Triangularization Method For Process Simulation And Optimization Based On Symbolic Computation

In the field of process systems,there are systems represented by polynomial structures,most of which are non-linear.When solving such kind of system,the numerical solution is applied,but the method will lead to the deviation between the results obtained and the reality due to the problems of processing accuracy and falling into the local optimal solution.On the basis of traditional numerical calculation,the following two kinds of problems are put forward:a.If the calculation of intermediate process can be reduced appropriately,the accuracy of the results can be improved to a certain extent;B.If the algorithm can give all feasible solutions,the global optimal solution that meets the requirements can be obtained from the feasible solution.In order to solve these two problems,this paper introduces the idea of symbolic computation.The main contents of this paper are as follows:1.To solve the process simulation problem,i.e.polynomial equations whose degree of freedom is 0,this paper proposes an improved method,which is based on Gr?bnerbasis method in symbolic computation and combining with system decomposition.Gr?bnerbasis is a method of transforming the system with a coupled structure to a triangular structure in the equivalent solution space,which is helpful for numerical calculation.The triangular structure obtained by this method can be solved sequentially in each calculation,and each sequence contains only one equation.However,the triangulation calculation of Gr?bnerbasis is limited by the interaction between model size and equations.In order to reduce the computational complexity of Gr?bnerbasis,system decomposition is introduced in this paper.From the view of process system and topological structure,the triangulation problem of large system is transformed into the triangulation problem of several subsystems in order to reduce the computational scale of Gr?bnerbasis and reduce the interaction of equations through subsystems.Through practical examples,we can see that the triangulation process is accelerated by the method of system decomposition.2.A polynomial projection-lifting algorithm is proposed for the process optimization problem,i.e.polynomial programming with non-zero degree of freedom.Based on the cylindrical algebra theory in symbolic mathematics,this algorithm introduces "projection operator" to transform the polynomial system from high dimension to low dimension in symbolic form,and then upgrades the polynomial system from low dimension to high dimension,checks the expansion of feasible region,and obtains the triangular expression of high dimension about low dimension.Depth-First-Search(DFS)algorithm is used in the expansion process,which extends from lower dimension to obtain the feasible range of each dimension.Considering the unique objective function value of optimization problem,this paper combines cylindrical algebra theory with optimization idea to design proj ection-lifting algorithm for polynomial programming.For practical examples,the specific solving process of the algorithm is given,and the feasibility of the algorithm is demonstrated by comparing the actual results.3.For solving integer programming problems,Gr?bnerbasis method and polynomial proj ection-lifting algorithm are used in this paper.For the non-polynomial part of integer programming,this paper introduces binary transformation and polynomial equations in equal solution space.The programming problem is transformed into the form of symbolic calculation method which can adopt Gr?bnerbasis algorithm and polynomial programming,and then the optimal solution of the problem can be obtained according to the optimality of the actual problem.This paper demonstrates the feasibility of these two algorithms in integer programming through practical examples,compares them with the integer programming algorithm of GAMS(The General Algebraic Modeling System)and proves the accuracy of these two algorithms.