Mathematical Model. We describe the changes in the concentrations of the components depending on the reaction time. To do this, the equations of chemical kinetics are used, which are a system of ordinary nonlinear differential equations (SONDE):

 (1)

(2)

Here are the initial conditions: yi(0) = yi0 at τ = 0; τ ∊ [0, τ*]; yi — reagent concentrations, mole fractions; τ — conditional contact time of the reaction mixture with the catalyst, kg∙min/mol; J — number of stages; I — number of substances; νij — stoichiometric matrix; wj — speed of the j-th stage, 1/min or mol/(kg∙min); kj, k–j — speed constants of the stages (given), 1/min; aij — negative elements of matrix νij; bij — positive elements νij; Aj, A–j — pre-exponential multipliers, 1/min; Ej+, Ej– — activation energies of direct and reverse reactions, kcal/mol; R — gas constant, 2 cal/(mol∙K); T — temperature, K; τ* — reaction duration, kg∙min/mol.

The model of catalytic hydrogenation of PAH takes into account the dynamics of the molar composition and volume of the reaction mixture. Therefore, changes in the concentration of components at each point in time are considered [3]:

 (3)

To describe a nonstationary reaction that occurs with a change in the volume of the reaction mixture, it is necessary to solve a system of nonlinear differential equations at each moment of time. The direct kinetic problem is the solution to SONDE (1)–(3).

In the process of hydrogenation of PAH, naphthenes are obtained from the source aromatic hydrocarbons, which have a higher density and can be used as rocket fuel. For this purpose, nickel catalysts are used, and the control or variable parameter is the process temperature, which should be within 200–500 K. Optimality criteria — maximizing the output of target naphthenes at the end of the reaction and maximizing the conversion of feedstock.

Research Algorithm. Multicriteria optimization is the selection of the best solution from a variety of alternatives, taking into account several criteria. The importance of each of them is determined by the weight (priority).

Suppose, f(x) — an objective function, and the constraints given in the form of equalities h1(x)…hm(x) and inequalities gm+1(x)…gp(x) are represented by a column vector of components x = [x1,…, xn]T in n-dimensional Euclidean space.

We formulate the problem of nonlinear programming [5].

Optimize

 (4)

with m linear or nonlinear constraints in the form of equalities

 (5)

and with (p – m) linear or nonlinear constraints in the form of inequalities

 (6)

Decomposition (4)–(6) is the formulation and solution of a linear and quadratic programming problem. Each of them is determined by the type of equations (4)–(6). Thus, in the case of quadratic function (4) and linear equations (5), (6) — the quadratic programming problem described below.

We define the extremum of function

 (7)

with constraints:

 (8)

In equations (7), (8) Q — nonnegatively defined quadratic symmetric matrix; a, b, c — coefficient matrices.

When setting multicriteria Pareto optimization problem, (4)–(6) will have the form:

 (9)

In equation (9),  — vector-function of optimality criteria f1 and f2. Set x* — desired solution to the problem in the region of variation parameters. Set  represents the desired solution to the problem in the region of optimality criteria, which is unimprovable in terms of Pareto approximation. Then, x* defines the Pareto set, F* — Pareto front.

Priori and posteriori Pareto approximation algorithms were used to solve (9). One of them is the ideal point method, which represents the best solution according to all criteria [6]. To find it, we must first determine the minimum and maximum values of each criterion of all the solutions under consideration. Then, for each criterion, the maximum value of all the minimum and the minimum of all the maximum values are selected.

However, this approach has some drawbacks. Firstly, it may be ineffective if the ideal point is outside the range of acceptable criteria values. In this case, other methods of solving multicriteria optimization problems are used. In addition, the ideal point method does not take into account the relationship between the criteria and may cause the selection of a compromise solution that is not optimal for all criteria. Therefore, when using this method, it is needed to additionally analyze and verify the optimality of the solutions obtained [7].

The lexicographic ordering method is also used in solving multicriteria optimization problems. In this case, the criteria are ordered by priority and considered sequentially. If the solutions cannot be sorted by the first criterion, then they are sorted by the next criterion, etc. [7]. The advantages of lexicographic ordering are simplicity and transparency. Using this method, it is possible to obtain a single optimal solution that is easily interpreted. As for the disadvantages, we note, firstly, the inability to take into account compromise solutions that may turn out to be optimal by all criteria. Secondly, there is a risk of selecting an unfavorable solution if the first criterion has a lot of weight, but is not the most important for this task [8].

The presented work provides a solution to the problem of multicriteria optimization of the hydrogenation process of polycyclic aromatic hydrocarbons through the well-known NSGA-II method. It is based on a genetic algorithm and uses several techniques to solve the problem of non-dominance [9]. The main steps of the algorithm are described below [10].

NSGA-II allows working accurately and efficiently with multicriteria optimization tasks. It effectively solves the problem of non-dominance, which provides getting optimal solutions for all criteria.

NSGA-II is based on a genetic algorithm with parental selection and survival. Individuals are selected along the fronts, while the front is divided if not all individuals can survive. Solutions in the split front are selected on the basis of the distance between them, which is the Manhattan distance in the criteria space [9]. The endpoints are saved at each generation and are assigned a conditionally infinite distance for use in subsequent iterations [11] (Fig. 1).

Figure 1 shows an example of a set of solutions for a multicriteria optimization problem using criteria f1 and f2. The red dots represent the Pareto front. The calculation of the cluster distance for solution i is shown — this is the average length of the side of the cuboid in which solution i is located (marked with a blue frame).

To enhance the impact on the selection of parents, NSGA-II uses binary tournament selection [9]. Each individual is first compared by rank, and then — by the distance between them.

Research Results. The program implementing the algorithm for solving the multicriteria optimization problem is written in Python.

In the course of the study, a system of differential equations (1–3) was solved. For the hydrogenation reaction of polycyclic aromatic hydrocarbons, it has the form [12]:

At τ=0, y1(0)=0.025; y2(0)=0.9; y6(0)=0.067; y9(0)=0.008; yi(0)=0, i=3–5,7,8,10–20; Q(0)=1.

Figure 2 shows the calculated set of solutions satisfying the constraints of the system and the Pareto front, where f1 — output of the target naphthenes, and f2 — conversion of the feedstock.

The controlled parameter is the process temperature [13]. The optimality criteria are maximizing the output of target naphthenes at the end of the reaction and maximizing the conversion of feedstock [14]. Calculations using the proposed algorithm gave the results of matching the optimality criteria and the values of the temperature parameter, which are summarized in Table 1.

Thus, as the temperature grows, the output of the target naphthenes increases, and the conversion of the feedstock decreases. Each of these solutions is unimprovable, and the selection of specific values depends on the decision maker.

For NSGA-II to work successfully, it is required to select the algorithm parameters correctly. Specifically, the criteria of population size, number of generations, probability of crossover and mutation, should be optimized. When launching the algorithm on the model of the hydrogenation process of polycyclic aromatic hydrocarbons, the following parameters were used: population size — 100, number of generations — 100. An insufficiently large population may cause premature convergence of the algorithm to the local optimum. Too large population can slow down the optimization process [15]. An excessive number of generations potentially leads to retraining the algorithm, whereas with an insufficient number of generations there may not be enough time to achieve optimal solutions.

Discussion and Solution. A program has been created that implements the NSGA-II multicriteria optimization algorithm. Working with the corresponding problem within the framework of this method includes solving a system of differential equations, visualizing a set of solutions satisfying the constraints of the system, and constructing a Pareto front. In addition, the values of the variable parameters were found to achieve optimization goals. For the PAH hydrogenation process, based on the kinetic model, a set of temperature values has been calculated that are optimal for obtaining the unimprovable values of two optimality criteria: naphthene output and feedstock conversion. With increasing temperature, the reaction rate and the output of naphthenes increases. However, the conversion of raw materials is decreasing. In addition, too high temperatures can cause adverse reactions and decomposition of products.

The data obtained in the framework of the presented research can be useful for optimizing the process of hydrogenation of PAH under industrial conditions. It is important to take into account the impact of temperature on the output of naphthenes and the conversion of raw materials when developing a production strategy. In addition, other parameters, on which the kinetics of the reaction depends, should be considered. These are, e.g., pressure, flow rate of reagents, and the role of catalysts.

Thus, the developed program and the proposed algorithm make it possible to simultaneously analyze several criteria for the optimality of the process based on a kinetic model, and generate a set of unimprovable values of variable parameters.