-dominated points, is determined from the set of cells.
Grid algorithm allows to get all the compromise solution to the problem of multicriteria optimization.

Modeling and optimization of the catalytic reaction conditions in the synthesis of benzylalkyl ethers
In [14], a kinetic model was developed for the reaction of benzylbutyl ether synthesis in the presence of a metal complex catalyst. The reaction produces the desired product. PhCH 2 OBu benzyl butyl ether and by-product PhCH 2 OCH 2 Ph dibenzyl ether. The introduction of the process into production requires determining the optimal reaction conditions in order to obtain the highest yield of the target product and the smallest by-product. Based on the kinetic reaction model, the formulation of the multicriteria optimization problem is possible. Chemical experiments were carried out at several temperatures from 140 to 175°С. In this temperature range is necessary to determine the extrema values to achieve the optimality criteria of the reaction. Then the variable parameters are temperature and reaction time, with corresponding physicochemical limitations.
The main parameters of the multicriteria optimization problem for the synthesis of benzylbutyl ether have the form [15][16][17]: -Vector of variable parameters where x 1 -reaction temperature, T; x 2 -reaction time, t*.
-Vector function of optimality criteria F(X)=(f 1 (X), f 2 (X)): ( The solution to the problem of a multicriteria optimization algorithm was conducted Paretoapproximation NSGA-II [18][19][20] in Matlab using parallelization [21,22]. A condition for exiting the algorithm was a minimal change in the value of the optimality criterion (less 10-6), in accordance with the experimental values. Figures 3 and 4 show the results of solving the multicriteria optimization problem for conditions of the synthesis in benzyl butyl ether in the presence of a metal complex catalyst.
According to the approximation of the Pareto front ( Fig. 3) with the yield of the target benzyl butyl ether, the output of the secondary dibenzyl ether also increases. The corresponding values of the varied parameters, i.e. the Pareto set is shown in Fig. 4.
If the reaction is carried out up to 600 min (as in a chemical experiment [14]), the optimum temperature is 140°C (fig. 4). However, a longer reaction time leads to an increase in the yield of the target and by-products. The increase in the duration of the process up to 800 minutes requires heating the mixture to 170°C to maintain an upward trend in product yields, as calculated.
The adequacy of the calculated values is confirmed by experimental data and assessing the quality of the approximation Pareto [23][24][25][26][27][28]: Uniformity of distribution of solutions (average distance between solutions).    Average dispersion is a measure of uniformity of distribution of solutions.
where 1 2 A = C + C -solution set of capacity. The values of evaluation criteria Pareto approximation (6), (7) are shown in Table. 1. These values correspond to the error in the experimental data.
The values of the parameters for assessing the quality of Pareto approximations from the range of variable parameters determine the genotype, and in the field of optimality criteria, the phenotype.

Conclusion
Thus, an information-computational system for multi-criteria optimization of catalytic reactions has been developed. Information and computing system combines subsystems for solving individual computing problems. Each subsystem is designed for separate calculations -direct kinetic problem, calculation of the inverse kinetic problem, multi-criteria optimization of the reaction conditions. For the catalytic reaction of the synthesis of benzylalkyl ethers, using the developed informationcomputing system, the problem of optimizing conditions for the solution has been solved. Based on the kinetic model, the optimal values of temperature and reaction time are determined. The values of the parameters for assessing the quality of the Pareto approximation are calculated.

Acknowledgments
This research was performed due to the Russian Science Foundation grant (project No. 19-71-00006).