Integrated Scheduling and Dynamic Optimization of Sequential Batch Processes with Online Implementation

An efficient decomposition method to solve the integrated problem of scheduling and dynamic optimization for sequential batch processes is proposed. The integrated problem is formulated as a mixed‐integer dynamic optimization problem or a large‐scale mixed‐integer nonlinear programming (MINLP) problem by discretizing the dynamic models. To reduce the computational complexity, we first decompose all dynamic models from the integrated problem, which is then approximated by a scheduling problem based on the flexible recipe. The recipe candidates are expressed by Pareto frontiers, which are determined offline by using multiobjective dynamic optimization to minimize the processing cost and processing time. The operational recipe is then optimized simultaneously with the scheduling decisions online. Because the dynamic models are encapsulated by the Pareto frontiers, the online problem is a mixed‐integer programming problem which is much more computationally efficient than the original MINLP problem, and allows the online implementation to deal with uncertainties. © 2013 American Institute of Chemical Engineers AIChE J, 59: 2379–2406, 2013     

基于遗传算法的高职化工院校排课系统的设计与实现

排课问题是一个有约束的、多目标的组合优化问题,并且已经被证明是一个NP完全问题。遗传算法是一种随机搜索算法,非常适合于解决NP问题。本文将遗传算法应用于求解课表问题,采用优化编码结构组合的方法减少排课冲突,降低算法的复杂度;对排课问题的关键参数进行量化分析,针对染色体编码完成各个遗传算子的设计和开发任务,最后集成排课的整体优化算法。     

Integration of scheduling and control with online closed-loop implementation: Fast computational strategy and large-scale global optimization algorithm

In this paper, we propose a novel integration method to solve the scheduling problem and the control problem simultaneously. The integrated problem is formulated as a mixed-integer dynamic optimization (MIDO) problem which contains discrete variables in the scheduling problem and constraints of differential equations from the control problem. Because online implementation is crucial to deal with uncertainties and disturbances in operation and control of the production system, we develop a fast computational strategy to solve the integration problem efficiently and allow its online applications. In the proposed integration framework, we first generate a set of controller candidates offline for each possible transition, and then reformulate the integration problem as a simultaneous scheduling and controller selection problem. This is a mixed-integer nonlinear fractional programming problem with a non-convex nonlinear objective function and linear constraints. To solve the resulting large-scale problem within sufficiently short computational time for online implementation, we propose a global optimization method based on the model properties and the Dinkelbach's algorithm. The advantage of the proposed method is demonstrated through four case studies on an MMA polymer manufacturing process. The results show that the proposed integration framework achieves a lower cost rate than the conventional sequential method, because the proposed framework provides a better tradeoff between the conflicting factors in scheduling and control problems. Compared with the simultaneous approach based on the full discretization and reformulation of the MIDO problem, the proposed integration framework is computationally much more efficient, especially for large-scale cases. The proposed method addresses the challenges in the online implementation of the integrated scheduling and control decisions by globally optimizing the integrated problem in an efficient way. The results also show that the online solution is crucial to deal with the various uncertainties and disturbances in the production system.     

二次求解具有控制切换结构的动态优化问题

基于多级表述策略,提出了二次求解具有控制切换结构动态优化问题的数值方法。基于常用的优化方法获得初始控制结构。动态优化问题根据控制结构进行分级,每一级对应一个特定的控制弧段,进而将原问题表述为一个多级动态优化问题。基于控制向量参数化(CVP),多级动态优化问题转化为一个非线性规划(NLP)问题进行求解。控制参数和级长作为优化变量。基于Pontryagin极大值原理,构造多级伴随系统,进而获得NLP求解器所需的梯度信息。仿真实例验证了方法的有效性。     

Method of calculating the bed combustion of a solid fuel coke residue

An approximate method of solving the steady-state problem of the bed combustion of a solid fuel coke residue is proposed. The method is based on splitting the initial problem into thermal and chemical problems. The former represents the problem on the hydrodynamics and heat transfer in a blown heat-generating granular bed. The latter is the problem on the chemical conversion of a mixture of reactants in a specified temperature field. These auxiliary problems are combined into an iteration cycle to match their solutions. The model problem on the combustion of a milled peat coke residue was calculated for the cocurrent and countercurrent process schemes. The bed combustion of coke is calculated for the experimental conditions published in the literature. The calculated results are compared with the experimental data obtained under these conditions.     

Studies of distributed parameter systems: Decoupling the state-parameter estimation problem

The problem of state-parameter estimation is considered in terms of decoupling the estimation procedure. First, the theoretical preliminaries necessary for the mathematical statement of the problem are defined. Then using the extended Kalman filter (EKF) approach, the state and parameter are estimated by applying the solution techniques to a distributed parameter system. Next, the state estimation problem is decoupled from the parameter estimation problem and by using a numerical example, the advantage of this decoupling procedure is demonstrated. The numerical results show that convergence can be improved when this decoupling procedure is employed. The effect of the location of the measurements on the estimation problem is also analysed in this work. The results show that the convergence of the problem depends on the location as well as the number of measurements.     

Integration of production scheduling and dynamic optimization for multi-product CSTRs: Generalized Benders decomposition coupled with global mixed-integer fractional programming

Integration of production scheduling and dynamic optimization can improve the overall performance of multi-product CSTRs. However, the integration leads to a mixed-integer dynamic optimization problem, which could be challenging to solve. We propose two efficient methods based on the generalized Bender decomposition framework that take advantage of the special structures of the integrated problem. The first method is applied to a time-slot formulation. The decomposed primal problem is a set of separable dynamic optimization problems and the master problem is a mixed-integer nonlinear fractional program. The master problem is then solved to global optimality by a fractional programming algorithm, ensuring valid Benders cuts. The second decomposition method is applied to a production sequence formulation. Similar to the first method, the second method uses a fractional programming algorithm to solve the master problem. Compared with the simultaneous method, the proposed decomposition methods can reduce the computational time by over two orders of magnitudes for a polymer production process in a CSTR.     

Comparison of certain MINLP algorithms when applied to a model structure determination and parameter estimation problem

The maximum likelihood method is frequently used in parameter estimation. If the structure of the model is unknown, the maximization of the likelihood function can be replaced by minimizing an information criterion. One criterion that allows this to be done is Akaike’s information criterion (AIC). Minimizing the AIC is a mixed integer non-linear programming (MINLP) problem. In this paper, three different MINLP algorithms are compared in the solution of a simultaneous model structure determination and parameter estimation problem by minimizing the AIC criterion. The problem considered appears in quantitative Fourier transformed infra red (FTIR) spectroscopy where concentration estimates of certain gas components are to be obtained from measured absorbances at different wave numbers. The resulting problem is a large MINLP problem containing several hundreds, or even thousands, of variables including a huge number of possible model structures. It is, however, found that the studied algorithms solve the considered problem in quite a small number of iterations and a reasonable CPU-time.     

Identification of parameters in systems of ordinary differential equations using quasilinearization and data perturbation

Given a set of observed data for a particular physical phenomenon, the problem of computing the “best fit” parameters for the mathematical model describing the phenomenon is a common problem in process or reaction mechanism identification. If the mathematical model comprises a set of non-linear ordinary differential equations, this leads to a non-linear boundary value problem. A very powerful way of attacking this class of problem uses an adaptation of the Newton-Raphson-Kantorovich procedure, called quasilinearization, which regards the non-linear problem as the limit of a sequence of linear problems. Starting from an initial trial solution, convergence if it does occur, occurs rapidly; further, convergence is assured if the initial guess is “close enough” to the true solution. The difficulty of making a good initial guess, a serious limitation of the method in the past, can in principle be overcome by the algorithm proposed. When a given vector may not be within the domain of convergence of the original problem, it must be within the domain of convergence of some other derived problem. The latter may then be perturbed towards the original problem in a finite number of steps. In the case of process identification, new data points are derived; these are subsequently adjusted until they coincide with the original data. The algorithm has been successfully applied to several examples from recent chemical engineering literature.     

An MILP-MINLP decomposition method for the global optimization of a source based model of the multiperiod blending problem

The multiperiod blending problem involves binary variables and bilinear terms, yielding a nonconvex MINLP. In this work we present two major contributions for the global solution of the problem. The first one is an alternative formulation of the problem. This formulation makes use of redundant constraints that improve the MILP relaxation of the MINLP. The second contribution is an algorithm that decomposes the MINLP model into two levels. The first level, or master problem, is an MILP relaxation of the original MINLP. The second level, or subproblem, is a smaller MINLP in which some of the binary variables of the original problem are fixed. The results show that the new formulation can be solved faster than alternative models, and that the decomposition method can solve the problems faster than state of the art general purpose solvers.     

Smooth penalty function method for rigorous optimization of distillation processes

Integer decisions on stage numbers and feed locations, and global optimality are still challenging for rigorous optimization of distillation processes. In the present article, we propose a smooth penalty function method to address both these problems. The proposed method is based on the relaxation of the integer decision problem into continuous nonlinear programming (NLP) problem by adopting the bypass efficiency model developed by Dowling and Biegler. A smooth penalty term (SPT) is proposed and added to the total annual cost (TAC) function to form a new objective function, namely, the smooth penalty function. Using the new objective function, the problem is initially solved with negative weight coefficients for the SPTs regarding each column section to get an optimum near the global optimum of the SPT. Then, starting from this solution, the problem is solved again iteratively by increasing the values of the weight coefficients until all the stage numbers become integers. The performance of the method is validated by an illustrating problem and in three case studies, including a reactive distillation optimization problem.     

On the tuning of predictive controllers: Application of generalized Benders decomposition to the ELOC problem

This work investigates the computational procedures used to obtain global solution to the economic linear optimal control (ELOC) problem. The proposed method employs the generalized Benders decomposition (GBD) algorithm. Compared to the previous branch and bound approach, a naive application of GBD to the ELOC problem will improve computational performance, due to less frequent calls to computationally slow semi-definite programming (SDP) routines. However, the reverse-convex constraints of the original problem will reappear in the relaxed master problem. In response, a convexification of the relaxed master constraints has been developed and proven to preserve global solution characteristics. The result is a multi-fold improvement in computational performance. A technological benefit of decomposing the problem into steady-state and dynamic parts is the ability to utilize nonlinear steady-state models, since the relaxed master problem is free of SDP type constraints and can be solved using any global nonlinear programming algorithm.     

A discretization-based approach for the optimization of the multiperiod blend scheduling problem

In this paper, we introduce a new generalized multiperiod scheduling version of the pooling problem to represent time varying blending systems. A general nonconvex MINLP formulation of the problem is presented. The primary difficulties in solving this optimization problem are the presence of bilinear terms, as well as binary decision variables required to impose operational constraints. An illustrative example is presented to provide unique insight into the difficulties faced by conventional MINLP approaches to this problem, specifically in finding feasible solutions. Based on recent work, a new radix-based discretization scheme is developed with which the problem can be reformulated approximately as an MILP, which is incorporated in a heuristic procedure and in two rigorous global optimization methods, and requires much less computational time than existing global optimization solvers. Detailed computational results of each approach are presented on a set of examples, including a comparison with other global optimization solvers.     

Optimal design of reverse osmosis‐based water treatment systems

We address the problem of optimal design of reverse osmosis (RO)‐based water treatment systems. A superstructure optimization method is proposed to solve the problem, where the superstructure for a RO system is structurally enhanced with additional features. We formulate the problem as mixed‐integer nonlinear program which is solved to yield optimal results. A case study on desalination is considered in this work, and the numerical results obtained using our approach are validated using a commercial simulation tool. We further extend the problem by considering the effects of degradation of membrane performance over time and solve it by representing the problem as a two‐stage stochastic program. This new approach is highly useful for identifying minimum cost robust designs for membrane‐based water purification systems, which are especially important in desalination applications. © 2012 American Institute of Chemical Engineers AIChE J, 2012     

纺织企业Flow—Shop调度问题的研究与实现

以某纺织企业纺纱车间为背景,针对Flow—Shop调度问题,以“最大完工时间最小化”为调度性能指标建立数学模型,应用微粒群算法（PSO）对该问题进行求解,通过仿真实现验证了其有效性。     

Trade-off optimal design of a biocompatible solvent for an extractive fermentation process

In this study, crisp and flexible optimization approaches are, respectively, introduced to design an optimal biocompatible solvent for an extractive fermentation process. The optimal design problem is formulated as a mixed-integer nonlinear programming model in which performance requirements of the compounds are reflected in the objective and the constraints. In general, the requirements for the objective and constraints are not rigid; consequently, the flexible or fuzzy optimization approach is applied to soften the rigid requirement for maximization of the extraction efficiency and to consider the mass flow rate and biocompatibility of solvent as the softened inequality constraints to the solvent design problem. Having elicited the membership function for the objective function and the constraint, the optimal solvent design problem can be formulated as a flexible goal attainment problem. Mixed-integer hybrid differential evolution is applied to solve the problem in order to find a satisfactory design.     

Short-term scheduling and recipe optimization of blending processes

The main objective of this paper is to develop an integrated approach to coordinate short-term scheduling of multi-product blending facilities with nonlinear recipe optimization. The proposed strategy is based on a hierarchical concept consisting of three business levels: Long-range planning, short-term scheduling and process control. Long-range planning is accomplished by solving a large-scale nonlinear recipe optimization problem (multi-blend problem). Resulting blending recipes and production volumes are provided as goals for the scheduling level. The scheduling problem is formulated as a mixed-integer linear program derived from a resource-task network representation. The scheduling model permits recipe changeovers in order to utilize an additional degree of freedom for optimization. By interpreting the solution of the scheduling problem, new constraints can be imposed on the previous multi-blend problem. Thus bottlenecks arising during scheduling are considered already on the topmost long-range planning level. Based on the outlined approach a commercial software system has been designed to optimize the operation of in-line blending and batch blending processes. The application of the strategy and software is demonstrated by a detailed case study.     

New algorithm for the flexibility index problem of quadratic systems

A new flexibility index algorithm for systems under uncertainty and represented by quadratic inequalities is presented. Inspired by the outer‐approximation algorithm for convex mixed‐integer nonlinear programming, a similar iterative strategy is developed. The subproblem, which is a nonlinear program, is constructed by fixing the vertex directions since this class of systems is proved to have a vertex solution if the entries on the diagonal of the Hessian matrix are non‐negative. By overestimating the nonlinear constraints, a linear min–max problem is formulated. By dualizing the inner maximization problem, and introducing new variables and constraints, the master problem is reformulated as a mixed‐integer linear program. By iteratively solving the subproblem and master problem, the algorithm can be guaranteed to converge to the flexibility index. Numerical examples including a heat exchanger network, a process network, and a unit commitment problem are presented to illustrate the computational efficiency of the algorithm. © 2018 American Institute of Chemical Engineers AIChE J, 64: 2486–2499, 2018     

Self-similar growth of a gas bubble induced by localized heating: the effect of temperature-dependent transport properties

A gas bubble growing in a saturated liquid under the action of an internal heat source is studied theoretically as a model problem for the growth of a bubble around a miniature spherical heater submerged in the liquid. In general, this problem does not admit a self-similar solution but it is shown that for a heat source with strength increasing in proportion to the square root of time, a self-similar solution exists even under temperature-dependent transport properties. In that case the bubble radius is also proportional to the square root of time with a proportionality constant which can be found from the solution of a boundary value problem. The effect of temperature dependence of gas-in-liquid diffusivity and liquid and gas thermal conductivities on bubble growth is examined in detail. Finally, approximating explicit procedures are proposed for the computation of bubble growth rate without resorting to the solution of the boundary value problem.     

A fuzzy multiobjective approach for minimization of injection molding defects

This paper describes a fuzzy multiobjective optimization approach for determining the set‐points of the injection molding processing parameters to minimize the defects formed on the molded parts. The severities of the defects are represented by membership functions using the fuzzy set theory. The minimization of these membership functions, which is a multiobjective optimization problem, is transformed into a substitute problem. The preference function in the substitute problem is original and is proposed specifically for characterizing the quality requirements of the injection molding defects. The formulated optimization problem is solved with design of experiments, in which the process behavior is approximated empirically by a set of quadratic polynomials that can be easily optimized. Experimental results are presented to emphasis the workability of the proposed methodology.