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Numerical solution of optimal control problems with explicit and implicit switches

OPTIMIZATION METHODS & SOFTWARE 2018年第3期33卷 450-474页

作者： Bock, Hans Georg Kirches, Christian Meyer, Andreas Potschka, Andreas Heidelberg Univ Interdisciplinary Ctr Sci Comp IWR Heidelberg Germany Tech Univ Braunschweig Inst Math Optimierung Braunschweig Germany

In this article, we present a unified framework for the numerical solution of optimal control problems (OCPs) constrained by ordinary differential equations with both implicit and explicit switches. We present the problem class and qualify different types of implicitly switched systems. This classification significantly affects opportunities for solving such problems numerically. By using techniques from generalized disjunctive programming, we transform the problem into a counterpart one wherein discontinuities no longer appear implicitly. Instead, the new problem contains discrete decision variables and vanishing constraints. Recent results from the field of mixed-integer optimal control theory enable us to omit integrality constraints on variables, and allow to solve a relaxed OCP. We use a first discretize, then optimize' approach to solve the problem numerically. A direct method based on adaptive collocation is used for the discretization. The resulting finite dimensional optimization problems are mathematical programs with vanishing constraints, and we discuss numerical techniques to solve sequences of this challenging problem class. To demonstrate the efficacy and merit of our proposed approach, we investigate three benchmark problems for hybrid dynamic systems.

关键词： switched systems generalized disjunctive programming optimal control mixed-integer optimal control direct transcription methods mathematical programs with vanishing constraints

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Optimal synthesis of work and heat exchangers networks considering unclassified process streams at sub and above-ambient conditions

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APPLIED ENERGY 2018年 224卷 567-581页

作者： Onishi, Viviani C. Quirante, Natalia Ravagnani, Mauro A. S. S. Caballero, Jose A. Univ Coimbra Dept Elect & Comp Engn Polo 2 P-3030290 Coimbra Portugal Inst Syst Engn & Comp Coimbra INESC Coimbra Polo 2 P-3030290 Coimbra Portugal Univ Alicante Dept Chem Engn Ap Correos 99 E-03080 Alicante Spain Univ Estadual Maringa Dept Chem Engn Ave Colombo 5790 BR-87020900 Maringa PR Brazil

Work and heat exchanger networks have recently drawn increasing attention due to their paramount importance in achieving energy savings. In this work, we introduce a new optimization model for the cost-effective synthesis and energy integration of work and heat exchanger networks considering unclassified process streams (ie., streams whose classification as hot or cold streams cannot be defined a priori). Our innovative modelling approach combines mathematical programming techniques and the pinch location method to obtain an optimal network design with minimal cost, while adjusting pressure and temperature levels of unclassified streams. We propose disjunctive operators for the selection of pressure manipulation equipment, and streams identity classification depending on energy requirements and process operating conditions. In addition, our approach addresses previous shortcomings by eliminating the need for: (i) assigning a specific route of pressure manipulation;and, (ii) classifying streams as low or high-pressure streams;which provides further flexibility to the system. Our methodology is also able to effectively deal with variable inlet and outlet streams temperatures to reach specific optimization goals. The model is solved to global optimality through the minimization of the process total annualized cost. Besides improved computational performance, results from energy analyses reveal that streams classification during process optimization can be greatly advantageous for both subambient and above-ambient applications. In the liquefied natural gas process, it reduces up to 89% the energy demand when compared to literature records.

关键词： Simultaneous optimization generalized disjunctive programming Mixed-integer nonlinear programming Work and heat integration Pinch location method Variable temperatures

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Global optimization of disjunctive programs

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JOURNAL OF GLOBAL OPTIMIZATION 2017年第2期69卷 283-307页

作者： Kirst, Peter Rigterink, Fabian Stein, Oliver Karlsruhe Inst Technol Inst Operat Res Karlsruhe Germany Univ Newcastle Sch Math & Phys Sci Newcastle NSW Australia

We propose a new branch-and-bound framework for global optimization of disjunctive programs with general logical expressions. We do not assume the logical expressions to be in any normal form, and, under slightly stronger assumptions, we allow the use of negations and implications. In contrast to the widely used reformulation as a mixed-integer program, we compute the lower bounds and evaluate the logical expression in one step. Thus, we reduce the size of the problem and work exclusively with continuous variables, which is computationally advantageous. We present preliminary numerical results as proof of concept.

关键词： Global optimization disjunctive programming generalized disjunctive programming Branch-and-bound

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Short-term scheduling of multiple source pipelines with simultaneous injections and deliveries

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COMPUTERS & OPERATIONS RESEARCH 2016年 73卷 27-42页

作者： Mostafaei, Hossein Castro, Pedro M. Ghaffari-Hadigheh, Alireza Azarbaijan Shahid Madani Univ Dept Appl Math Tabriz Iran Univ Lisbon Ctr Matemat Aplicacooes Fundamentais & Invest Ope Fac Ciencias P-1749016 Lisbon Portugal

This paper addresses the optimal scheduling of straight pipelines featuring multiple intermediate nodes acting as dual-purpose stations, with a continuous-time Mixed-Integer Linear programming formulation partly derived from generalized disjunctive programming. The new model allows for an intermediate station to act as an output and input terminal at the same time so as to reduce the number of segment switches between active and idle, and consequently decrease operating costs. Contrary to previous approaches, decisions related to batch sizing, batch sequencing and timing are determined in a single step. Several examples of growing complexity are solved to illustrate the effectiveness and computational advantage of the proposed model in both solution quality and CPU time. (C) 2016 Elsevier Ltd. All rights reserved.

关键词： Mathematical modeling Mixed-integer linear programming generalized disjunctive programming Multiproduct pipelines

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A modeling strategy for integrated batch process development based on mixed-logic dynamic optimization

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COMPUTERS & CHEMICAL ENGINEERING 2016年 94卷 287-311页

作者： Moreno-Benito, Marta Frankl, Kathrin Espuna, Antonio Marquardt, Wolfgang UCL Ctr Proc Syst Engn Torrington Pl London WC1E 7JE England Rhein Westfal TH Aachen Proc Syst Engn Turmstr 46 D-52056 Aachen Germany Univ Politecn Cataluna Dept Chem Engn ETSEIB Av Diagonal 647 E-08028 Barcelona Spain

This paper introduces an optimization-based approach for the simultaneous solution of batch process synthesis and plant allocation, with decisions like the selection of chemicals, process stages, task-unit assignments, operating modes, and optimal control profiles, among others. The modeling strategy is based on the representation of structural alternatives in a state-equipment network (SEN) and its formulation as a mixed-logic dynamic optimization (MLDO) problem. Particularly, the disjunctive multistage modeling strategy by Oldenburg and Marquardt (2008) is extended to combine and organize single-stage and multistage models for representing the sequence of continuous and batch units in each structural alternative and for synchronizing dynamic profiles in input and output operations with material transference. Two numerical examples illustrate the application of the proposed methodology, showing the enhancement of the adaptability potential of batch plants and the improvement of global process performance thanks to the quantification of interactions between process synthesis and plant allocation decisions. (C) 2016 Elsevier Ltd. All rights reserved.

关键词： Batch process synthesis Plant allocation Dynamic optimization generalized disjunctive programming Synchronization Multistage modeling

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Cutting Plane Algorithm for Convex generalized disjunctive Programs

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INFORMS JOURNAL ON COMPUTING 2016年第2期28卷 209-222页

作者： Trespalacios, Francisco Grossmann, Ignacio E. Carnegie Mellon Univ Pittsburgh PA 15213 USA

In this work, we propose a cutting plane algorithm to improve optimization models that are originally formulated as convex generalized disjunctive programs. generalized disjunctive programs are traditionally reformulated as mixed-integer nonlinear programming (MINLP) problems using either the big M (BM) or the hull reformulation (HR). The former yields a smaller MILP/MINLP problem, whereas the latter yields a tighter one. The HR can be further strengthened by using the concept of basic step from disjunctive programming. The proposed algorithm uses the strengthened formulation to derive cuts for the big-M formulation, generating a stronger formulation with small growth in problem size. We test the algorithm with several instances. The results show that the algorithm improves generalized disjunctive programming convex models, in the sense of providing formulations with stronger continuous relaxations than the BM formulation, with few additional constraints. In general, the algorithm also leads to a reduction in the solution time of the problems.

关键词： generalized disjunctive programming mixed-integer nonlinear programming

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Algorithmic Approach for Improved Mixed-Integer Reformulations of Convex generalized disjunctive Programs

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INFORMS JOURNAL ON COMPUTING 2015年第1期27卷 59-74页

作者： Trespalacios, Francisco Grossmann, Ignacio E. Carnegie Mellon Univ Pittsburgh PA 15213 USA

In this work, we propose an algorithmic approach to improve mixed-integer models that are originally formulated as convex generalized disjunctive programs (GDPs). The algorithm seeks to obtain an improved continuous relaxation of the mixed-integer linear and mixed-integer nonlinear programming (MILP/MINLP) model reformulation of the GDP while limiting the growth in the problem size. There are three main stages that form the basis of the algorithm. The first one is a presolve, consequence of the logic nature of GDP, which allows us to reduce the problem size, find good relaxation bounds, and identify properties that help us determine where to apply a basic step. The second stage is the iterative application of basic steps, selecting where to apply them and monitoring the improvement of the formulation. Finally, we use a hybrid reformulation of GDP that seeks to exploit both of the advantages attributed to the two common GDP-to-MILP/MINLP transformations, the Big-M, and the Hull reformulation. We illustrate the application of this algorithm with several examples. The results show the improvement in the problem formulations by generating models with improved relaxed solutions and relatively small growth of continuous variables and constraints. The algorithm generally leads to reduction in the solution times.

关键词： generalized disjunctive programming mixed-integer nonlinear programming

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Logic hybrid simulation-optimization algorithm for distillation design

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COMPUTERS & CHEMICAL ENGINEERING 2015年 72卷 284-299页

作者： Caballero, J. A. Univ Alicante Dept Chem Engn E-03080 Alicante Spain

In this paper, we propose a novel algorithm for the rigorous design of distillation columns that integrates a process simulator in a generalized disjunctive programming formulation. The optimal distillation column, or column sequence, is obtained by selecting, for each column section, among a set of column sections with different number of theoretical trays. The selection of thermodynamic models, properties estimation, etc. is all in the simulation environment. All the numerical issues related to the convergence of distillation columns (or column sections) are also maintained in the simulation environment. The model is formulated as a generalized disjunctive programming (GDP) problem and solved using the logic based outer approximation algorithm without MINLP reformulation. Some examples involving from a single column to thermally coupled sequence or extractive distillation shows the performance of the new algorithm. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： Distillation generalized disjunctive programming Simulation Optimization

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Tightening piecewise McCormick relaxations for bilinear problems

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COMPUTERS & CHEMICAL ENGINEERING 2015年 72卷 300-311页

作者： Castro, Pedro M. Univ Lisbon Fac Ciencias Ctr Invest Operac P-1749016 Lisbon Portugal Lab Nacl Energia & Geol P-1649038 Lisbon Portugal

We address nonconvex bilinear problems where the main objective is the computation of a tight lower bound for the objective function to be minimized. This can be obtained through a mixed-integer linear programming formulation relying on the concept of piecewise McCormick relaxation. It works by dividing the domain of one of the variables in each bilinear term into a given number of partitions, while considering global bounds for the other. We now propose using partition-dependent bounds for the latter so as to further improve the quality of the relaxation. While it involves solving hundreds or even thousands of linear bound contracting problems in a pre-processing step, the benefit from having a tighter formulation more than compensates the additional computational time. Results for a set of water network design problems show that the new algorithm can lead to orders of magnitude reduction in the optimality gap compared to commercial solvers. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： Optimization Mathematical modeling Nonlinear programming generalized disjunctive programming Water minimization

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Integration of different models in the design of chemical processes: Application to the design of a power plant

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APPLIED ENERGY 2014年 124卷 256-273页

作者： Caballero, Jose A. Navarro, Miguel A. Ruiz-Femenia, Ruben Grossmann, Ignacio E. Univ Alicante Dept Chem Engn E-03080 Alicante Spain Carnegie Mellon Univ Dept Chem Engn Pittsburgh PA 15213 USA

With advances in the synthesis and design of chemical processes there is an increasing need for more complex mathematical models with which to screen the alternatives that constitute accurate and reliable process models. Despite the wide availability of sophisticated tools for simulation, optimization and synthesis of chemical processes, the user is frequently interested in using the 'best available model'. However, in practice, these models are usually little more than a black box with a rigid input output structure. In this paper we propose to tackle all these models using generalized disjunctive programming to capture the numerical characteristics of each model (in equation form, modular, noisy, etc.) and to deal with each of them according to their individual characteristics. The result is a hybrid modular equation based approach that allows synthesizing complex processes using different models in a robust and reliable way. The capabilities of the proposed approach are discussed with a case study: the design of a utility system power plant that has been decomposed into its constitutive elements, each treated differently numerically. And finally, numerical results and conclusions are presented. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： Process synthesis generalized disjunctive programming Utility systems Modular optimization Kriging

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