Mixed-integer Dynamic Optimization Problem Research Articles

Hybrid simulation-optimization based approach for the optimal design of single-product biotechnological processes

In this work, we present a systematic method for the optimal development of bioprocesses that relies on the combined use of simulation packages and optimization tools. One of the main advantages of our method is that it allows for the simultaneous optimization of all the individual components of a bioprocess, including the main upstream and downstream units. The design task is mathematically formulated as a mixed-integer dynamic optimization (MIDO) problem, which is solved by a decomposition method that iterates between primal and master sub-problems. The primal dynamic optimization problem optimizes the operating conditions, bioreactor kinetics and equipment sizes, whereas the master levels entails the solution of a tailored mixed-integer linear programming (MILP) model that decides on the values of the integer variables (i.e., number of equipments in parallel and topological decisions). The dynamic optimization primal sub-problems are solved via a sequential approach that integrates the process simulator SuperPro Designer ® with an external NLP solver implemented in Matlab ®. The capabilities of the proposed methodology are illustrated through its application to a typical fermentation process and to the production of the amino acid L-lysine.

Inferential Control of Reactive Destillation Columns – An Algorithmic Approach

AbstractDecentralized control system design comprises the selection of a suitable control structure and controller parameters. Here, mixed integer optimization is used to determine the optimal control structure and the optimal controller parameters simultaneously. The process dynamics is included explicitly into the constraints using a rigorous nonlinear dynamic process model. Depending on the objective function, which is used for the evaluation of competing control systems, two different formulations are proposed which lead to mixed‐integer dynamic optimization (MIDO) problems. A MIDO solution strategy based on the sequential approach is adopted in the present paper. Here, the MIDO problem is decomposed into a series of nonlinear programming (NLP) subproblems (dynamic optimization) where the binary variables are fixed, and mixed‐integer linear programming (MILP) master problems which determine a new binary configuration for the next NLP subproblem. The proposed methodology is applied to inferential control of reactive distillation columns as a challenging benchmark problem for chemical process control.

Optimal Allocation of Dynamic VAR Support Using Mixed Integer Dynamic Optimization

Dynamic security of power systems is a major concern during contingencies, especially in present environment when there is inadequate transmission line capacity and dynamic reactive power support. This paper addresses dynamic VAR allocation to ensure acceptable transient voltage performance and short-term stability against severe contingency. To filter out severe contingency, severity indices are proposed. An algorithm based upon trajectory sensitivity analysis is used to select candidate VAR locations. The overall problem is formulated (in dynamic framework) as mixed integer dynamic optimization (MIDO). To solve MIDO problem, it is converted into mixed integer nonlinear problem (MINLP) by complete discretization using simultaneous discretization approach. The effectiveness of proposed approach is illustrated using New England 39-bus system.

Discontinuous Modeling Formulations for the Optimal Control of Partial Shutdowns

AbstractA control strategy is proposed for the optimal operation of multi-unit system undergoing a partial shutdown. The strategy entails manipulating the degrees-of-freedom available during and after a shutdown such that production is restored in a cost-optimal fashion while meeting safety and operational constraints. In this work, we propose several efficient modeling formulations for handling model discontinuities such as shutdown thresholds, induced shutdowns and minimum shutdown duration. The problem is cast as a mixed-integer dynamic optimization problem (MIDO) and is embedded within a feedback control framework based on model predictive control (MPC). A two-tiered regularization approach is presented.

Simultaneous Scheduling and Control of Multiproduct Continuous Parallel Lines

In this work, we propose a simultaneous scheduling and control optimization formulation to address both optimal steady-state production and dynamic product transitions in multiproduct parallel continuous stirred tank reactors (CSTRs). The simultaneous scheduling and control problem for multiproduct parallel continuous reactors is cast as a mixed-integer dynamic optimization (MIDO) problem. The reactor dynamic behavior is described by a set of nonlinear ordinary differential equations (ODEs) that are combined with the set of mixed-integer algebraic equations representing the optimal scheduling production model. The proposed simultaneous scheduling and control approach avoids suboptimal solutions that are obtained when both problems are solved in a decoupled way. Hence, the proposed optimization strategy can yield improved optimal solutions. The simultaneous approach for addressing the solution of dynamic optimization problems, based on orthogonal collocation on finite elements, is used to transform the set...

Control vector optimization and genetic algorithms for mixed-integer dynamic optimization in the synthesis of rice drying processes

Rice drying synthesis is an essential operation that has to be done carefully and cost-effectively. Rice is harvested at high moisture content and hence must be dried within 24 h for safe storage. However, improper drying can cause fissuring in the rice grain, and thus greatly reduce its market value. Multi-pass drying systems are therefore used to gradually bring moisture content to desired level. The problem of rice synthesis, considered in this study, seeks the configuration of units and their corresponding operating conditions that maximize rice quality. This problem is formulated as a mixed-integer dynamic optimization problem. The integer part of the problem reflects process alternatives while the dynamic part originates from nonlinear differential-algebraic equations describing the drying behavior of a rice grain. Clearly such a formidable problem is not easy to solve. Hence, we propose an approach that makes use of two algorithms: a genetic algorithm to search for the best configuration of units and a control vector parameterization approach that optimizes the operating conditions for each configuration. We demonstrate the effectiveness of the approach on a case study.

DOTcvpSB, a software toolbox for dynamic optimization in systems biology

BackgroundMathematical optimization aims to make a system or design as effective or functional as possible, computing the quality of the different alternatives using a mathematical model. Most models in systems biology have a dynamic nature, usually described by sets of differential equations. Dynamic optimization addresses this class of systems, seeking the computation of the optimal time-varying conditions (control variables) to minimize or maximize a certain performance index. Dynamic optimization can solve many important problems in systems biology, including optimal control for obtaining a desired biological performance, the analysis of network designs and computer aided design of biological units.ResultsHere, we present a software toolbox, DOTcvpSB, which uses a rich ensemble of state-of-the-art numerical methods for solving continuous and mixed-integer dynamic optimization (MIDO) problems. The toolbox has been written in MATLAB and provides an easy and user friendly environment, including a graphical user interface, while ensuring a good numerical performance. Problems are easily stated thanks to the compact input definition. The toolbox also offers the possibility of importing SBML models, thus enabling it as a powerful optimization companion to modelling packages in systems biology. It serves as a means of handling generic black-box models as well.ConclusionHere we illustrate the capabilities and performance of DOTcvpSB by solving several challenging optimization problems related with bioreactor optimization, optimal drug infusion to a patient and the minimization of intracellular oscillations. The results illustrate how the suite of solvers available allows the efficient solution of a wide class of dynamic optimization problems, including challenging multimodal ones. The toolbox is freely available for academic use.

Integrated Design and Control Using a Simultaneous Mixed-Integer Dynamic Optimization Approach

In this work, we address the simultaneous design and control of a binary distillation column. The problem is first formulated as a mixed-integer dynamic optimization problem that is then transformed into a mixed-integer nonlinear programming problem using the simultaneous dynamic optimization approach (i.e., full discretization). This formulation is capable of designing the optimal feed tray location, tray sizing, optimal operating steady states, the optimal open-loop trajectory between them, and also the best controller paring and parameters that does the best tracking of the open-loop trajectory. For solving the complex mixed-integer dynamic optimization problem, an optimization decomposition strategy is proposed. The solution strategy is based on solving relaxed versions of the optimization problem and using the results to initialize complex problem versions. The full space problem was solved with the Bonmin solver. Two cases were analyzed, and in both the solver and the proposed decomposition strategy...

An evolutionary approach for multi-objective dynamic optimization applied to middle vessel batch distillation

In this paper, a new approach is presented to perform multi-objective dynamic optimizations of novel batch distillation utilizing an evolutionary algorithm. The contribution is divided into two major parts. First, the development of an efficient hybrid evolutionary algorithm covering multi-objective mixed integer dynamic optimization problems is presented. The efficiency of the optimization solver is proven by several complex test problems. Second, the application of the algorithm is shown by the optimization of a middle vessel batch distillation. The challenging non-linear dynamic model, which takes the start-up phase into account, is solved in Aspen Custom Modeler. It could be proven that the proposed evolutionary algorithm can be applied to complex mathematical problems. Likewise the algorithm was found to successfully handle the optimization of middle a vessel batch distillation. The results show the feasibility of the proposed methodology and a significant increase in profitability of the process.

Simultaneous design, scheduling, and optimal control of a methyl‐methacrylate continuous polymerization reactor

AbstractThis work presents a mixed‐integer dynamic optimization (MIDO) formulation for the simultaneous process design, cyclic scheduling, and optimal control of a methyl methacrylate (MMA) continuous stirred‐tank reactor (CSTR). This article makes two specific contributions. The first consists of incorporating process dynamics into grade transitions to determine changeover times and profiles, as opposed to using fixed changeover times as in previous works dealing with design, scheduling, and dynamic optimization. A second contribution is that the steady states that describe each polymer grade are left as functions of decision variables, which has not been done in the integration of design, scheduling, and dynamic optimization. Also, this work incorporates uncertainty in product demands. The corresponding mathematical formulation includes the differential equations that describe the dynamic behavior of the system, resulting in a MIDO problem. The differential equations were discretized using the simultaneous approach, based on ortogonal collocation on finite elements, rendering a mixed integer non‐linear programming (MINLP) problem, where a profit function is to be maximized. The objective function includes product sales, some capital and operational costs, inventory costs, and transition costs. The optimal solution to this problem involves design decisions: flow rates, feeding temperatures and concentrations, equipment sizing, variables values at steady state; scheduling decisions: grade productions sequence, cycle duration, production quantities, inventory levels; and optimal control results: transition profiles, durations, and transition costs. The problem was formulated and solved in two ways: as a deterministic model and as a two‐stage stochastic programming problem with hourly product demands as uncertain parameters described by discrete distributions. © 2008 American Institute of Chemical Engineers AIChE J, 2008

Optimal delivery of chemotherapeutic agents in cancer

Cancer refers to a set of diseases where normal cells of the body lose mechanisms which are responsible for controlling their growth and motility. Chemotherapy aims to kill these abnormal or cancer cells, however normal cells and healthy tissues are also adversely affected. Systematic modelling and optimization approaches can be employed to obtain effective chemotherapeutic protocol that minimizes the tumour cell population while limiting the damage to the normal cells. In this paper, two models for cancer chemotherapy are considered and the effect of the time of drug administration on the final tumour size is analyzed. The first model is cell cycle non-specific and considers all the cancer cells as belonging to one compartment whereas the second model is cell cycle specific and takes into account the effect of the drugs that act on cells at certain specific stages of the cell cycle. An optimal control problem is also formulated and solved for both the models so as to obtain the chemotherapeutic schedule which minimizes the final tumour size while taking into account the constraints on drug resistance and toxicity. For the first model a mixed integer dynamic optimization problem and for the second model a dynamic optimization problem are solved. The optimal chemotherapeutic drug administration profiles and the variation of number of cells with time are obtained and analyzed in detail.

Simultaneous cyclic scheduling and optimal control of polymerization reactors

AbstractThe simultaneous solution of the optimal sequencing and optimal dynamic transitions of two multigrade polymerization continuous stirred‐tank reactors (CSTRs) is presented. The simultaneous formulation leads to a mixed‐integer dynamic optimization (MIDO) problem. The profiles of the state variables during dynamic transitions are discretized using orthogonal collocation on finite elements, transforming the MIDO problem into a mixed‐integer nonlinear programming (MINLP) problem. The objective of this formulation is to maximize the profit of the manufacturing operation, considering polymer sales, inventory costs and transition costs. The transition cost term is determined by the duration of the transition stages and the manipulated variable profile during transitions. Two industrial reaction systems are used to analyze the optimal solution and compare it to other solutions obtained using different methods. Finally, a sensitivity analysis is carried out to describe the impact of changes in certain problem parameters. © 2007 American Institute of Chemical Engineers AIChE J, 2007

Integrated scheduling and dynamic optimization of grade transitions for a continuous polymerization reactor

This paper presents a modeling and numerical solution method for an integrated grade transition and production scheduling problem for a continuous polymerization reactor. The optimal sequence of production stages and the transitions between them is supposed to be determined for producing a given number of polymer grades at certain amounts and quality specifications in the most economical way. The production schedule has to satisfy due dates for specific orders. This operational problem is cast into a mixed-integer dynamic optimization problem. Disjunctions and logical constraints are combined with a validated differential–algebraic model describing the polymer process during the production of a specific grade as well as along a transition between two different grades. The modeling and solution approach proposed by Oldenburg et al. [Oldenburg, J., Marquardt, W., Heinz D., & Leineweber, D. B. (2003)] is tailored to this problem class to provide an efficient solution technique. An industrial example process serves as an example to illustrate the modeling and solution techniques suggested.

Simultaneous mixed-integer dynamic optimization for integrated design and control

We consider strategies for integrated design and control through the robust and efficient solution of a mixed-integer dynamic optimization (MIDO) problem. The algorithm is based on the transformation of the MIDO problem into a mixed-integer nonlinear programming (MINLP) program. In this approach, both the manipulated and controlled variables are discretized using a simultaneous dynamic optimization approach. We also develop three MINLP formulations based on a nonconvex formulation, the conventional Big-M formulation and generalized disjunctive programming (GDP). In addition, we compare the outer approximation and NLP branch and bound algorithms on these formulations. This problem is applied to a system of two series connected continuous stirred tank reactors where a first-order reaction takes place. Our results demonstrate that the simultaneous MIDO approach is able to efficiently address the solution of the integrated design and control problem in a systematic way.

Simultaneous Cyclic Scheduling and Control of a Multiproduct CSTR

In this work, we propose a simultaneous scheduling and control formulation by explicitly incorporating into the scheduling model process dynamics in the form of differential/algebraic constraints. The formulation takes into account the interactions between scheduling and control and is able to handle nonlinearities embedded into the processing system. The simultaneous scheduling and control problems is cast as a mixed-integer dynamic optimization (MIDO) problem where the simultaneous approach, based on orthogonal collocation on finite elements, is used to transform it into a mixed-integer nonlinear programming (MINLP) problem. The proposed simultaneous scheduling and control formulation is tested using three multiproduct continuous stirred tank reactors featuring difficult nonlinearities.

Global Methods for Dynamic Optimization and Mixed-Integer Dynamic Optimization

An overview of global methods for dynamic optimization and mixed-integer dynamic optimization (MIDO) is presented, with emphasis placed on the control parametrization approach. These methods consist of extending existing continuous and mixed-integer global optimization algorithms to encompass solution of problems with ODEs embedded. A prerequisite for so doing is a convexity theory for dynamic optimization as well as the ability to build valid convex relaxations for Bolza-type functionals. For solving dynamic optimization problems globally, our focus is on the use of branch-and-bound algorithms; on the other hand, MIDO problems are handled by adapting the outer-approximation algorithm originally developed for mixed-integer nonlinear problems (MINLPs) to optimization problems embedding ODEs. Each of these algorithms is thoroughly discussed and illustrated. Future directions for research are also discussed, including the recent developments of general, convex, and concave relaxations for the solutions of nonlinear ODEs.

Global mixed‐integer dynamic optimization

AbstractRecent advances in process synthesis, design, operations, and control have created an increasing demand for efficient numerical algorithms for optimizing a dynamic system coupled with discrete decisions; these problems are termed mixed‐integer dynamic optimization (MIDO). In this communication, we develop a decomposition approach for a quite general class of MIDO problems that is capable of guaranteeing finding a global solution despite the nonconvexities inherent in the dynamic optimization subproblems. Two distinct algorithms are considered. On finite termination, the first algorithm guarantees finding a global solution of the MIDO within nonzero tolerance; the second algorithm finds rigorous bounds bracketing the global solution value, with a substantial reduction in computational expense relative to the first algorithm. A case study is presented in connection with the optimal design and operation of a batch process consisting of a series reaction followed by a separation with no intermediate storage. The developed algorithms demonstrate efficiency and applicability in solving this problem. Several heuristics are tested to enhance convergence of the algorithms; in particular, the use of bounds tightening techniques and the addition of cuts resulting from a screening model of the batch process are considered. © 2005 American Institute of Chemical Engineers AIChE J, 2005

Simultaneous optimal configuration, design and operation of batch distillation

AbstractA methodology for simultaneous determination of optimal batch distillation configuration, design and operation is presented. The configuration design methodology utilizes a mixed integer dynamic optimization (MIDO) formulation approach, where the optimal batch distillation system is obtained automatically, based on its maximum overall profitability for a given separation duty. Using rigorous models, the MIDO problem is solved using a practical stochastic solution approach of genetic algorithm and penalty function. The feasibility of this methodology is demonstrated for both binary and multicomponent separation scenarios. In the binary separation case study, the effect of feed composition for different binary mixtures on the optimal configurations, that is, regular vs. inverted columns, is investigated and discussed. The regular column was found to be more profitable for feeds with a high fraction of the light component, whereas the inverted column is optimal for heavier feeds. The optimality of a particular configuration over another is, however, case study specific, depending on, for example, how easy the mixture is to separate. In the multicomponent separation case study, the results obtained highlight the superiority of the multivessel configuration over the regular and inverted configurations. © 2005 American Institute of Chemical Engineers AIChE J, 2005

Short-Term Scheduling of Batch Processes. A Comparative Study of Different Approaches

The approaches to scheduling problem formulation in chemical engineering can be broadly classified into two categories, namely, the standard recipe approach (SRA) and the overall optimization approach (OOA). In the SRA, first the recipes are standardized either empirically or via single-batch optimization (SBO) and then the production scheduling problem is formulated on the basis of these standardized recipes. However, the standardization of recipes removes degrees of freedom from the system, and because of this, the solutions obtained with this approach can be suboptimal, whereas in the OOA, the process dynamics are directly included in the scheduling problem formulation instead of the standardized recipes. This restores the additional degrees of freedom of the system, and therefore this approach can yield a better solution. However, direct inclusion of the process dynamics in the scheduling problem formulation results in a mixed-integer dynamic optimization (MIDO) problem, the solution of which can be a...

Optimal design and operation of continuous ultrafiltration plants

In this paper, an approach for the optimal design and operation of a continuous ultrafiltration installation is presented. The approach is based on a dynamic model which takes into account fouling and cleaning issues and consists of a set of integral, differential and algebraic equations comprising integer and continuous variables. An economic objective function which includes capital and operating costs, is used and appropriate operating constraints are imposed. The overall problem is formulated as a mixed integer dynamic optimisation problem and a solution procedure is proposed. Using this method, a cheese whey ultrafiltration plant involving the use of commercially available tubular ceramic membrane modules, is studied. Using the proposed solution procedure, the optimum number of stages, the number of membrane modules per stage, the feed and circulation pumps sizing, the work pressure profile and the timing of the operating and cleaning tasks, are determined.