Department of Chemical Engineering

Solution of algebraic systems of disjunctive equations

Ignacio E. Grossmann et al. — Wed, 09 May 2012 14:09:47 PDT

Abstract: "This paper considers the solution of systems of equations that are expressed by the two sets of equations: a global rectangular system of equations involving more variables than equations, and a set of conditional equations that are expressed as disjunctions. The set of disjunctions are given by equations and inequalities, where the latter define the domain of validity of the equations. In this way the solution of such a system is defined by variables x satisfying the rectangular equations, and exactly one set of equations for each of the disjunctions. This paper focuses mainly in the solution of systems of linear disjunctive equations. Using a convex hull representation of the disjunctions, the disjunctive system of equations is converted into an MILP problem. A sufficient condition is presented under which the model is shown to be solvable as an LP problem. The extension of the proposed method to nonlinear disjunctive equations is also discussed. The application of the proposed algorithms are illustrated with several examples."

A successive linear programming approach for initialization and reinitialization after discontinuities of differential algebraic equations

Vipin Gopal et al. — Wed, 09 May 2012 14:09:37 PDT

Abstract: "Determination of consistent initial conditions is an important aspect of the solution of differential algebraic equations (DAEs). Specification of inconsistent initial conditions, even if they are slightly inconsistent, often leads to a failure in the initialization problem. In this paper, we present a Successive Linear Programming (SLP) approach for the solution of the DAE derivative array equations for the initialization problem. The SLP formulation handles roundoff errors and inconsistent user specifications among others and allows for reliable convergence strategies that incorporate variable bounds and trust region concepts. A new consistent set of initial conditions is obtained by minimizing the deviation of the variable values from the specified ones. For problems with discontinuities caused by a step change in the input functions, a new criterion is presented for identifying the subset of variables which are continuous across the discontinuity. The LP formulation is then applied to determine a consistent set of initial conditions for further solution of the problem in the domain after the discontinuity. Numerous example problems are solved to illustrate these concepts."

Optimal control of a reactive distillation column

Stefan Walter — Wed, 09 May 2012 14:09:27 PDT

Interior point SQP strategies for structured process optimization problems

Joao S. Albuquerque — Wed, 09 May 2012 14:09:17 PDT

Abstract: "Successive Quadratic Programming (SQP) has been the method of choice for the solution of nonlinear programming problems. For the solution of large problems with SQP based codes, the combinatorial complexity associated with active set methods for the treatment of inequalities has been a bottleneck in exploiting the problem structure. In this paper, we examine the merits of incorporating an interior point method within an SQP framework and the significant computational savings are demonstrated on problems drawn from optimal control. We also provide a novel interpretation of the popularly used predictor-corrector interior point method using the principles of reduced Hessian SQP (rSQP)."

Dynamic optimization in planning under process model uncertainty

Tarun Bhatia et al. — Wed, 09 May 2012 14:09:08 PDT

Abstract: "Planning for multi-product batch plants involves decisions for facility design, product scheduling, inventory control etc. These are usually addressed for recipe based processes with fixed stage processing times. There is significant benefit in incorporating detailed dynamic process models while planning, and addressing the overall problem simultaneously (Bhatia and Biegler [1]). It thus becomes desirable to have sufficiently accurate process models, or develop methods that deal with process model uncertainty. This work addresses dynamic processing and planning decisions simultaneously for multi product batch plants, now with uncertainty in process parameters. Instances of uncertain parameters are used to construct planning scenarios that are addressed through a multiperiod planning formulation. Dynamic processing under uncertainty is addressed via a closed loop state feedback based correction strategy. Control parameters in the closed loop implementation of processing decisions are then determined in an open loop manner, and treated as invariant design variables in the multiperiod problem formulation."

A case study for reactor network synthesis : the vinyl chloride process

Ajay Lakshmanan et al. — Wed, 09 May 2012 14:08:58 PDT

Abstract: "A key objective of the integrated reactor network synthesis approach is the development of waste minimizing process flowsheets (Lakshmanan and Biegler, 1994). With increasing environmental concerns in process design, there is a particularly strong need to avoid the generation of wasteful or harmful byproducts within the reactor network. This also avoids expensive treatment and separation costs downstream in the process. In this study, we focus on the application of integrated reactor network synthesis concepts for the vinyl chloride process. Vinyl chloride is currently produced by a balanced process from ethylene, chlorine and oxygen with three separate reaction sections: oxychlorination of ethylene, direct chlorination of ethylene and pyrolysis of ethylene dichloride, with the hydrogen chloride produced in the pyrolysis reactor used completely in the oxychlorination reactor. Each of these reaction sections generate chlorinated hydrocarbons and carbon oxides as byproducts. Detailed kinetic models for the three reaction sections are used to develop optimal reactor networks which improve the conversion of ethylene to vinyl chloride product and minimize the production of by-products. This case study presents an application of the mixed integer nonlinear programming based reactor network synthesis strategy (Lakshmanan and Biegler, 1996a). A candidate flowsheet is proposed based on these results and a set of recommendations is given to improve the selectivity of vinyl chloride production."

Analysis of a reversed Coleman-Conn reduced SQP method

Lorenz T. Biegler — Wed, 09 May 2012 14:08:48 PDT

Abstract: "We propose a quasi-Newton algorithm for solving large optimization problems with nonlinear equality constraints. It is designed for problems with few degrees of freedom, and is motivated by the need to use sparse matrix factorizations. The algorithm incorporates a correction vector that approximates the cross term Z[superscript T]WYp[subscript y] in order to estimate the curvature in both the range and null spaces of the constraints. The algorithm can be considered to be, in some sense, a practical implementation of an algorithm of Coleman and Conn. We give conditions under which local and superlinear convergence is obtained."

Multilevel-hierarchical MNLP [i.e., MINLP] synthesis of process flowsheets

Z Kravanja et al. — Wed, 09 May 2012 14:08:40 PDT

Abstract: "The objective of this contribution is to propose a multilevel-hierarchical approach to the MINLP synthesis of process flowsheets. Following a hierarchical strategy, the designer can postulate the superstructure at different levels of representation of flowsheet alternatives and model it at the corresponding level of aggregation and complexity. By the use of the prescreening procedure the superstructure is optimized more effectively and reliably. The approach enables one to address different process operations like reactions, connectivity and species allocation, separation, energy and heat integration and HEN through simultaneous superstructure optimization."

MNLP [i.e., MINLP] model for optimal synthesis and operation of utility plants

J. C. Bruno — Wed, 09 May 2012 14:08:30 PDT

Abstract: "This paper presents a mixed-integer nonlinear programming (MINLP) model for performing structural and parameter optimization of utility plants that satisfy given electrical, mechanical and heating demands of industrial processes. In this model a non-linear objective function that accounts for the cost of equipment and operation is minimized. The proposed approach allows for the simultaneous optimization of the configuration, and selection of flowrates, enthalpies and steam turbine efficiencies. All major conventional utility plant equipment are included in the superstructure for the MINLP model. The proposed approach is not only useful for synthesis, but also for analyzing different design alternatives. The model has been implemented in the computer package STEAM, and several applications are reported to illustrate the program capabilities, including a comparison with a simplified MILP model."

LOGMIP : a discrete continuous nonlinear optimizer

Aldo Vecchietti et al. — Wed, 09 May 2012 14:08:20 PDT

Abstract: "Discrete-continuous non-linear optimization models are frequently used to formulate problems in Process System Engineering. Major modeling alternatives and solution algorithms include generalized disjunctive programming and MINLP. Both have advantages and drawbacks depending on the problem they are dealing with. In this work, we describe the theory behind LOGMIP, a new computer code for disjunctive programming and MINLP. We discuss a hybrid modeling framework which combines both approaches, allowing binary variables and disjunctions for expressing discrete choices. An extension of the Logic-Based OA algorithm has been implemented to solve the proposed hybrid model. Computational experience is reported on several examples, which are solved within disjunctive, MINLP and hybrid approaches."

Nonsmooth dynamic simulation with linear programming based methods

Vipin Gopal et al. — Wed, 09 May 2012 14:08:11 PDT

Abstract: "Process simulation has emerged as a valuable tool for process design, analysis and operation. In this work, we extend the capabilities of iterated linear programming (LP) for dealing with problems encountered in dynamic nonsmooth process simulation. A previously developed LP method is refined with the addition of a new descent strategy which combines line search with a trust region approach. This adds more stability and efficiency to the method. The LP method has the advantage of naturally dealing with profile bounds as well. This is demonstrated to avoid the computational difficulties which arise from the iterates going into physically unrealistic regions. A new method for the treatment of discontinuities occurring in dynamic simulation problems is also presented in this paper. The method ensures that any event which has occurred within the time interval in consideration is detected and if more than one event occurs, the detected one is indeed the earliest one. A specific class of implicitly discontinuous process simulation problems, phase equilibrium calculations is also looked at. A new formulation is introduced to solve multiphase problems."

Dynamic operations in the planning and scheduling of multi-products batch plants

Tarun Bhatia et al. — Wed, 09 May 2012 14:08:02 PDT

Abstract: "Execution of tasks in dynamic batch units provides additional operating freedom via transient control profiles. When considered at the design and scheduling stage, this freedom can stretch the limits of profitability under strict market, facility and time constraints. The work in this paper incorporates dynamic processing conditions for products in a multi-product batch plant, as opposed to fixing the process by recipes, in the broader context of equipment design, production planning, scheduling and inventory considerations. The objective is a general function of fixed design costs, operating costs, production revenues etc. Decisions include stage processing times for products, transient stage operating policies, continuous design parameters, production capacity and production schedules. The infinite dimensional optimal control problem for each operation is solved using collocation over finite time elements ([6], [7]). Scheduling, with its combinatorial complexity, is addressed in the scope of flowshop plants for specific transfer policies using the Aggregated Scheduling model in [3] and [4]. Two examples are solved via sequential and simultaneous solution approaches. The smaller first example allows transient control at the reaction stage for problems with relevant objectives in planning and scheduling. The second example allows transient control at the reaction and high purity separation stage for a general objective function. Considerable savings achieved in most situations are reported, along with moderate computational requirements for solving the examples."

Optimal multiperiod operational planning for utility systems

Ramaswamy Iyer et al. — Wed, 09 May 2012 14:07:52 PDT

Abstract: "In this paper, the operational planning problem for utility systems is formulated as a mixed-integer linear program (MILP). For multiperiod operation with varying demands for utilities, the optimal choice of units for each period is determined. The objective function accounts for both the operating costs for each period and changeover costs for startup/shutdown of units between periods of operation. A two-stage approach is proposed that requires the solution of MILP subproblems coupled with a shortest path algorithm, resulting in orders of magnitude reduction in computation time as compared to a direct MILP solution using branch and bound enumeration. The computational requirements of the algorithm are linear with respect to the number of periods and global solution of the MILP is guaranteed. Solution of a test problem shows savings of the order of 3% in total annual cost of operation with the main advantage being the simplicity of the proposed plan (few start-ups and shutdowns). The solution method is also extended to the case for ramp function change in demands."

A mixed integer programming model for stochastic scheduling in new product development

Craig W. Schmidt et al. — Wed, 09 May 2012 14:07:42 PDT

Abstract: "This paper presents a new, real-world scheduling problem concerning the New Product Development process of an agricultural chemical or pharmaceutical company. A Research and Development (R&D) department must schedule the tasks needed to bring a new product to market, in the face of uncertainty about the costs and durations of the tasks, and in the income resulting from introducing the new product. There is a risk that a product will fail a mandatory task, such as an environmental or safety test, and never reach the market. The objective of the schedule is to maximize the expected Net Present Value of the research. A model of this problem initially has a nonlinear, nonconcave objective. The objective is convexified and linearized by appropriate transformations, giving a Mixed Integer Linear Program (MILP). The model uses a continuous time representation and discrete distributions for the stochastic parameters. Different representations of the disjunctive scheduling constraints are discussed. A small numerical example is presented, followed by some conclusions."

Quadratic programming methods for tailored reduced Hessian SQP

Claudia Schmid et al. — Tue, 01 May 2012 11:47:41 PDT

Global optimization of process networks with multicomponent flows

Ignacio Quesada et al. — Tue, 01 May 2012 11:47:33 PDT

Abstract: "This paper deals with the global optimization of networks consisting of splitters, mixers and linear process units and which involve multicomponent streams. Examples include pooling and blending systems and sharp separation networks. A reformulation-linearization technique is first applied to concentration and flow based models in order to obtain a relaxed LP formulation that provides a valid lower bound to the global optimum. This formulation is then used within a spatial branch and bound search. The application of this method is considered in detail for sharp separation systems with single feed and mixed products. Numerical results are presented on twelve test problems to show that only few nodes are commonly required in the branch and bound search."

MILP model for scheduling and design of a special class of multipurpose batch plants

V. T. Voudouris et al. — Tue, 01 May 2012 11:47:24 PDT

Abstract: "In this paper we propose a method for the integrated scheduling and design for a special class of multipurpose batch processes. The type of plants considered are the ones where not all the products use the same processing stages, and manufacturing of the products can be characterized through production routes. A novel representation for cyclic schedules is proposed that has the effect of aggregating the number of batches for each product. It is shown that the no-wait characteristics of subtrains can be exploited with a reduction scheme that has the effect of greatly decreasing the dimensionality of the problem. This reduction scheme can be complemented with a tight formulation of the underlying disjunctions in the MILP to reduce the computational expense. The proposed MILP model for scheduling is extended to design problems in which the potential existence of intermediate storage in the production paths is also considered. In addition to the rigorous scheduling of the process, the sizes of the equipment constituting the various production stages are determined. By using exact linearization schemes it is shown that the problem can be reformulated as an MILP model and solved rigorously to global optimality. Application of the proposed model is illustrated with several example problems."

Algorithmic approaches to process synthesis : logic and global optimization

Christodoulos A. Floudas et al. — Tue, 01 May 2012 11:47:15 PDT

Abstract: "This paper presents an overview on two recent developments in optimization techniques that address previous limitations that have been experienced with algorithmic methods in process synthesis: combinatorics and local optima. The first part deals with the development of logic based models and techniques for discrete optimization which can facilitate the modelling of these problems as well as reducing the combinatorial search. It will be shown that various levels can be considered for the integration of logic in mixed-integer optimization techniques. The second part deals with the development of deterministic optimization methods that can rigorously determine the global optimum in nonconvex optimization models. It will be shown that this can be effectively accomplished with algorithms that exploit identifiable nonlinear structures. Examples are presented throughout the paper and future research directions are also briefly discussed."

Synthesis of optimal chemical reactor networks

Ajay Lakshmanan et al. — Tue, 01 May 2012 11:47:07 PDT

Abstract: "The reactor is the most influential unit operation in many chemical processes. Reaction systems and reactor design often determine the character of the flowsheet. However, research in reactor network synthesis has met with limited success because of nonlinear reaction models, uncertain rate laws, and numerous possible reactor types and networks. In this paper we take advantage of attainable region properties derived from geometric targeting techniques to design a concise reactor modules [sic] for reactor network synthesis. The reactor module is made up of a continuous stirred tank reactor (CSTR) and a plug flow reactor (PFR) (for two dimensional targeting) or a differential sidestream reactors (DSR) (for higher dimensions). These modules are used to synthesize the optimal reactor network target with respect to a specific objective function and the problem is formulated as a compact MINLP. This new algorithm overcomes many of the drawbacks of existing algorithms. Finally, we solve several example problems to illustrate the feasibility of the proposed algorithm and discuss applications to simultaneous reactor network and process synthesis."

Synthesis of chemical reactor networks

Diane Hildebrandt et al. — Tue, 01 May 2012 11:46:59 PDT

Abstract: "An interesting problem in chemical reactor theory is finding bounds or targets on a given performance index in a reacting system. Moreover, performance of the reactor subsystem has a key impact on the design of other processing subsystems. It determines the recycle structure of the process, the separation sequence and has a strong influence on the energy and environmental considerations. However, this area of process synthesis has seen relatively little development when compared to heat integration and separation synthesis. As with the design of heat exchanger networks, this approach has evolved into the (discrete and continuous) optimization of network superstructures as well as the performance targeting of the optimal network prior to its construction. In this study we review both methods for reactor network synthesis but concentrate on advances with the latter approach. The targeting approach is based on geometric interpretations of reaction and mixing. It uses a constructive approach to find the attainable region; that is, it effectively captures all possible reactor structures and finds the bounds on the performance of a reacting system. The approach also generates reactor structures which are candidates for the optimal system. It is however severely limited by the dimensionality of the problem and in practice only 2 and 3 dimensional problems have been solved. Nevertheless, insights gained from this geometric approach have led to an understanding of more general properties of optimal reactor structures. In particular the reactors that make up optimal structures are parallel -- series systems of plug flow reactors, CSTR's and differential side stream reactors. Furthermore, the number of parallel structures is related to the dimensionality of the problem. In addition, these properties can be embedded within optimization formulations in order to deal with more complex problems. In particular, we describe several formulations that incorporate simpler properties derived from attainable region concepts. At this point, this approach is not as rigorous as the geometric approach but readily extendable to more complex reaction systems. In addition, it can be integrated with other process subsystems and allows for simultaneous approaches for heat integration, separation structures and reactor network design. In this way, trade-offs resulting from different parts of the process are properly taken into account in the optimization. All of these concepts will be illustrated with numerous examples. Finally, future work will concentrate on the extension of geometric concepts to more general reactor systems as well as to separation systems. These will also lead to more compact optimization formulations and the consideration of larger and more complex process problems."