| Process analysis and optimization are the main subjects of process systems engineering(PSE).A key to process analysis and optimization is solving nonlinear equations.Numerical computation is the most important approach to deal with such problems,for its advantages in fast solution speed and wide range of applications.However,as an approximate solution method,the convergence and reliability cannot be guaranteed because of the instabilities.Another scientific computation method,symbolic computation,can directly address the expressions with unknown variables.Apply symbolic computation methods to make up for the inherent drawbacks of numerical computation in PSE and study the numerical-symbolic hybrid algorithms to improve the solution accuracy,efficiency and reliability are the goals of this thesis.In this thesis,for several typical numerical computation issues on the system analysis and optimization,the corresponding numerical-symbolic algorithms are proposed.Model triangularization is the main research basis,and the analyses of model structure,solution space and parametric space are the research objects.The methods of system analysis are eventually extended to the process optimization problems.The original contributions can be summarized as follows:(1)Propose a triangularizing reformulation and accelerating solution method.When solving the process simulation problems,the traditional EO method suffers from difficulties in variable initialization,and the SM method has slow convergence since tear process.A numerical-symbolic algorithm is developed to improve these two methods.The graph theory is first applied for system decomposition in which the simoutaneous subsystems are triangularized by Grobner basis method,and the final reformulated model can be solved sequentially without any iterative tearing process.The reformulated model can keep the same solution space,significantly increase the solving efficiency and improve the robustness of initialization.(2)Propose a multi-solution analysis and solution method.In the practical process systems,the process variables are always defined in a physical region.Due to the complexity of the models,a multiplicity of solutions occurs in some systems.A numerical-symbolic algorithm is developed to handle the constrained process simulation model.By triangularzing the model,finding solutions of a constrained multivariable polynomial system is converted to the one of a series of constrained univariate polynomials sequentially.The multi-solution features can be explicitly expressed.According to the constraints,all physical solutions can be accurately located and solved by using real root isolation and local numerical computation.(3)Propose a deterministic global optimization method for polynomial programming.Finding the global minima of polynomial programming is known to be NP-hard,and because of the multi-extremum feature,it is hard to find all global minima.A deterministic.global optimization method based on the KKT system is developed.The high-dimensional solution space can be mapped to one-dimensional space of objective function.A univariate polynomial equation of the objective function is derived.Its minimum value and the corresponding global minimum points can be obtained by solving the triangular system through the real root isolation method.(4)Propose a triangularizing and analytical method for operational flexibility analysis.The main purpose of operational flexibility analysis is to deterimine and describe the flexibility space.The existing numerical analysis methods only can estimate the envelope of the space.From the view of symbolic computation,the original flexibility analysis model is regarded as an existential quantifier model.Through the cylindrical algebraic decomposition method,the quantifiers can be eliminatied,and the model can be mapped to the flexibility space of uncertain parameters.This method not only can describe the flexibility space analytically,but explicitly express the steady-state operations to handle the uncertainties. |
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