Differential Evolution in Chemical Engineering, Developments and Applications (2017)

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Advances in Process Systems Engineering – Vol. 6

DIFFERENTIAL EVOLUTION

IN CHEMICAL ENGINEERING

Developments and Applications

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Techniques and Applications in Chemical Engineering

edited by Gade Pandu Rangaiah

Vol. 2: Stochastic Global Optimization:

Techniques and Applications in Chemical Engineering

Vol. 3: Recent Advances in Sustainable Process Design and Optimization

edited by D. C. Y. Foo, M. M. El-Halwagi and R. R. Tan

Vol. 4: Computation of Mathematical Models for Complex Industrial Processes

by Yu-Chu Tian, Tonghua Zhang, Hongmei Yao and Moses O. Tadé

Vol. 5: Multi-Objective Optimization:

Techniques and Applications in Chemical Engineering (Second Edition)

Vol. 6: Differential Evolution in Chemical Engineering: Developments and Applications

edited by Gade Pandu Rangaiah and Shivom Sharma

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NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO

World Scientific

editors

Gade Pandu Rangaiah

National University of Singapore

Shivom Sharma

École Polytechnique Fédérale de Lausanne, Switzerland

Advances in Process Systems Engineering – Vol. 6

DIFFERENTIAL EVOLUTION

IN CHEMICAL ENGINEERING

Developments and Applications

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Library of Congress Cataloging-in-Publication Data

Names: Rangaiah, Gade Pandu, editor. | Sharma, Shivom, editor.

Title: Differential evolution in chemical engineering : developments and applications / edited by Gade Pandu Rangaiah (NUS, Singapore),

Shivom Sharma (Ecole Polytechnique Fâedâerale de Lausanne, Switzerland). Description: [Hackensack] New Jersey : World Scientific, [2017] |

Series: Advances in process systems engineering ; volume 6 Identifiers: LCCN 2016056698 | ISBN 9789813207516 (hc : alk. paper) Subjects: LCSH: Chemical engineering--Mathematics. | Evolution equations. | Mathematical optimization.

Classification: LCC TP184 .D54 2017 | DDC 660--dc23 LC record available at https://lccn.loc.gov/2016056698

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

Desk Editor: Herbert Moses Typeset by Stallion Press

Email: [email protected] Printed in Singapore

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Preface

Humans inherently optimize their activities in daily life. Optimization can also be seen in natural phenomena and evolution. In chemical engineer-ing, optimization is required in many areas such as model development, design, operation and control. The optimization problem can be for one or more objectives, depending on the requirements of the application. Many optimization problems in chemical engineering are complex in nature due to the presence of non-linearity, large number of decision variables and constraints. Hence, numerical optimization methods are used to solve such optimization problems. Among these, stochastic or metaheuristic methods are simple and useful for solving any optimization problem.

An effective optimization method should be able to find the best possible value of the performance criterion by varying decision variables within their bounds whilst also satisfying the constraints in the problem. Differential evolution (DE), proposed by Storn and Price in the year 1995, is a simple and effective method for solving difficult optimization problems. Employing a population of individuals (trial solutions), it iteratively improves them for the given performance criterion through mutation, operation and selection operations. Over the last two decades, DE algorithm has been continually improved in population initialization, mutation and crossover operations, adaptation of parameters, adaptation for multiple objectives and hybridiza-tion with other optimizahybridiza-tion techniques.

This book on DE presents the recent developments in DE and its applica-tions in chemical engineering. It consists of 13 chapters, grouped into three

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parts. Introduction chapter (Part I) provides an overview to optimization, DE and chemical engineering applications. Part II (Chapters 2–5) presents a review of DE applications in chemical engineering and DE programs for single and multiple objectives. Finally, Chapters 6–13 (Part III) describe the use of DE for the optimization of chemical engineering applications of importance. The applications covered in Parts II and III, according to the chapter sequence, pertain to chemical reaction engineering, thermo-dynamics, oil and gas industry, heat exchanger network, separation pro-cesses, petrochemicals, fermentation process, metabolic engineering and polymerization reaction engineering. A number of chapters in this book employ DE programs in FORTRAN, MS Excel, R, MATLAB and GAMS, for single and multi-objective optimization. These programs and other rel-evant materials for many chapters are available on the book’s website (http://www.worldscientific.com/worldscibooks/10.1142/10379).

Differential Evolution in Chemical Engineering will be useful for prac-titioners, researchers and students interested in process optimization and DE. This book offers an overview to process optimization, detailed DE algorithm and use of DE programs for optimizing chemical engineering and related applications. Readers familiar with the basics of optimization and chemical engineering can read any chapter of interest independent of other chapters in this book. In general, we recommend all readers to go through the Introduction chapter. Many chapters in this book can be used as supplementary material in optimization courses for senior undergradu-ate and postgraduundergradu-ate students, and the associundergradu-ated exercises can be used as assignments or projects. Students can use available programs for solving the exercises in this book or any other optimization problem.

Two experts anonymously reviewed each submission/chapter in this book. Then, contributors thoroughly revised, and finally editors read the revised version for consistency. We are grateful to all contributors and reviewers for their cooperation in the timely completion of these activities for a high quality and useful book. Our special thanks to Prof. A. Bonilla-Petriciolet, for handling reviews of two chapters co-authored by both of us. We acknowledge Lim Swee Cheng, Steven Patt, Herbert Moses and D. Rajesh Babu of World Scientific, for their suggestions and assistance during the preparation of this book.

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Preface vii

I thank my research students and co-authors for their contributions to this book. I am grateful to all my family members for their deep affection and unwavering support. I am particularly grateful to my wife (Krishna Kumari) for taking care of me and our family.

I express my sincere gratitude to my research advisors: Prof. G.P. Rangaiah, Prof. F. Maréchal and Dr. N. Bhandari. I would like to thank my parents, parents-in-law, wife (Dr. Vaishali Gaur), brother and sisters for their everlasting love and support.

Gade Pandu Rangaiah Shivom Sharma

NUS, Singapore EPFL, Switzerland

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About the Editors

Dr. Gade Pandu Rangaiah has been with the National University of Singapore (NUS) since 1982, in the Department of Chemical & Biomolecular Engineering. He received his Bachelor, Masters and Doctoral degrees, all in chemical engineering, from Andhra Univer-sity, IIT Kanpur and Monash UniverUniver-sity, respec-tively. He worked in Engineers India Limited for two years before his Doctoral study. Dr. Rangaiah’s teaching was recognized with many awards, including the NUS Annual Teaching Excellence Award for four consecutive years. Dr. Rangaiah’s research interests are in modeling, optimization, design and control of chemical and related processes. He supervised 50 graduate theses including 22 doctoral theses. Dr. Ranga-iah edited six books, and the second edition of the first book is also published. The three recent books are: “Plant-Wide Control: Recent Developments and Applications” (with V. Kariwala), “Multi-Objective Optimization in Chemical Engineering: Developments and Applications” (with A. Bonilla-Petriciolet), and “Chemical Process Retrofitting Revamp-ing: Techniques and Applications”. He contributed many chapters to these and other books. Dr. Rangaiah published 180+ journal papers and 130+ conference papers in the research area of Process Systems Engineering. For more details on his research and publications, browse http://cheed.nus.edu.sg/stf/chegpr/homefinal.html.

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Dr. Shivom Sharma is currently working as a post-doctoral fellow at EPFL (École Polytech-nique Fédérale de Lausanne), Switzerland. He received his Bachelor degree from U. P. Techni-cal University, Lucknow. After finishing his Mas-ters degree in Chemical Engineering from I.I.T. Roorkee, he joined the University of Petroleum and Energy Studies, Dehradun and worked there for almost a year. He received his PhD degree (with thesis entitled ‘multi-objective differential evolution: modifications and applications to chemical processes’) from the Department of Chemical & Biomolecular Engineering, National Uni-versity of Singapore. After completing his PhD under the supervision of Prof. G.P. Rangaiah, he worked as a research fellow in the same research group, for two and half years. Dr. Sharma’s research interests include multi-objective optimization methods, differential evolution, bio-ethanol process, bio-diesel process, water networks and bio-refineries. He has published 12 journal papers, 11 book chapters and presented his work at 13 interna-tional/regional conferences. For more details on his research work, browse https://sites.google.com/site/shivomsharmanus/.

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List of Contributors

Bonilla-Petriciolet A., Department of Chemical & Biochemical Engi-neering, Instituto Tecnológico de Aguascalientes, Aguascalientes, México, 20256.

Chen S.Q., Department of Chemical & Biomolecular Engineering, National University of Singapore, Singapore 117585.

Contreras-Zarazúa G., Universidad de Guanajuato, Campus Guanaju-ato, Departamento de Ingeniería Química, División de Ciencias Naturales y Exactas, Noria Alta s/n, 36050, Guanajuato, Gto., México.

Corazza M.L., Department of Chemical Engineering, Universidade Fed-eral do Paraná, Curitiba, Brasil 81531-970.

Curteanu S., Gheorghe Asachi Technical University of Iasi, Faculty of Chemical Engineering and Environmental Protection, Department of Chemical Engineering, Iasi, Romania 700050.

Dragoi E.N., Gheorghe Asachi Technical University of Iasi, Faculty of Chemical Engineering and Environmental Protection, Department of Chemical Engineering, Iasi, Romania 700050.

Errico M., Department of Chemical Engineering Biotechnology and Environmental Technology, Southern Denmark University, Campusvej 55, Odense M 5230, Denmark.

Hamedi N., Department of Chemical Engineering, School of Chemical and Petroleum Engineering, Shiraz University, Shiraz, Iran.

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Jaime-Leal J.E., Department of Chemical Engineering, Universidad de Guanajuato, Mexico, 36050.

Koop L., Department of Chemical Engineering, Universidade Federal do Paraná, Curitiba, Brasil 81531-970.

Maréchal F., Industrial Process and Energy Systems Engineering, École Polytechnique Fédérale de Lausanne, CH-1951 Sion, Switzerland. Quiroz-Ramírez J.J., Universidad de Guanajuato, Campus Guanajuato, Departamento de Ingeniería Química, División de Ciencias Naturales y Exactas, Noria Alta s/n, 36050, Guanajuato, Gto., México.

Rahimpour M.R., Department of Chemical Engineering, School of Chemical and Petroleum Engineering, Shiraz University, Shiraz, Iran. Ramírez-Márquez C., Universidad de Guanajuato, Campus Guanaju-ato, Departamento de Ingeniería Química, División de Ciencias Naturales y Exactas, Noria Alta s/n, 36050, Guanajuato, Gto., México.

Rangaiah G.P., Department of Chemical & Biomolecular Engineering, National University of Singapore, Singapore 117585.

Rong B.G., Department of Chemical Engineering Biotechnology and Environmental Technology, Southern Denmark University, Campusvej 55, Odense M 5230, Denmark.

Sánchez-Ramírez E., Universidad de Guanajuato, Campus Guanajuato, Departamento de Ingeniería Química, División de Ciencias Naturales y Exactas, Noria Alta s/n, 36050, Guanajuato, Gto., México.

Segovia-Hernández J.G., Universidad de Guanajuato, Campus Guanaju-ato, Departamento de Ingeniería Química, División de Ciencias Naturales y Exactas, Noria Alta s/n, 36050, Guanajuato, Gto., México.

Sharma S., Industrial Process and Energy Systems Engineering, École Polytechnique Fédérale de Lausanne, CH-1951 Sion, Switzerland. Singh A., Department of Chemical & Biomolecular Engineering, National University of Singapore, Singapore 117585.

Sreepathi B.K., Department of Chemical & Biomolecular Engineering, National University of Singapore, Singapore 117585.

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List of Contributors xiii

Srinivas M., Department of Chemical & Biomolecular Engineering, National University of Singapore, Singapore 117585.

Torres-Ortega C.E., Department of Chemical Engineering Biotechnology and Environmental Technology, Southern Denmark University, Campusvej 55, Odense M 5230, Denmark.

Voll F.A.P., Department of Chemical Engineering, Universidade Federal do Paraná, Curitiba, Brasil 81531-970.

Wang F.S., Department of Chemical Engineering, National Chung Cheng University, Chiayi 62102, Taiwan.

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3

Differential Evolution in Chemical Engineering 9in x 6in b2817-ch01

Chapter 1 Introduction

Shivom Sharma1_{and Gade Pandu Rangaiah}2,*

1_{Industrial Process and Energy Systems Engineering}

École Polytechnique Fédérale de Lausanne, CH-1951 Sion, Switzerland

2_{Department of Chemical and Biomolecular Engineering}

National University of Singapore, 117585 Singapore

*_{Corresponding author: [email protected]}

1.1 Process Optimization

Optimization is an approach to find the best possible solution in the domain of interest while satisfying relevant constraints (restrictions). Optimization problems can be found everywhere, from engineering to economics and from daily life to holiday planning. For example, holiday planning optimization finds the place(s) to visit, when to visit, how to travel and duration of stay (which are all decision variables) to achieve the most happiness (which is the objective function or performance criterion); this may have constraints on budget, travel dates and places to visit as well as other objectives such as safety.

Optimization has been fruitfully applied to improve the performance and/or understanding in diverse areas such as science, engineering, business and economics. The goal of optimization is to find the values of decision variables, which will maximize or minimize the value of a given objective function (performance criterion) without violating specified constraints. Mathematically, an optimization problem can be stated as follows.

Objective function: Minimize or maximize f1(x) (1.1a)

Decision variables: x ≡ x1, x2… xn (1.1b)

Constraints: xL_{< x < x}U _(1.1c)

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g(x) < 0 (1.1d)

h(x) = 0 (1.1e)

Here, f1(x) is the given objective function, x is the set of decision variables with xL_{and x}U_{as the lower and upper bounds, and g and h are} the set of inequality and equality constraints, respectively. Many application problems have more than one decision variable and a number of inequality and/or equality constraints.

An optimization problem is generally assumed to have only one

objective function as in equation (1.1a); such problems belong to single-objective optimization (SOO). Each of these problems will have

one or more optimal solutions. Note that optimization refers to both minimization and maximization, and an optimum can be either a minimum or a maximum. A minimization objective can be transformed into a maximization objective by multiplying with –1 or taking reciprocal (with a suitable modification to avoid division by zero). Similarly, a minimization method can easily be modified to a maximization method. Many books describe optimization for minimization, and we follow the same in this chapter. In other words, optimization and optimum are used as synonymous with minimization and minimum, respectively.

In the literature, numerous chemical engineering application problems have been optimized for single objective (e.g., see Himmelblau, 1972; Edgar et al., 2001; Ravindran et al., 2006; Rangaiah, 2010; Floudas, 2013). For example, optimization has been successfully applied in the design and operation of chemical and refinery processes, biotechnology, food technology, pharmaceuticals, fuel cells, power plants and bio-fuel production. Capital/equipment cost, operating cost, profit, net present value, energy consumption, efficiency, conversion, yield, selectivity, eco-indicator 99, global warming potential and CO2 emissions are the commonly used objective functions in process optimization problems. 1.2 Classification of Optimization Methods

Optimization problems and methods can be classified in various ways using the characteristics summarized in Table 1.1. Some of these are briefly described in the following sub-sections. Many chemical engineering application problems have more than one variable and bounds on variables. Also, they often contain constraints arising from governing equations (such as mass and energy balances, and rate equations) and from process limitations (such as on maximum

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Introduction 5 Differential Evolution in Chemical Engineering 9in x 6in b2817-ch01

temperature, pressure and flow rate for safety and due to material of construction).

Table 1.1 Characteristics and classification of optimization problems and methods

Characteristic Classification

Number of variables: one or more Single variable or multivariable optimization Type of variables: real, integer or mixed Nonlinear, integer or mixed (nonlinear)

integer programming

Nature of equations: liner or nonlinear Linear or nonlinear programming Constraints: no constraints (besides

bounds) or with constraints

Unconstrained or constrained optimization Number of objectives: one or more Single-objective or multi-objective

optimization

Derivatives: without or using derivatives Direct or gradient search optimization Optimum: local or global in the search

space

Local or global optimization Random numbers: without or using

random numbers

Deterministic or stochastic optimization methods

Trial points/solutions: one or more in each

iteration Single point (also known as trajectory) or population based methods 1.2.1 Use of derivatives

Optimization methods can be classified based on the use of derivate information. If the objective function and constraints are continuous and differentiable, then derivative-based methods such as steepest descent, quasi-Newton and successive quadratic programming (SQP) methods based on gradient vector can be used. These methods are computa-tionally efficient, and give the same solution in different runs if the initial point is the same. Derivative-free methods (e.g., Nelder-Mead or downhill simplex) are used when the objective function or constraints have discontinuities. Both gradient-based and gradient-free methods can be used for solving SOO problems. Some of them are for unconstrained optimization whereas others for problems with constraints. For example, Nelder-Mead, steepest descent and quasi-Newton methods are for problems without constraints, whereas simplex, generalized reduced gradient (GRG) and SQP methods are for constrained optimization. For details on these methods, see Edgar et al. (2001) and Ravindran et al. (2006).

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1.2.2 Local and global methods

A given optimization problem may have more than one optimum. Fig. 1.1 illustrates this situation for both minima and maxima; in this figure, x-axis represents the search space in one or many decision variables, and the objective function (y-axis) can be for minimization or maximization. There are three local minima, two global minima, four local maxima and one global maximum in Fig. 1.1. By definition, a local minimum is the minimum in its nearby region whereas a global minimum is the lowest minimum over the entire search region (within bounds and satisfying constraints, if any). The objective function in Fig. 1.1 is neither convex nor concave over the entire region, and it is said to be multi-modal.

Fig. 1.1 Local and global optima of an optimization problem

Based on their search capability, optimization methods can be classified into local and global methods. Local search methods generally converge to an optimum in the neighborhood of the initial/starting point. Nelder-Mead, steepest descent, quasi-Newton, GRG and SQP methods are local search methods. These methods require an initial point or solution for starting the search, and converge to a nearby minimum, which can be local or global minimum. On the other hand, global methods search the entire search space, have the capability to escape from the local optimum and to find the global optimum. Multi-start is a simple strategy for searching global optimum using local methods in

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conjunction with a number of initial points. Global optimization methods, particularly stochastic methods, may not be successful in every run, and they need more computation time compared to local optimization methods. Randomization is an important component of stochastic search methods. There are many global methods, and they are introduced in the next sub-sections.

1.2.3 Deterministic or stochastic methods

Optimization methods can also be classified into deterministic and stochastic methods. In deterministic optimization methods, the search for optimum is not random, i.e., it does not depend on random numbers; rather, the search is determined according to the algorithm, optimization problem and initial point. Hence, the new solution found in each iteration does not depend on random numbers. The final/converged solution by a deterministic method depends on the initial point. Examples of deter-ministic optimization methods are Nelder-Mead (also known as downhill simplex), steepest descent, quasi-Newton, GRG and SQP methods, which are all local methods. They are computationally efficient and can locate the optimum precisely. If the optimization problem is multi-modal as in Fig. 1.1, they are likely to converge to a local minimum near the initial point/solution, thus failing to find the global minimum. Note that deterministic methods require continuous and differentiable objective function and constraints.

Stochastic optimization methods, on the other hand, employ random numbers in their search strategies and are more likely to find the global optimum but they generally require more computational time and give less precise optimum compared to the deterministic methods. Almost all of them do not require continuity and differentiability of equations in the optimization problem as well as the initial/starting solution. Hence, stochastic methods can be applied to any type of optimization problems including black-box problems, wherein only the effect of decision variables on the objective and/or constraints is known (and not the underlying mathematical equations and their nature). Since they use random numbers in their search, they may converge to slightly different solutions in different runs. Stochastic optimization methods are based on search using a single point/solution or population of points/solutions, and they are inspired by logic, physical and/or natural phenomena. Simulated annealing (SA), genetic algorithms (GA), differential evolution (DE), particle swarm optimization (PSO) and ant colony optimization (ACO)

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are stochastic SOO methods. They are also known as metaheuristics, further discussed in Section 1.3. Although stochastic methods are time consuming, they have become popular due to their applicability to any type of optimization problems, reliability in locating the global optimum, relative simplicity of algorithms and easy adaptability for multi-objective optimization (MOO), discussed in Section 1.4.

1.2.4 Number of individuals

Another important classification of optimization methods is based on the use of single or multiple individuals at a time. A single point method uses a single individual/solution at a time, which follows a path in the search space as the iterations continue. Hill-climbing and SA are single point optimization methods. Conversely, population-based methods use a number of individuals, and these individuals interact with one another for modifying their respective paths. GA, DE, PSO and ACO are some important population-based methods. Deterministic methods are single point methods.

1.3 Metaheuristics

Metaheuristics are designed to solve complex optimization problems for global optimality. The word ‘metaheuristic’ was coined by Fred Glover in 1986. By dictionary meaning, heuristic means to develop rules based on common understanding, and metaheuristic means high level heuristic. Thus, metaheuristicsare iterative algorithms incorporating suitable operations and strategies for efficiently finding the global optimum. They use random numbers (stochastic nature), and incorporate exploration and exploitation strategies. A good balance between exploration and exp-loitation of search space is critical for reaching the global optimum efficiently. Exploration is useful in searching the entire search space and in escaping local optimum, whereas exploitation is required for faster convergence. Metaheuristics can locate the global optimum in reasonable computation time, but they will need infinite time for guaranteed conver-gence to the global optimum. In the chemical engineering literature, GA, DE, ACO, PSO, SA and tabu search (TS) are commonly used meta-heuristics for solving SOO problems. Fig. 1.2 presents a classification of these and some other metaheuristics. Many of them are population-based, and some population-based metaheuristics are collectively known as evolutionary methods as they are based on nature-inspired concepts.

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Fig. 1.2 Classification of common and recent metaheuristics

Among the evolutionary methods, GA is inspired by the natural evolution of different species (Holland, 1975). Originally, binary strings (or chromosomes) were used to implement GA, where each chromosome in the population represents a trial solution. Each of the chromosomes in the population is randomly initialized within the bounds on decision variables. These chromosomes undergo selection, crossover and mutation operations. Selection operation ensures diversity of mating pool with higher chances of selecting better individuals for crossover and mutation. Crossover operation exchanges information between parent chromosomes, whereas mutation operation adds new random information into the offspring. Subsequently, GA was also implemented using real numbers (Deb, 2001).

DE was proposed by Storn and Price (1995) for solving optimization problems over continuous search space. Section 1.5 in this chapter provides more details on DE. ACO was proposed to solve routing problems (Dorigo and Gambardella, 1997), and it has been used to solve job-shop scheduling, batch scheduling and combinatorial problems. ACO works on the principle of self-organization and transfer of information among individual ants through pheromones. Ants always search shortest

Metaheuristics Genetic Algorithm Genetic Programming Evolutionary Strategy Evolutionary Programming Differential Evolution Population-based

Ant Colony Optimization Particle Swarm Optimization

Harmony Search Bees Algorithm Firefly Algorithm Cuckoo Search Simulated Annealing Tabu Search Local Search Guided Local Search

Single point

Evolutionary Others

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path between nest and available food. PSO mimics the social behavior of swarms (Kennedy and Eberhart, 1995); in this, particles or swarms search possible solutions in their neighborhood, and share their experiences with other particles.

SA uses the concept of annealing (i.e., slow cooling) process in metallurgy (Kirkpatrick et al., 1983). In each iteration, a trial solution is generated in the neighborhood of the current solution, and the current solution is replaced by the trial solution if the latter has better objective value or satisfies Metropolis criterion to avoid trapping of search in local optimum regions. These steps are repeated many times before reducing the temperature. Tabu search (TS) maintains a short memory of trial solutions, and uses them to prohibit reverse moves (Glover, 1986) while searching for a better solution in the neighborhood.

In the 21st_{century, several new metaheuristics were proposed. Geem}

et al. (2001) developed music inspired harmony search. Pham et al.

(2005) proposed bees algorithm, which mimics food foraging behavior of honey bee colonies. Yang (2008) developed firefly algorithm, which is inspired by flashing behavior of fireflies. Yang and Deb (2009) introduced cuckoo search, which is inspired by obligate brood parasitism of some cuckoo species.

1.4 Multi-objective Optimization Problems and Their Solutions Many applications are likely to have two or more objective functions; such problems belong to MOO. For example, an MOO problem with three objectives is as follows.

Objective functions: Minimize or maximize f1(x) (1.2a) Minimize or maximize f2(x) (1.2b) Minimize or maximize f3(x) (1.2c) Decision variables: x ≡ x1, x2… xn (1.2d) Constraints: xL_{< x < x}U _(1.2e) g(x) < 0 (1.2f) h(x) = 0 (1.2g) Here, f1(x), f2(x) and f3(x) are the given three objective functions. Solution of an MOO problem, having conflicting objectives, gives many optimal solutions, which are called Pareto-optimal front or non-dominated solutions. See Bhaskar et al. (2000), Rangaiah (2009), Rangaiah and Bonilla-Petriciolet (2013) and Rangaiah et al. (2015) for

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more details on MOO and its numerous applications in chemical engineering.

Weighted sum and ε-constraint are two classical methods for converting an MOO problem into an SOO problem. In order to find the set of non-dominated solutions for MOO problem, the converted SOO problem is solved many times, obtaining one non-dominated solution each time. In the weighted sum method, a scalar weight is assigned to each objective function, and different values of weights are used to obtain different non-dominated solutions. The ε-constraint method optimizes the MOO problem for one objective function, while other objective functions are considered as additional inequality constraints in the SOO problem.

Weighted sum and ε-constraint methods cannot accommodate preferred values for different objective functions (Deb, 2001; Rangaiah, 2009). Goal programming and compromise programming can accommodate such preference of the decision maker. In these, the desirable solution is that having the smallest difference between objective functions and their respective goals. Normal boundary intersection and normalized normal constraint are other methods to convert MOO problems into SOO problems. The converted SOO problem can be solved using a gradient-based method (e.g., GRG and SQP) or metaheuristics (e.g., GA, DE, ACO, PSO, SA and TS).

All the metaheuristics (stochastic optimization methods) were proposed for solving SOO problems. Later, researchers adapted many of them to solve the MOO problem in a single run. MOO methods like non-dominated sorting genetic algorithm-II (Deb, 2002), multi-objective differential evolution (Sharma and Rangaiah, 2013a; Chapter 5 in this book) and multi-objective particle swarm optimization (Coello and Salazar, 2002) can generate the complete Pareto-optimal front in a single run. ACO (Mariano and Morales, 1998), SA (Serafini, 1994; Ramteke and Gupta, 2009) and TS (Gandibleux et al., 1997) have also been adapted successfully for multiple objectives.

1.5 Differential Evolution

Differential evolution (DE) was proposed by Storn and Price (1995, 1997) for solving SOO problems over continuous search space. Fig. 1.3 presents a simple pseudocode for classic DE. An initial population is randomly generated inside the bounds on decision variables, and values of objective functions and constraints are calculated for each and every

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individual in the initial population. In each generation, a mutant individual is created for each target individual in the initial/current population, by using three or more other individuals randomly chosen from the initial/current population. After that, target and mutant individuals produce a trial individual by crossover operation. Finally, selection is performed between the target and trial individuals based on objective function value, to select the better individual for the subsequent generation. There are three parameters in classic DE algorithm: (1) population size (NP) with a recommended value of 5 to 20 times number of decision variables, (2) crossover probability (Cr), which can be between 0 and 1, and (3) mutation factor (F), which has a recommended value between 0 and 2. In addition, the user has to choose maximum number of generations (MNG) or iterations. See Section 1.7 for an illustration of multi-objective DE algorithm with a simple application.

Fig. 1.3 A simple pseudocode for classic DE

Since its proposal in the year 1995, DE has undergone many changes and developments, and is now one of the popular and effective global optimization techniques. Its code and concepts are readily available on the internet (http://www.icsi.berkeley.edu/~storn/code.html). Researchers have improved classic DE in various aspects of the algorithm such as population initialization, mutation, crossover and selection operations (Price et al., 2005; Brest et al., 2006; Rahnamayan etal., 2008).A review

Specify DE parameters (population size, maximum number of generations, crossover probability and mutation factor)

Initialize the population randomly inside the bounds on decision variables

Evaluate objective function and constraints for all individuals in the initial population Generation loop

For i = 1 to population size

Select ith_{individual as target individual}

Generate a mutant individual using three random individuals in the population Generate trial individual using target and mutant individuals

Calculate objective function and constraints for trial individual Selection between target and trial individuals, for the next generation Repeat the generation loop for maximum number of generations.

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of these developments in DE until the year 2009 is available in Chapter 2 of this book. Recent advances in DE are summarized by Das et al. (2016).

In particular, since DE performance for solving application problems depends on the values of its parameters, several studies have focused on the adaption of these parameters (Zhang and Sanderson, 2008; Huang

et al., 2010; Wang et al., 2010). Further, DE has been successfully

adapted for multiple objectives (Kukkonen and Lampinen, 2009; Dong and Wang, 2009; Gong and Cai, 2009; Gujarathi and Babu, 2010; Ali et

al., 2012; Chen et al., 2014). Chapter 5 in this book describes one

adaptation of DE for multiple objectives and an MS Excel-based program for it. Instead of commonly used maximum number of generations, new termination criteria have been proposed and studied in conjunction with DE (Srinivas and Rangaiah, 2007; Zhang and Rangaiah, 2011; Fernández-Vargas et al., 2016 for SOO; Sharma and Rangaiah, 2013c for MOO).

1.6 Applications of Differential Evolution in Chemical Engineering Chapter 2 in this book presents developments and applications of DE in chemical engineering, until the year 2009. Recently, Dragoi and Curteanu (2016) reviewed DE applications for solving chemical engineering problems. In order to provide an overview of DE applications in chemical engineering in recent years, Scopus database was searched using the keyword ‘differential evolution’ in the article title and keywords. Other criteria for narrowing the Scopus search are as follows: fields — article or review article, date range — January 2008 to April 2016, subject area — chemical engineering, energy and environmental science, language — English. In total, 904 articles were found, and 216 articles were short-listed based on their relevance to DE and chemical engineering. From these short-listed articles, Fig. 1.4 shows the number of journal papers on DE applications in chemical engineering and related areas over the years 2008 to 2016, and Table 1.2 summarizes number of articles published in different journals. It can be seen that nearly 70 papers are in energy-related journals (Table 1.2).

DE applications in chemical engineering can be divided into two broad categories: (1) modeling or parameter estimation, and (2) process design and optimization. About 60% studies on DE applications in chemical engineering, energy and environmental science from the year

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2008 to April 2016 are on process design and optimization, and the remaining are on parameter estimation, mainly for estimating reaction kinetics. Most of the studies considered single objective function; only about 10% of studies considered two or more objective functions. Tables 1.3 and 1.4 summarize applications of DE in chemical engineering, and the commonly used objectives in these applications. Energy Conversion and Management journal has attracted the most number of DE studies (20 papers) followed by the Energy journal with 14 papers.

Fig. 1.4 Number of reported DE applications in chemical engineering and related areas in recent years; number for 2016 is until April only

Some recent studies on DE applications have also considered modifications in DE algorithm aspects. Important modifications in DE algorithm are summarized below. The modified DE algorithms have been used for optimization of chemical engineering related applications.

 Adaptation of mutation factor, crossover probability and/or mutation strategy

 Population initialization, different sub-populations with different mutation strategies

 Termination criteria, equality and inequality constraints handling  DE with PSO, DE with free search, DE with group search, DE with

tabu list, DE with chaotic search, DE with sequential simplex, DE with clustering, DE with conjugate gradient method

 Permutation based DE, Discrete binary DE

0 5 10 15 20 25 30 35 40 2008 2009 2010 2011 2012 2013 2014 2015 2016 No. of Journal Papers Year of Publication

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Table 1.2 Number of reported DE applications in different journals (N - number of journal papers): January 2008 to April 2016

Name of Journal (J) N

Adsorption Science and Technology AIChE J, Biochemical Engineering J, Bioprocess and Biosystems Engineering, Chemical Engineering Transactions, Energy and Fuels, International J of Heat and Mass Transfer, J of Environmental Chemical Engineering, J of Renewable and Sustainable Energy, Powder Technology, Water Science and Technology

1

Biotechnology and Bioprocess Engineering, Canadian J of Chemical Engineering, Chemical Industry and Chemical Engineering Quarterly, Computer Aided Chemical Engineering, Frontiers in Energy, J of Chemical and Engineering Data, J of Chemical Technology and Biotechnology, Korean J of Chemical Engineering, Petroleum Science and Technology, Renewable and Sustainable Energy Reviews, Separation and Purification Technology

2

Chemical Engineering and Technology, Chemical Engineering Communications, Chemical Engineering J, Ecological Modelling, Indian Chemical Engineer, International J of Energy Research, J of Petroleum Science and Engineering

3

Chemical Engineering Science, Chemical Product and Process Modeling, International J of Chemical Reactor Engineering, J of Chemical Engineering of Japan, J of Industrial and Engineering Chemistry, Renewable Energy

4

Asia-Pacific J of Chemical Engineering, Chemical Engineering Research

and Design, Water Resources Research 5

Applied Thermal Engineering, Chemical Engineering and Processing: Process Intensification, J of the Taiwan Institute of Chemical Engineers

6

Computers and Chemical Engineering 8

Chinese J of Chemical Engineering, Fluid Phase Equilibria 9

Applied Energy 10

J of Natural Gas Science and Engineering 11

Industrial and Engineering Chemistry Research, International J of Hydrogen

Energy 12

Energy 14

Energy Conversion and Management 20

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Table 1.3 Recent applications of DE in process modeling or parameter estimation

Reaction Engineering: Jet stirred reactor combustor, double moving beds coupled

reactors, pyrolysis, industrial fluid catalytic cracking, 3-phase trickle bed reactor, batch and fed-batch fermentation, reactive system, ethylene oxide reactor, coal-water slurry gasifier, reaction network, CSTR, radial flow packed bed reactor, FT synthesis slurry reactor, reactive extraction, terephthalic acid production, ammonia synthesis.

Power Generation: Photovoltaic, power system, dispatch problem, wind power, solar

cell/power, SOFC and PEM fuel cells, reheat regenerative power cycle

Other Applications: Reservoir, cooling tower, Cr(VI) removal from wastewater, ionic

liquids, biochemical system, partition coefficient estimation, solubility parameter estimation, PID controller tuning, phase equilibrium and stability, VLE modeling, model discrimination, soft sensor, slurry flow

Common Objective Functions: Misfit, error, relative error, total relative error,

absolute error, mean squared error, sum of squared error, root mean square error, mean absolute percentage error, integral error, integral time weighted absolute error, fitting error, Gibbs free energy, deviation from set point, error-in-variable, standard deviation of error, cross correlation of error.

Table 1.4 Recent applications of DE in process design and optimization

Reactors: Fluidized and fixed bed reactors, methanol synthesis, CO2 conversion to

methanol, dimethyl ether production, (packed bed) membrane reactor, thermally coupled reactors, thermally coupled dual membrane reactor, copolymerization reactor, naphtha reformer, naphtha pyrolysis, ethylene cracking furnace, alkylation process, Williams-Otto process, ethylene oxide reactor, Fischer-Tropsch reactor, cyclohexane dehydration, auto-thermal ammonia reactor, hydrogen production, styrene reactor

Fermentation Processes: Fermentation, fed-batch fermentation,

fermentation-pervaporation and fermentation-extraction processes

Separation Processes: Bixin extraction, filtering hydrocyclone, intensified distillation

column, extractive distillation, middle vessel batch distillation, ethanol dehydration, solvent design for ethanol extractive fermentation, lactic acid production/recovery

Power Generation: Distributed generation, electric power network, solar plant with

storage, hydrothermal power system, supercritical CO2 Brayton cycles, renewable

distributed generation, coal fired power plant, PEM fuel cell, power cycle with waste heat and LNG (liquefied natural gas) cold energy, solar thermal refrigeration systems

Other Applications: Heat exchangers, HEN retrofitting, photovoltaic water pumping,

water distribution, scheduling, blending process, pressure vessel design

Common Objective Functions: Expected global cost, life cycle cost, total (annual)

cost, profit, operating cost, efficiency, fuel cost, utility cost, primary energy consumption, emissions, production rate, purity, effectiveness, yield, productivity, conversion, recovery, selectivity, fugacity, activity, outlet mole fractions

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1.7 A Simple Application of DE

A simple heat exchanger network (HEN), illustrated in Fig. 1.5, consists of one cold feed, one hot effluent stream and a heat exchanger (HE), a heater and a cooler. The cold feed stream enters at 30C (= TC,in) and it has to be heated to 125C (= TC,out) while the hot effluent stream enters at 115C (= TH,in) and it has to be cooled to 40C (= TH,out). The mass flow rate and heat capacity of the feed stream are respectively 16 kg/s and 4 kJ/(kg.K), and the mass flow rate and heat capacity of the effluent stream are respectively 20 kg/s and 3.8 kJ/(kg.K).

Fig. 1.5 Schematic of a heat exchanger network (HEN) for heat recovery and reuse number

HE in Fig. 1.5 helps to recover and reuse thermal energy, thus reducing both steam required in the heater and cooling water required in the cooler. This reduces the operating cost. However, inclusion of HE is likely to increase the required investment (also known as capital cost). Thus, there will be a trade-off between operating cost and investment, which is the case in many engineering applications. The design question is whether HE should be included and, if so, what should be the size of HE. This can be formulated as an optimization problem with one or two objectives, and DE can be used to solve it. This section describes the formulated optimization problem with two objectives, and then presents results obtained using IMODE program (described in Chapter 5), which is based on multi-objective DE and is implemented in MS Excel environment. Note that optimization problem formulation requires background and knowledge in the relevant discipline. For example, HEN

Heat Exchanger (HE) Heater (QH) Steam Feed T2 TC,out = 125oC TC,in = 30 o_C Cooler (QC) T1 TH,in = 115oC _T H,out = 40oC Cooling Water Effluent

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problem formulation (described below) is simple and easy to understand by chemical and mechanical engineers.

HE, cooler and heater in Fig. 1.5 are all assumed to be counter-current type, which is the most common. Temperatures of the effluent and feed streams after HE are denoted as T1 and T2 respectively. If there is no HE, then T1 = TH,in and T2 = TC,in. HEN under consideration is at the design stage, and so the area of the heat exchanger, heater and cooler (denoted by AHE, AH and AC respectively) can be varied for optimization. The governing (model) equations and the optimization problem for the HEN can be developed based on heat transfer principles and suitable cost correlations/data for the investment and operating cost. One of the governing equations is the energy balance, which, for HE, is given by:

16 × 4 × T − T _, = 20 × 3.8 × (T _, − T ) (1.3)

The above equation can be re-arranged to find T as follows:

T = T _, + × .

× × T , − T (1.4)

The two objective functions are:

Minimize Investment ($) = IH + IHE + IC (1.5a)

Minimize Operating Cost ($/year) = CSteam + CCW (1.5b)

Here, IH, IHE and IC are respectively the investment for the heater, HE and cooler. Each of them is given by:

Ii = 38000 + 520Ai0.9 _{for i = Heater, HE and Cooler (1.6)}

Here, Ai is the heat transfer area of equipment i, which is given by the following equations obtained from the heat transfer rate equation involving log mean temperature difference.

A = × ×( , )× , , ( ) (1.7a) A = × . ×( , )× , , , , (1.7b) A = × . ×( , )× , ( ) , (1.7c)

In the above equations, overall heat transfer coefficients for heater, HE

and cooler are assumed to be UH (= 0.78 kW/m2_{-K), UHE (= 0.50 kW/}

m2_{-K) and UC (= 0.50 kW/m}2_{-K), respectively. Further, saturated steam}

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(heating medium) enters the heater at 180o_{C and leaves as condensate at}

the same temperature, thus giving the latent heat of condensation for

heating purpose. Cooling water enters and exits the cooler at 30o_{C and}

40o_{C, respectively, due to heat transferred from the effluent.}

In the operating cost equation (1.5b), steam cost (CSteam) and cooling water cost (CCW) are given by:

CSteam = 400 × 16 × 4 × (T , − T ) (1.8a)

CCW = 25 × 20 × 3.8 × (T − T _, ) (1.8b)

Here, utility costs are taken to be 400 US$/(kW.y) for steam and 25 US$/(kW.y) for cooling water.

In equations 1.3 to 1.8, TH,in = 115o_{C, TH,out = 40}o_{C, TC,in = 30}o_{C and} TC,out = 125o_{C from the problem statement. Further, knowing T1, all other} unknown quantities in these equations can be calculated (i.e., T2 from equation 1.4, Ai from equation 1.7, investment from equation 1.6 and operating cost from equation 1.8). Hence, HEN optimization in Fig. 1.5 has only one decision (independent) variable, namely, T1. Lower bound for this variable is TH,out = 40o_{C, which still ensures a minimum approach} temperature of 5o_{C, and upper bound is 115}o_{C (i.e., TH,in in the absence} of HE). In addition, the calculated/dependent variable, T2 is also constrained between 30o_{C (in the absence of HE) and 110}o_{C (= 115 - 5 to}

ensure a minimum approach temperature of 5o_C).

In summary, the optimization problem is simultaneous minimization of two objectives in equation 1.5 with respect to T1 between 40 and

115o_{C subject to two inequality constraints to keep T2 between 30 and}

110o_{C. Note that equations 1.4, 1.6, 1.7 and 1.8 are essentially equality} constraints but they can be solved easily one by one. The latter approach is better since stochastic optimizers are not effective in handling equality constraints (Sharma and Rangaiah, 2013b).

HEN optimization problem is entered into the Objectives & Constraints worksheet of IMODE program (Fig. 1.6a), and it is linked with the Main Program Interface worksheet (Fig. 1.6b). Algorithm parameters are the default values along with population size of 40 and maximum number of generations of 100. HEN design problem is solved by clicking ‘Run IMODE’ icon. Optimal values of objectives (Invest-ment and Operating Cost), decision variable (T1) and two inequality constraints on T2 are plotted in Fig. 1.7. This figure also contains plots showing the variation of Operating Cost with area of heater, heat exchanger and cooler, respectively.

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Fig. 1.6a HEN optimization problem entered into the Objectives & Constraints worksheet of the IMODE program

Fig. 1.6b Main program interface of the IMODE program for the HEN optimization problem

A

Worksheet for Providing the Decision Variables, Functions and Constraints (Scroll down for instructions on calculation and linking of objectives and constraints.)

HEN example Decision Variable and its Bounds

Description Investment Operating Cost 290 k$ 30 110 k$/year 798.0 95.31 68.78 35.42 24.48 341.46 14.43 210.72 50891 137086 102170 760000 38000

Objectives & Constraints MOOSetup Intermediate Results Results at ChiTC Results at SSTC Results after MNG

Value

Lower Bound Upper Bound

T1 T2 LMTD of Heater A of Heater LMTD of HE A of HE LMTD of Cooler A of Cooler Invesment for Heater Investment for HE Investment for Cooler Cost of Steam Cost of CW Quantity 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 T2 > T2 < 60 40 115 Objective Constraints Calculated Values B C D E F G H I J K A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 B C D E F G

Main Program Interface

IMODE (Improved Multi-Objective Differential Evolution) for problems with continuous/integer variables and inequality constraints

Algorithm: Objective Functions: Value Cell Goal NumF Value Cell Minimum Maximum Type of Variables NumX Set Cell Compare Type Constraint Limit NumC NP Cr 88.06 40 115 Continuous T1 95.31 Greater 30 T₂ MNG F TLS TR TC MNG Rcrit δGD λ1 λ2 λ3 δSP 95.31 Lesser 110 T₂ 35.42 Greater 0 AHeater 341.46 Greater 0 AHE 210.72 Greater 0 A_Cooler 290.15 Minimize F1 798 Minimize F2 Add Objective Functions Add Decision Variables Add Constraints 1 5 40 0.5 100 0.5 Run IMODE

Objectives & Constraints MOOSetup Intermediate Results Results at ChiTC Results at SSTC Results after MNG 20 20 0.01 0.0003 0.1 SSTC Parameters 0.1 0.1 0.1 0.9 Design Variables: Inequality Constraints: Algorithm Parameters:

Generation Interval: Saving Intermediate Result

(This should be less than the maximum number of generations)

H I J K L M N

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As expected and as can be seen in Fig. 1.7, Investment and Operating Cost objectives are conflicting. More importantly, MOO provides quantitative trade-off between Investment and Operating Cost, and also many optimal solutions along with the optimal values of decision variable, constraints and dependent variables (namely, areas of heater, heat exchanger and cooler). All these give a deeper insight into the process on hand and for selecting one of the optimal solutions based on the preferences of the decision maker and other considerations.

Before using MOO results, it is desirable to analyze and explain qualitatively the trends of objectives, decision variables and other quantities, in order to ensure their validity. For example, increasing T1 leads to increased energy recovery in the heat exchanger, thus resulting in the decreased requirement of steam (in the heater) and cooling water (in the cooler), which reduces Operating Cost. In terms of equipment size, heat exchanger area increases, and areas of the heater and cooler decrease as T1 increases. These variations are consistent with the expectations based on the knowledge in the heat transfer field. The overall outcome of area changes is the increased investment cost.

Application described in this section, although realistic, is relatively simple. More complex and realistic applications are described and discussed in many chapters of this book. For example, IMODE program and its application are presented in Chapter 5, retrofitting of large HENs is covered in Chapter 6, and Chapter 10 describes optimization of bioethanol separation by the hybrid process of distillation and vapor permeation. The next section outlines scope and contributions of all chapters in this book.

1.8 Scope and Outline of Chapters

After this Introduction chapter in Part I Overview, the subsequent chapters in this book are organized in Parts II and III. Chapters 2 to 5 form Part II on DE developments, and Chapters 6 to 13 in Part III cover many chemical engineering applications of DE. An outline of these chapters is presented in this section. Many of the chapters contain exercises at the end for practice by the interested readers. Moreover, these exercises can also be adapted as projects for students in optimization courses.

In Chapter 2, Chen et al. describe DE, its parameters and their values. They summarize the proposed modifications to various components of

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Fig. 1.7 Pareto-optimal front of Operating Cost and Investment in the top left plot, and the corresponding decision variable and dependent variables in other plots

3000 2500 2000 1500 1000 Min. Oper

ating Cost (k$/y

ear)

500 0

0 200 400 600 800 1000 1200 100 150 200 250 300 350 400 Area of Heat Exchanger (m2) Area of Cooler (m2)

3000 2500 2000 1500 1000 Min. Oper

ating Cost (k$/y

ear) 500 0 30 40 50 60 70 80 90 100 110 10 20 Area of Heater (m2₎ 30 40 50 60 70 80 90 T₂ (ºC) 3000 2500 2000 1500 Min. Oper

ating Cost (K$/y

ear) 1000 500 0 230 270 310 350 390 Min. Investment (k$) T1 (ºC) 430 40 55 70 85 100 115

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DE; these modifications have improved DE capabilities for tackling a variety of optimization problems. Further, Chen et al. provide an overview of chemical engineering applications of DE reported until the year 2009. In particular, DE has found many applications for parameter estimation and modeling in addition to process design and operation. Recently, Dragoi and Curteanu (2016) reviewed DE applications for solving chemical engineering problems. Given these developments and applications, DE is attractive and useful both as a simple general optimizer and as a sophisticated tool to solve complex chemical engineering applications.

Chapter 3 by Rahimpour and Hamedi provides an overview on DE applications in chemical reaction engineering. DE is a simple and robust technique suitable for optimization of nonlinear and complicated models resulting from mass and energy balances combined with intricate thermodynamic and other auxiliary equations. Considerable effort has been made to optimize reactor operating conditions using DE. In Chapter 3, DE algorithm is first explained, and then different aspects of reactor and kinetic modeling are introduced. Afterwards, key decision variables and objective functions are discussed. Based on previous publications, the decision variable section is divided into temperature, pressure, flow rate, membrane thickness, reactor size, feed concentration and kinetic parameters. Chapter 3 summarizes the main aspects of using DE in chemical reaction engineering, thus providing an overview for future researchers.

Srinivas and Rangaiah describe two versions of DE with a Tabu List (DETL) in Chapter 4. One version incorporates the concept of avoiding revisits during the search, using tabu check in the generation step of DE. Another version implements the same in the evaluation step of DE. These versions are evaluated on benchmark and phase stability problems. Benchmark problems consist of 2 to 20 decision variables and a few to hundreds of local minima whereas phase stability problems involve multiple components and comparable minima. Further, a new benchmark problem with characteristics similar to phase stability problems is proposed and used. The results show that the performance of the two versions of DETL is comparable, and it is better than DE in number of function evaluations and better than TS in reliability.

Chapter 5 by Sharma et al. presents the Integrated Multi-Objective Differential Evolution (IMODE) program in MS Excel, useful for solving MOO problems. The algorithm in this program has four main

Differential Evolution in Chemical Engineering, Developments and Applications (2017)

DIFFERENTIAL EVOLUTION

IN CHEMICAL ENGINEERING

World Scientific

editors

Gade Pandu Rangaiah

National University of Singapore

Shivom Sharma

École Polytechnique Fédérale de Lausanne, Switzerland

DIFFERENTIAL EVOLUTION

IN CHEMICAL ENGINEERING

Developments and Applications

Preface

About the Editors

List of Contributors

Contents

Supplementary Materials

Chapter 1

Introduction