A Hybrid EvolutionaryâDeterministic Optimization Approach for

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A hybrid evolutionary-deterministic optimization approach for conceptual design Mirko Skiborowski, Marcel Rautenberg, and Wolfgang Marquardt Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b01995 • Publication Date (Web): 02 Oct 2015 Downloaded from http://pubs.acs.org on October 6, 2015

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Industrial & Engineering Chemistry Research

A hybrid evolutionary-deterministic optimization approach for conceptual design Mirko Skiborowski,∗,†,‡ Marcel Rautenberg,†,¶ and Wolfgang Marquardt†,§ AVT - Process System Engineering, RWTH Aachen University, 52064 Aachen, Germany E-mail: [email protected]

Abstract Most optimization-based approaches in conceptual process design either focus on global optimization using simplified process models or utilize some kind of metaheuristic to optimize by means of repetitive runs of a detailed simulation model. Since even the design of a single distillation column model results in a non-convex large-scale and mixed-integer optimization problem, if rigorous thermodynamic models are applied, deterministic optimization is still mostly limited to local optimization. In order to investigate and improve the solution quality of previously developed efficient local optimization approaches, this paper proposes a hybrid evolutionary-deterministic optimization approach. The resulting memetic algorithm does not only allow to optimize the initial process structure, but also facilitates discrete decision making that severely complicates a deterministic optimization due to the resulting discontinuities. The proposed approach does not only ease the application by reducing the necessary user input for the initialization, but also strengthens the confidence in the quality of ∗

To whom correspondence should be addressed AVT.PT, RWTH Aachen University ‡ Current address: Department of Biochemical and Chemical Engineering, Laboratory of Fluid Separations, TU Dortmund University, 44227 Dortmund, Germany ¶ Current address: BASF SE, GTE/AB - O925, 67056 Ludwigshafen, Germany § Current address: Forschungszentrum Juelich GmbH, 52425 Juelich, Germany †

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the results, since it provides an extensive screening of the design space. Several case studies, including utility and entrainer selection, demonstrate the performance of the hybrid optimization approach and suggest that even more complex design problems can be solved efficiently.

1

Introduction

Chemical engineering problems, which aim at an accurate description of process performance, based on rigorous thermodynamics, reaction kinetics or mass and heat transfer modeling, result in highly nonlinear and non-convex large-scale problems. The computational effort for the solution of these problems increases exponentially with the number of discrete variables and deterministic optimization methods are mostly limited to local optimization 1 . The performance of local solution procedures depends strongly on the initialization, which cannot only result in local optima of low quality, but may even impede the determination of a feasible solution. While major progress has been made in recent years in the field of global optimization, the application of global optimization methods is still limited in terms of problem size and complexity 2 . Local optimization approaches, based on sophisticated initialization strategies, problem decomposition and incremental model refinement at least facilitate an efficient computation of local optimal solutions. However, the required initial design specifications leave some degree of uncertainty concerning the quality of the local optimal solution. The requirement of a smooth model formulation also severely complicates the gradient-based optimization of discrete decisions, such as the choice of utilities or of auxiliary components. In order to overcome these limitations, a novel hybrid optimization approach is presented, which integrates the efficient local optimization approaches presented in previous publications 3,4 . These and other deterministic approaches, as well as metaheuristics and hybrid optimization approaches are briefly reviewed in Section 2 in the context of chemical engineering problems. Based on an analysis of the limitations of different approaches, the developed 2 ACS Paragon Plus Environment

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novel hybrid approach is introduced in Section 3. Its application is further demonstrated for three different case studies in Section 4, illustrating the possible benefits from the application of the hybrid approach and verifying the good performance of the local optimization. Finally, Section 5 presents conclusions and suggestions for further application and extension. The content of the current article is based on the PhD-thesis of Skiborowski 5 .

2

Optimization-based conceptual design methods

As already mentioned in the introduction, the consideration of rigorous thermodynamics, reaction kinetics or mass and heat transfer modeling results in non-convex, large-scale optimization problems with several degrees of freedom. While some of them, like pressure or heat duties represent continuous values, others, like the number of trays in a distillation column or the choice of utilities represent discrete decisions. The latter can be introduced as disjunctions, resulting in the formulation of the mathematical optimization in form of a generalized disjunctive programming (GDP) problem, which can either be solved by a specialized solution approach, as e.g. the logic-based outer-approximation algorithm introduced by T¨ urkay and Grossmann 6 , or the GDP can be reformulated into a mixed-integer nonlinear programming (MINLP) problem 7 . Independent of the specific model type, an efficient solution of such discrete optimization problems generally requires sophisticated model formulations 8 and good initial values and bounds to converge 9,10 . The major obstacles of the optimization are the solution of the nonlinear process model and the non-convexities, which impede the determination of the global optimum, without the utilization of convex reformulation and interval techniques or scattering initial values through the complete search space. Optimization approaches for these kind of problems can be classified as either deterministic gradient-based optimization approaches, so-called metaheuristics, which are generally non-deterministic, and the combination of both types of approaches in form of hybrid optimization approaches.



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2.1

Gradient-based optimization

Gradient-based optimization approaches require a smooth formulation of the mathematical optimization problem, but the final solution of the optimization represents a proven (at least locally) optimal solution. These approaches are generally considered computationally much more efficient in comparison to metaheuristics . Detailed descriptions of the different deterministic optimization methods in the context of chemical engineering are provided in the textbooks of Edgar et al. 11 or Kallrath 1 , while up-to-date reviews are presented in the articles of Trespalacios and Grossmann 12 and Biegler 13 . For the optimization-based design of distillation-based separation processes, including rigorous thermodynamics and equilibrium-tray or rate-based models for process evaluation, so far only local deterministic optimization approaches have been applied. Those approaches that are capable of solving the constraint nonlinear problems rely on sophisticated initialization strategies based on problem decomposition and incremental model refinement 14–19 . In comparison to MINLP approaches, a direct logic-based solution of a GDP problem offers the advantage that only reduced NLP subproblems have to be solved, which contain the complex thermodynamic models only for disjunctions that are true. Despite the benefits of such a logic-based solution of a GDP problem formulation, which have been illustrated for distillation process design by Yeomans and Grossmann 20 , there is yet no logic-based solver readily available in optimization modeling software systems like GAMS 21 or AIMMS 22 . Kraemer et al. 18 and Caballero 10 present elaborate reviews of the history of rigorous tray-by-tray optimization models and optimization approaches. The novel hybrid evolutionary-deterministic optimization approach presented in this article builds on recently introduced approaches that decouple the complex property and equilibrium calculations and which have demonstrated computationally robust and efficient means for the optimization of homogeneous and heteroazeotropic distillation as well as membrane-assisted distillation processes 3,4 . Since these approaches only provide locally optimal solutions there remains some degree of uncertainty concerning the quality of the determined solution, which may depend on the provided ini4 ACS Paragon Plus Environment

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tialization. The calculation of the proven global optimum is computationally much more demanding and the complexity increases exponentially with the size of the problem and the degree of nonlinearity of the mathematical problem 23 . Although global optimization approaches have been considerably improved in the last decades 2,24 the solution to complex nonlinear optimization problems, such as separation processes for azeotropic mixtures, remains a challenging task 12,25 .

2.2

Metaheuristics

Metaheuristics are general algorithmic frameworks, which are designed to solve complex optimization problems 26 . Since all metaheuristics constitute derivative-free algorithms, they can be applied to discontinuous and multi-modal optimization problems. While different metaheuristics have been used for the optimization of complex chemical processes, like, e.g., simulated annealing 27 , tabu search 28 or particle swarm optimization 29 , the most prominent metaheuristics are evolutionary algorithms (EA). EAs can be classified into genetic algorithms (GA), genetic programming, evolutionary programming, differential evolution and evolutionary strategies (ES) 30 . Although they differ in the exact formulation and interpretation, they are all inspired by biological evolution and try to determine the optimal solution by constantly creating new generations based on a recombination of a few selected individuals with the best fitness and a mutation of the resulting offspring. However, optimization with an EA is ambiguous, since in addition to the optimization problem, also the parameters of the EA need to be tuned for the given problem 31 . Rios and Sahinidis 32 provide a review of several derivative-free algorithms and a comparison of available software implementations. They conclude that the determination of the best solution is challenging even for small problems for most current derivative-free solvers. For strongly nonlinear constrained optimization problems the direct application of metaheuristics becomes infeasible 33 . Thus, in case of complex chemical engineering problems, a 5 ACS Paragon Plus Environment


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simulator is integrated as a black box model into the evaluation step of the metaheuristic, which then operates only on the design degrees of freedom (DDoF). Such an approach, which was first proposed by Gross and Roosen 34 , has several advantages, but also problems that need to be considered. Usually the simulation software offers a multitude of process models, such that no tedious implementation is necessary 35 . In addition to the solution of the simulation model, the simulator may also facilitate the integration of additional inequality constraints, e.g., purity specifications, by means of design specifications or other inherent capabilities. However, in case of complex process models, not all constraints can be handled by the simulator and constraint violations need to be considered by means of the metaheuristic 29,35 . This is especially important, since failure in the simulator convergence can result in a failure of the optimization 10 . The most prominent approach in EA and other population-based methods is the use of penalty functions, which deteriorate the fitness of infeasible individuals 30 . Despite, the determination of an infeasible design by means of the simulator can become the bottleneck of stochastic optimization 36 . From the perspective of the optimizer, a lack of convergence of the simulator and infeasible design specifications are indistinguishable 37 . While some authors propose to assume that a non-convergent simulation corresponds to an infeasible solution 28,38 , this may limit the optimizer to sub-optimal solutions due to lacking convergence of the simulator. Consequently, a high convergence ratio, i.e., the ratio of converged solutions for the evaluation of a set of feasible DDoF, is an inevitable prerequisite for a meaningful optimization 37 . According to Henrich et al. 39 , only a problem-specific tailored EA with specifically adapted parameter settings can be used as sole optimization technique for complex chemical processes. Their approach utilizes problem-specific adapted evolutionary operations, a simplified process model for an initialization in a pre-simulation step and an analysis and penalizing strategy in a post-simulation step. Silva and Salcedo 37 present a multi-step procedure for the initialization of reaction rates, composition and temperature profiles for the optimization


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of a reactive distillation column model in the simulation software. In conclusion, the optimization of complex engineering design problems by means of metaheuristics is not straight forward. It requires a tailored formulation, a careful selection of the parameters of the metaheuristics as well as the implementation of an initialization strategy in order to successfully solve the complex simulation problems. High computational effort, i.e., multiple hours up to several days, as reported in recent publications (e.g. VázquezOjeda et al. 40 , Koch et al. 41 , Domingues et al. 42 ), further impede the integration into the engineering workflow, in which recalculations due to new information are required frequently.

2.3

Hybrid optimization algorithms

Hybrid optimization algorithms, representing a combination of metaheuristics and deterministic local optimization, can considerably improve the performance of the optimization in terms of solution quality and computational time 36 . Various combinations have been proposed in literature for different kinds of optimization problems. As summarized by Kallrath 1 , the development of hybrid optimization algorithms is not subject to specific rules, but rather a form of art with the target to determine high quality solutions as fast as possible. The most simple hybrid algorithms are based on a sequential application of the different algorithms (e.g. Munawar and Gudi 43 , Staudt and Soares 44 ). In a first exploration phase, a metaheuristic is used to explore the vast design space and determine the neighborhood in which a very good solution is located. In the subsequent exploitation phase, an efficient deterministic optimization algorithm is used for local optimization. Alternatively, the local optimization approach can be applied to all feasible solutions determined by the metaheuristic 45 . However, in such sequential approaches both algorithms solve exactly the same problem and there is no integration between them. The combination of two different optimization algorithms in a nested optimization with an inner and outer loop presents a more sophisticated form of a hybrid algorithm. Athier et al. 46 first proposed such an optimization approach for heat exchanger network (HEN) synthesis. 7 ACS Paragon Plus Environment


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In the outer loop, a metaheuristic is used to determine the discrete decisions regarding the HEN structure, while a deterministic algorithm is used to solve the continuous DDoF for the exchanged heat duties in the inner loop. Multiple studies have proven that especially a hybridization of an EA with other optimization techniques can greatly improve the efficiency of the optimization 47 . The concept of such hybrid algorithms was termed memetic algorithm (MA) by Moscato 31 . A MA performs a local optimization of each individual and can be interpreted according to the evolutionary theory of Lamarck 30 . The embedded local optimization represents a lifetime learning procedure 48 . While EAs try to emulate biological evolution, MAs try to mimic cultural evolution, which depends much less on the aspect of mutation and is focused on a directed transfer of problem specific knowledge 31 . MAs have become the preferred methodology for many real-world applications 47 and have also been considered for some complex engineering problems. These include process design of reactive distillation columns 33,49 in combination with an additional side-stream reactor 50 , the design of distillation-based separation processes 51 and early stage synthesis of complete processes under uncertainties 52 While these publications demonstrate that MAs can facilitate the optimization of complex chemical engineering problems, the local optimization approach is generally restricted to the optimization of continuous DDoF. However, considering also discrete DDoF in the local optimization could further increase the efficiency. In addition, neither of these publications provides information on specific initialization strategies. As for the combination of an EA and a simulator, the EA cannot distinguish between non-converged and infeasible designs. However, the local optimizer may also report an infeasible solution due to an insufficient initialization. Consequently, the solution of MINLP problems in the local optimization step and high convergence ratios are desirable features, which are obtained by the novel hybrid optimization approach presented in the following section.


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3

Novel hybrid optimization approach

The novel hybrid optimization approach is based on the combination of an EA and the efficient local optimization approaches presented in previous publications 3,4,18 . Similar to the presented MAs, the deterministic local optimization is used to evaluate the fitness of the individuals of each generation within the EA. However, the interpretation of the DDoF of a specific design problem and their assignment to the stochastic and deterministic solver differ significantly. In contrast to previous approaches, the evaluation of the fitness of an individual is not restricted to the optimization of the operating point for a given process structure, but also covers DDoF represented by integer decisions to simultaneously decide on the process structure. Hence, a MINLP is solved in every evaluation step of the EA. The EA only fixes those structural design decisions that would result in model discontinuities, which cannot be handled efficiently by a continuous reformulation in a deterministic optimization. All other integer DDoF are handled by deterministic optimization and are only initialized by the EA. Consequently the number of trays and the location of feed and recycle trays are handled by the deterministic optimization, making use of the SR-MINLP approach introduced by Kraemer et al. 18 (cf. Section 3.2). Decisions on the selection of utilities, which introduce a discontinuity in the cost function, temperature difference calculation and pressure limits, or the choice of an entrainer, which affects the equilibrium model on each tray, as well as enthalpy calculations, are determined directly by the EA and fixed in the local deterministic optimization. Therefore, the hybrid approach utilizes the full potential of deterministic (mixed-integer) local optimization. The multistage solution procedure for the local optimization problem also results in an improved convergence ratio. The general solution strategy is illustrated in Figure 1. As for the classical EA, the hybrid algorithm begins with a randomly initialized population, which is however used as a first generation for the hybrid approach. The initialization is further elaborated in Section 3.1. As characteristic for a memetic algorithm, the fitness of the individuals of each generation is determined by means of a local optimization. In this step, the hybrid approach benefits from 9 ACS Paragon Plus Environment


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the computational robustness of the local optimization approach, which is further described in Section 3.2. After evaluation, the hybrid approach performs the sequence of three characteristic operations of an EA, which are selection, as well as recombination and mutation, in order to create the next generation. During selection, a group of individuals with the highest fitness is elected as parents for the next generation (cf. Section 3.3). The vector of DDoF of each new individual is constructed by a first recombination of the parent vectors and a subsequent random mutation of the resulting vector (cf. Section 3.4). In addition, tabu zones are constructed for all evaluated individuals. In order to account for the local optimization in the evaluation step, new individuals for the next generation are only accepted, if their DDoF vector is not covered by tabu zones of preceding individuals. The new generation is then processed in a recursive manner by the same operators and new generations are constructed and evaluated until some termination criterion is met (cf. Section 3.6).

Figure 1: Schematic of the hybrid optimization approach. In order to account for different types of variables, which represent the DDoF of a specific design problem, in the different operators, the variables are classified into ordinal integer, nominal integer and continuous variables analogous to Emmerich et al. 53 . While ordinal integer variables are characterized by a clear order of the potential values, there exists no 10 ACS Paragon Plus Environment

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characteristic order in case of nominal integer values. Several examples for each of the classifications and the differences concerning the recombination and mutation operators are listed in Table 1. Table 1: Characterization of the different DDoF classifications. ordinal integer

nominal integer

continuous

examples

max. number of trays utility initial intermediate initial feed tray entrainer product specification recombination discrete selection discrete selection intermediate (value of one parent) (value of one parent) (mean value) mutation geometrically distributed random change normally distributed random change in range random change

The different parameters that need to be specified for the EA are listed in Table 2. The different phases of the EA are further described in the following subsections, including the adaption of some of the parameters in Section 3.5. Table 2: Summary of the different parameters for the EA. parameter general λinit λ µ κ DDoF specific up dlo i , di dtabu i cj (d) mutation pi si termination ∆fmin ng,max nf,max tmax

description number of individuals in the first generation number of individuals per generation number of parents per generation maximum age (generations) of parents considered during selection lower and upper bound for DDoF i tabu zone diameter for DDoF i constraints containing one or more DDoFs mutation probability of DDoF i mutation scale (inverse of expected value) for DDoF i minimum improvement in fitness function maximum number of generation without significant improvement maximum number of failed attempts for generation of new individuals maximum time limit



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3.1

Initialization

The population of the first generation with λinit individuals is generated randomly. However, it has to meet a number of requirements that are characterized by means of lower up and upper bounds on the DDoF (dlo i , di ), tabu zones and additional problem specific con-

straints. While the variable bounds restrict the search space for the EA, tabu zones, which are quantified by means of a range dtabu around each DDoF, prohibit that individuals with i DDoF vectors in a close vicinity to each other are generated. The tabu zones account for the improvement in the local optimization and prohibit the EA from getting stuck in a local refinement search. The application of similar concept of tabu zones was already proposed by Urselmann and Engell 54 in the context of a memetic algorithm. The application of the tabu zones has the advantage that the search space for the EA is reduced and the distribution of the initial solution is promoted. The evaluation of the tabu zones can also be used as a termination criterion (cf. 3.6). In addition, any kind of algebraic constraint cj (d) can be integrated in order to prohibit infeasible specifications. A typical example is the restriction of the feed location in a distillation column below the maximum number of trays. Additional restrictions can be considered in order to improve the diversity of the initial population. A uniform distribution of nominal integer DDoF can be enforced in order to guarantee that each possible value occurs equally often in the initial population.

Figure 2: Different steps for the initialization of an individual of the initial population.


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The different steps for the generation of an individual of the initial population are illustrated in Figure 2. At first the nominal integer DDoF are set, either randomly or uniformly distributed. The remaining DDoF are divided in high and low priority DDoF which are set accordingly. The prioritization eliminates a possible bias from the constraints. If all DDoF were treated equally, the restriction of feed locations below the maximum number of trays would induce a strong bias towards higher columns. By specifying the maximum number of stages as high and the initial feed locations as low priority DDoF, the latter are first reevaluated several times, before the maximum number of trays is reevaluated. Alternatively, initial feed locations could be specified relative to the maximum number of trays, which would however require the specification of a continuous value. Once all DDoF are specified and the constraints are satisfied, the DDoF vector is checked against the tabu zones of the previously generated individuals. If the DDoF vector is not covered by the tabu zones of previous individuals, the new individual is generated, otherwise the algorithm starts again with setting the high priority DDoF in order to recreate the indiviudal. While the application of constraints presents a straightforward approach for the reduction of infeasible combinations of DDoF, alternative approaches are possible. The application of repair routines, as e.g. used in the memetic algorithm of Urselmann et al. 49 , represents one alternative. However, repair routines are also problem specific and need to be tailored accordingly 50 .

3.2

Evaluation

In the memetic algorithm of Urselmann et al. 49 , the genome constitutes all relevant DDoF. For a simple distillation column the genome represents the number of trays, the feed tray location and the heat duties. For the evaluation, these DDoF are at first fixed and a simulation of the individual is performed. After this simulation a local optimization is started, for which all continuous DDoF are optimized in order to improve the specified objective function. Finally, the individuals in the current population are replaced by the corresponding 13 ACS Paragon Plus Environment


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local optima. Consequently the local optimization refines the individuals, which are generated by the EA. While the current approach also utilizes a local optimization approach for the evaluation of the individuals no such update is preformed. The results from the local optimization are rather stored in a separate vector, while the resulting objective value is used to represent the fitness of the individual. The difference between the approaches can be explained by the differences in the genome of individuals and the initialization strategies. In contrast to the approach of Urselmann et al. 49 , the EA in the current approach optimizes only the initial values (e.g. nominal integer DDoF) or determines discrete decisions (e.g. nominal integer DDoF), which are fixed in the local optimization. For a simple distillation column, the EA only optimizes the initial structure, such that the genome represents only the maximum number of trays and the initial feed location. Afterwards a sophisticated deterministic solution strategy is utilized for the local optimization, which has been described in detail for distillation processes by Skiborowski et al. 4 . For the sake of comprehensibility a short generic version for distillation processes is represented in Figure 3 and will be described in the following. In the first step of the initialization, the compositions and temperatures for each column tray are either initialized based on feed flash calculation, or linearized compositions profiles, e.g. determined from previous shortcut calculations. Column designs with a maximum number of equilibrium trays are further determined by first solving only mass balances, equilibrium calculations and summation constraints (MES), before integrating the enthalpy balances (MESH). As a first optimization step the optimal operating point of the process with a fixed structure is determined by minimization of the energy duties w.r.t. given purity constraints. Finally, a cost-optimal process design is determined by means of a simultaneous optimization of the process structure and the operating point. Therefore, instead of solving the MINLP problems directly, the problem is solved as a series of successively refined problems, which each are solved as continuous nonlinear programming (NLP) problems. While the discrete decisions on process structure


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Figure 3: Solution strategy for the efficient local optimization of distillation processes. are fixed in the first steps of the solutions strategy, they are completely relaxed in the final step, which is solved as a series of relaxed MINLP (SR-MINLP) problems by means of a NLP solver, following the strategy proposed by Kraemer et al. 18 . So-called NCP-functions (NCP = Nonlinear Complementary Problem) are introduced as additional nonlinear constraints in order to finally obtain a discrete solution to the relaxed binary variables 55 . Refer to Kraemer et al. 18 as well as Skiborowski et al. 4 for further details. Therefore, in contrast to the MA of Urselmann et al. 49 , for which the EA operates on the full set of DDoF, the EA of the current approach only operates on a reduced subset of the DDoF. For distillation processes these are only the initial values for the number of trays and the location of feed and recycle trays, which are further subject to optimization in the local optimization approach, as well as additional nominal integer values, like utilities and materials. For a simple distillation column only two integer DDoF for initial values



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[Ntrays,max , Nf eed,init ], instead of the full set of DDoF [Ntrays , Nf eed , QB , QC ] contribute to the search space of the EA. Since the number of trays and the feed location (Ntrays , Nf eed ) are also degrees of freedom in the local optimization approach, tabu zones can efficiently be used (see Section 3.1 and 3.4) to decrease the search space of the EA. The continuous DDoF, like e.g. the heat duties, are only handled in the local optimization, where they are subject to optimization in every single step of the solution strategy. Consequently, another difference to the MA of Urselmann et al. 49 is that no single simulation (all DDoF are fixed) is performed in the current hybrid optimization approach. While the local optimization approach can in general be considered computationally robust, experience has shown that there are exceptional cases, in which the solver requires an unusual amount of time for the solution of a subproblem or finally terminates due to resource or iteration limits. In those cases it can be expected that the final solution is suboptimal or the local solver terminates with an infeasible solution. In order to avoid a large delay in the evaluation of a generation, time limits can be defined for the evaluation of each individual, after which the local optimization is terminated and the individual is considered infeasible. These time limits should be carefully selected based on the evaluation of a first trial generation. All infeasible solutions are neglected in the subsequent selection.

3.3

Selection

The selection of parents for the next generation is based on the (µ, λ, κ)-selection, as proposed for the (µ, λ, κ, ρ)-ES by Schwefel and Rudolph 56 . The best µ individuals are selected from the λ individuals of the current population and previous parents whose age has not exceeded κ generations. The maximum age κ allows to scale between the (µ, λ) strategy, where no parents are considered in the selection, and the (µ + λ) strategy, where the parents of all former generations are considered. This makes it possible to set a trade-off between losing good solutions and premature convergence. The (µ, λ, κ)-selection was also used in other hybrid algorithms that employ an EA 39,49,53 . 16 ACS Paragon Plus Environment

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3.4

Recombination and Mutation

Recombination and mutation strategies in EA follow in principle two different approaches. While mutation is the primary driving force in ES, GA are primarily based on the principle of recombination and mutation could even be neglected completely 57 . Most of the existing hybrid algorithms focus on the approach of ES. Since the approach of GA, which predominantly operates on binary genomes, is not well suited for the application to different kinds of continuous and discrete variables, the current approach also focusses on the ES strategies. In ES two parents are randomly selected from the pool of parents for each new individual. Based on the genomes of the parents the genome of the child is constructed by means of intermediate or discrete recombination. In case of intermediate recombination the value of the DDoF of the child is determined as the intermediate value of both parents. In case of discrete recombination the value of the DDoF of the child reflects one of the values of two parents. The genome of the child is further altered by means of mutation. Therefore, with a certain probability either a random variable is added to each degree of freedom or the actual value is overwritten with a random value out of the range of possible values.

Figure 4: Different steps for the creation of an individual for a new generation. Which recombination and mutation operator is applied depends on the type of the DDoF. While discrete recombination is applied for ordinal and nominal integers, intermediate recombination is applied for continuous variables. The mutation operators for the different DDoF correspond to the description of Emmerich et al. 53 . For ordinal integers a geometrically dis17 ACS Paragon Plus Environment


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tributed random variable with mutation scale si (inverse of expected value) and random sign up is added to the current value. In case this violates the upper or lower bound (dlo i , di ) the

value is set to the corresponding limit. For nominal integers the current value is replaced by a randomly selected feasible value. For continuous variables a normally distributed random variable with mutation scale si and random sign is added to the current value. In case the limits are violated, a reflection is performed, i.e., the amount of the violation is subtracted from the upper or added to the lower bound to yield the new value. Each of these mutations is performed with a respective probability pi , that has to be defined in advance. In order to account for constraints and tabu zones a similar strategy as introduced for the initial population is applied (cf. Section 3.1). The creation of a new generation is summarized in Figure 4. Note, although recombination and mutation operators for continuous variables have been implemented, there are no continuous variables handled by the EA in the case studies in the current article. They are however mentioned here, in order to illustrate how they can be applied, if e.g. intermediate product compositions for larger flowsheets are to be optimized by the EA.

3.5

Parameter adaption

In order to perform a transition from an early exploration to a later exploitation phase of the EA, the mutation parameters can be adjusted after the evaluation of each generation. Therefore, larger mutation scales or probabilities are applied at first, in order to cover a high ratio of the search space, while the mutation parameters are further tightened in subsequent generations, such that the most promising regions are exploited thoroughly. This is accomplished by providing an array of values for pi and si . For each generation the value in the according position is used, while the final value is also used for all subsequent generations. As an alternative to such a deterministic parameter-control, so-called adaptive or selfadaptive parameter-control approaches, where the mutation strengths for all DDoF are themselves variables that are subject to mutation and recombination could be applied 58 . 18 ACS Paragon Plus Environment

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While these adaption strategies showed superior performance in several model applications, the definition of efficient adaption strategies remains a difficult task 30 , which is an active research field in MAs 59 . Self-adaption is e.g. applied in the memetic algorithm of Urselmann et al. 49 , while it is disregarded in the hybrid algorithm of Henrich et al. 39 , who state that the number of evaluations necessary to adapt the parameters is too high. Since the number of individuals evaluated in the hybrid algorithm presented in this work is also small in comparison to classical EA, self-adaption is not considered.

3.6

Termination

Deterministic global optimization methods, like spatial branch-and-bound methods, rely on the computation of a convex relaxation of the original problem, in order to quantify the gap between the best current solution of the original problem and the best solution of the convex relaxation. The algorithm terminates if the gap is reduced to a pre-defined value, quantifying an acceptable distance to the global optimum. Such a precise criteria cannot be derived in case of a metaheuristic. The common criteria for termination is based on a stagnation in the evolution of the fitness of the best individual over a period of several generations. In that case it is expected that there will be no further improvement. Besides a maximum number of generations ng,max without a significant improvement ∆fmin , two additional termination criteria are considered. The first is a simple time limit tmax , which can be set for a first analysis of the performance, or due to time restrictions for the optimization. The second is based on the creation of new individuals and the use of the tabu zones. As illustrated in Figure 4, the genome of a potential individual is only accepted, if it is not covered by tabu zones of previously evaluated individuals. In case it is covered by such a tabu zone the individual is recreated by repeating the process of selection and recombination and mutation. A maximum number of such recreations of a single new individual nf,max is considered as third termination criteria, as it correlates with the coverage of the search space and avoids a continuous loop in case of a full coverage. 19 ACS Paragon Plus Environment


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3.7

Parallelization

The parallelization of the evaluation step is an important aspect, since the local optimization represents by far the computationally most expensive part of the hybrid algorithm. However, parallelization of EA is comparably simple, since all individuals of the same generation can be evaluated in parallel. However, not all individuals will require the same computational time for their evaluation. In order to account for the different evaluation times a flexible parallelization scheme by means of multi-tasking was implemented. Following a masterslave architecture implemented in Matlab, the EA acts as the master, starting the local optimization of single individuals by means of several GAMS entities, which act as slaves (cf. Figure 1). The approach operates dynamically, since the EA keeps track of the single evaluations and starts the evaluation of new individuals whenever a previous run terminates. This implementation does not require exclusive access to CPU cores and can be applied on any stand-alone computer with multiple cores. The parallelization results in a significant reduction of computation time compared to a sequential evaluation of the single individuals (cf. Figure 7).

4

Case studies

To illustrate the suggested methodology, this section presents different case studies of various complexity. For each of the case studies the fitness (f ) of the individuals is represetned by the total annualized costs (TAC). First, the optimization of a single distillation column is investigated in order to analyze the solution quality of the local optimization approach and the potential improvement of providing different initial structures. Afterwards the optimization of a more complex pressure swing-distillation process is investigated with and without the use of an additional entrainer. Besides the initial structure, the utilities are optimized by the EA, who specifies upper and lower bounds for the pressure in the high pressure column. Finally, the optimization of an extractive distillation process is investigated, for which the


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EA determines the optimal solvent from a set of 6 candidates, setting up the initial structure and the VLE model for the local optimization according to the selected solvent. Property and economic models are described in the Supplementary material, together with further R information on optimized designs. All calculations were performed on a PC with two Intel⃝ TM

Core

4.1

Xeon E5 2630 v2 hexa-core processors with 2.6 GHz, utilizing GAMS 23.1 and 23.6.

Single distillation column

As a first example the optimization of a single distillation column for the separation of acetone from a ternary mixture of acetone, IPA and water is investigated. The specifications are listed in Table 3. Table 3: Specifications for acetone, IPA and water separation.

feed distillate bottom

flowrate [ mol ] s

acetone [ mol ] mol

20.40

0.443 ≥ 0.995 ≤ 0.001

IPA [ mol ] water [ mol ] mol mol 0.053

0.504

The separation corresponds to the first column of an extractive distillation process that was investigated by means of a local optimization approach by Skiborowski et al. 60 . It is particulary complex, as the lower bound for the distillate purity necessitates a large minimum energy duty related to a tangent pinch formation close to the acetone-water edge in the rectifying section (cf. Figure 6). For the optimization of the single column, the EA only provides an initial and maximum value for the number of equilibrium trays Ntrays,max and the position of the feed tray Nf eed,init . Based on this initial structure, the local optimization and solution of the MINLP problem is performed according to the initialization and solution procedure described in Section 3.2. The parameters for the EA are listed in Table 4. In order to evaluate the performance and reliability of the hybrid approach, 25 consecutive optimization runs were performed, each utilizing five instances of GAMS in parallel. None of the runs required more than 5 generations. While 56% of the runs terminated due to an 21 ACS Paragon Plus Environment


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Table 4: Specifications of the EA for the single distillation column. general

Ntrays,max Nf eed,init

λinit 15

λ 9

µ 3

dlo dup i i 30 130 5 125

dtabu i 1 3

κ 6

∆fmin 10 ea pi 1 1

ng,max 3

nf,max 15

tmax 1800 s

si [0.03 0.1 0.2 0.5] [0.03 0.1 0.2 0.5]

insufficient improvement of the objective function over the specified number of three generations, the remaining 44% terminated due to too many failed attempts for the generation of novel individuals, which were not covered by previous tabu zones. Overall, a local optimum was successfully determined for 92% of all individuals. An overview of the performance data is provided in Table 5. The table lists the minimum (min), average (∅) and maximum value (max) for the number of generations (ng ), the total time of the optimization run (ttot ) and the convergence ratio for all 25 consecutive runs. In addition the minimum, average and maximum values for the TAC, the time required for the local optimization (topt ) and the DDoF [Ntrays , Nf eed , QB ] are listed for the best individuals of the 25 consecutive runs. Note that the single values are not listed according to the individuals with minimum, average or maximum total cost, but show the variation in the single values for the best individuals of the 25 consecutive runs. and is characterized by a total number of The overall best solution has a TAC of 219.419 ke a 61 trays with the feed located on tray 55 and a heat duty of 900 kW in the reboiler. As can be seen from Table 5, each run determines a solution, which is at most 0.02% worse than the overall best determined solution and the DDoF for the best individuals of the single runs also show little variation. Therefore, it can be concluded that the hybrid optimization approach generally determines an optimal solution for the single column optimization. One reason for this good performance is, that for the single distillation column, the results of the local optimization are already in close vicinity to the best solution, for a large variety of initial solutions. This is illustrated by means of the arrow-plot in Figure 5 (left), which depicts the changes from the initial DDoF specified by the EA to the final values determined by 22 ACS Paragon Plus Environment

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Table 5: Performance evaluation of the 25 runs for the single distillation column.

ng min ∅ max

4 4.16 5

ttot

overall [s] convergence [%]

588 1000 1646

80.95 91.44 100.00

[ ke ] best individual per run TAC a topt [s] Ntrays Nf eed QB [kW] 219.419 219.422 219.445

15 85 201

60 61 62

53 54 55

900 902 905

the local optimizer for a single run. Obviously, the majority of the arrows points to the same location. Only those individuals, for which the initial DDoF result in a too small column or which are initialized with a feed close to the top result in an inferior design. Consequently there is no distinct evolution over the different generations, as close to optimal solutions are directly determined in the first generation, resulting in the termination after the minimum number of generations.

Figure 5: Optimization results of a single run for the optimization of the single distillation column. Evolution of the DDoF in the local optimization (left) and cost distribution after the local optimization (right). While there is in principle no need for the hybrid optimization approach in determining a close to optimal single column design, the cost distribution for the single individuals in Figure 5 (right) illustrates that there are different configurations with close to optimal design. According to the approximation of the Pareto front a similar TAC can be obtained with a variable share of 15% that can be shifted between a larger investment or operating cost.



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This variability is expressed by design variants with 60 − 74 trays and heat duties between 879 − 906 kW, which vary by less than 1% in TAC. This variability is not only interesting from a financial perspective, but also provides the potential to evaluate additional criteria, e.g., concerning the controllability of the process. Although not further pursued in the current work, several key indicators for process controllability analysis 61 could be evaluated based on the results of the local optimization.

Figure 6: Illustration of the composition profile in the distillation column for the best determined process design. The composition profile of the best solution is illustrated in Figure 6. Obviously, most of the required equilibrium trays are utilized in the rectifying section in order to purify the acetone in the distillate. The composition profile shows strong nonlinearity and the required energy duty only slightly exceeds the minimum energy duty required to cross the pitchfork distillation boundary (PDB) in the direction of the total reflux boundary (TRB). Finally, Figure 7 illustrates the utilization of the single GAMS instances in the parallel optimization approach. Obviously the parameter settings specified in Table 4 result in a well-balanced utilization of all five GAMS instances with few idle times in between the different generations. 24 ACS Paragon Plus Environment

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Figure 7: Utilization of the five GAMS instances in the parallel optimization approach.

4.2

Entrainer-enhanced pressure-swing distillation

The second case study deals with the dehydration of ethanol by means of pressure-swing distillation in two variants. First, a variant without the addition of an auxiliary compound, as considered in a prior publication 3 , and second an entrainer-enhanced version with acetone as entrainer, as analyzed on a shortcut level in the publication 62 , are investigated.

Figure 8: Superstructure for the (entrainer-enhanced) pressure-swing distillation.



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The general superstructure that is utilized in both variants and the DDoF handled by the EA are illustrated in Figure 8. Similar to the case of a single column, the EA initializes col1 col2 the maximum number of equilibrium trays (Ntrays,max ,Ntrays,max ), as well as the initial feed col2 col1 (Nfcol1 eed,init ,Nf eed,init ) and the recycle location (Nrecycle,init ), whereas the EA prioritizes the total

number of trays over the feed and recycle locations in the generation of new individuals. In addition to these ordinal integer values also the utilities for the intermediate heat exchanger col2 hx ) represent nominal ) and the reboiler of the second distillation column (Nutility (HX) (Nutility

integer DDoF, which are fixed by the EA. The EA can choose between low (LP), medium (MP) and high pressure (HP) steam, which are defined according to Table 6, which also lists the associated cost values. Table 6: Available utilities and associated costs, adopted from Luyben 63 . LP

MP

HP

Tsteam [◦ C] 160 $ csteam [ GJ ] 7.78

184 8.22

254 9.83

While these values differ from those in Supplementary material, the remaining economic model is not altered. The material-specific correction factors for the calculation of the investment costs are adjusted to the pressure level in the local optimization approach. It turned out to be beneficial to have a bias towards a large number of trays in the initial generation, for which also a uniform distribution of the utility DDoF is assumed. Based on the decisions of the EA the number of trays and feed locations, as well as the operating pressure of the second column and the energy duty of the intermediate reboiler are optimized by the local optimization approach. The operating pressure in the second column is limited by the choice of utility, as well as an additional upper limit of 60 bar to account for the critical pressure of ethanol.


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4.2.1

Pressure-swing distillation without entrainer

The specifications for the ethanol dehydration without additional entrainer are adopted from Skiborowski et al. 3 and listed in Table 7. Table 7: Specifications for ethanol dehydration. flowrate [ mol ] ethanol [ mol ] s mol feed ethanol product water product

150.76

water [ mol ] mol

0.4236 ≥ 0.9948

0.5764 ≥ 0.999

The parameters of the EA are listed in Table 8. While the initial population and the time limit are increased, the general EA parameters are quite similar to the optimization of the single column and the adaption of the mutation scales for the ordinal integer values is performed accordingly. Since the nominal integer values are changed randomly in the mutation, there are no mutation scales and the probabilities are adapted. Table 8: Specifications of the EA for the pressure-swing distillation. general

λinit 20

λ 9

µ 3

κ ∆fmin 6 10 ea

col1 Ntrays,max Nfcol1 eed,init col1 Nrecycle,init col2 Ntrays,max Nfcol2 eed,init

dlo dup i i 30 130 10 125 5 120 30 130 5 120

dtabu i 1 3 3 1 3

pi 1 1 1 1 1

col2 Nutility hx Nutility

dlo i 1 1

dup i 3 3

dtabu i 0 0

ng,max 4

nf,max 15

[0.03 [0.03 [0.03 [0.03 [0.03

si 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2

pi [0.5 0.3 0.1 0.01] [0.5 0.3 0.1 0.01]

tmax 10800 s 0.5] 0.5] 0.5] 0.5] 0.5] si − −

The local optimization is performed in accordance with the initialization and solution strategy presented in Section 3.2 and further detailed in Skiborowski et al. 4 . In order to provide reasonable product specifications for the initialization of the single columns, the operating pressure for the second column is in the beginning fixed to 10 bar and the distillate compo27 ACS Paragon Plus Environment


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sitions are determined based on the azeotrope compositions at 1 atm (x1atm az,EtOH ) and 10 bar (x10bar az,EtOH ) by means of 10bar 10bar x1atm x1atm az,EtOH + xaz,EtOH az,EtOH − xaz,EtOH + , 2 10 10bar 10bar x1atm x1atm az,EtOH + xaz,EtOH az,EtOH − xaz,EtOH = − . 2 10

xcol1 D,EtOH =

(1)

xcol2 D,EtOH

(2)

After the single columns are interconnected, the operating pressure in the second column is a DDoF in the optimization, same as the product purities in both columns. In order to evaluate the performance and reliability of the hybrid approach, 15 consecutive optimization runs were performed, each utilizing up to ten instances of GAMS in parallel. The runs terminated after 5-10 generations. An overview of the performance data is provided in Table 9. The table lists the minimum (min), average (∅) and maximum value (max) for the number of generations (ng ), the total time of the optimization run (ttot ) and the convergence ratio for all 15 consecutive runs. In addition the minimum, average and maximum values obtained col1 col1 for the TAC, the time for the local optimization (topt ) and the DDoF [Ntrays , Nfcol1 eed , Nrecycle , col2 col1 col2 HX Ntrays , Nfcol2 eed , QB , QB , QB ] are listed for the best individuals of the 15 consecutive runs.

Variation of the pressure complicates the optimization and results in larger variations of the T AC of the best individuals and the number of generations in the different runs. Table 9: Performance evaluation of the 15 runs for the pressure-swing distillation.

min ∅ max

min ∅ max

overall convergence [%]

ng

ttot [s]

5 7.13 10

4394 5794 7122

col1 Ntrays

Nfcol1 eed

col1 Nrecycle

66 78 91

63 76 89

58 72 86

best individual per run [ ] TAC Me topt [s] a

73.21 83.84 98.23

4.856 4.861 4.870

238 382 669

best individual per run col2 Nfcol2 Qcol1 [MW] Qcol2 [MW] QHX [MW] Ntrays B B B eed 67 74 86

43 48 60

13.01 13.11 13.20


6.16 6.35 6.75

2.37 2.90 3.10

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The overall best solution has TAC of 4.856 Me and is characterized by a total number of 73 a trays in the first column, with feed tray 70 and recycle tray 66, a total number of 70 trays in the second column with tray 45 as feed tray, HP steam as utility for the second column and MP steam in the intermediate HX. Each run determines a solution, which is at most 0.3% worse in TAC than the best determined solution. All these solutions have in common, that HP steam is used in the second column and MP steam is used in the intermediate HX. While there is only little variance in the overall required heat duties (< 2%), the column sizes and the absolute positions of the feed trays however vary strongly. This can be explained by small effect of the investment cost on the TAC.

Figure 9: Evolution of the fitness (TAC) over the generations (left) and cost distribution (right) for a single run. Figure 9 (left) illustrates the evolution of the fitness over the generations and the cost distribution for the evaluated individuals (right) of a single run. As obvious from the cost distribution, the annual operating cost represent the major share of the TAC. The incentive for the reduction of the total number of trays in the optimization is therefore rather small. Consequently, there are again a lot of different designs with close to optimal fitness, all of which have HP steam as the utility in column two. Such designs are determined in every iteration (cf. Figure 9 (left)), confirming the good performance of the local optimization approach. The EA mainly performs the selection of the optimal utilities, which is finished after a few generations, while generating different configurations of the number of equilibrium 29 ACS Paragon Plus Environment


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trays and feed/recycle locations with close to optimal fitness. The significant difference in the performance of the processes with different utilities for the second column results from the different operating windows. While each optimized process operates at the maximum pressure for the given utility, the processes operated with HP steam in column two have the highest ethanol content in the top products of both columns, resulting in a decreased recycle stream, which is the key factor for a more economic operation.

Figure 10: Cost distribution of individuals with HP steam in column two (left) and ratio of the number of trays and the feed/recycle trays in both columns (right). Figure 10 (left) illustrates the annual operating and investment costs for those individuals with HP steam in column two in more detail. Obviously the different configurations represent an approximation of the Pareto front. This does not only provide the opportunity to select between different configurations or introduce an additional objective for controllability, but also strengthens the confidence in the results. The influence of the utility selection for the intermediate HX can also be observed in Figure 10 (left). As the use of HP steam results in higher operating cost, it is only considered for a few individuals, for which the intermediate HX is not utilized in the local optimization. However, there is not much of a difference between the utilization of LP and MP steam. Finally, Figure 10 (right) illustrates the ratios of the final feed or recycle locations to the total number of equilibrium trays in the respective column. While the total number of trays varies by a factor of two for the different individuals, the relative location of the feed and recycle trays is approximately constant. 30 ACS Paragon Plus Environment

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4.2.2

Pressure-swing distillation with acetone as entrainer

The specifications for this separation are similar to those considered for the shortcut calculations performed by Br¨ uggemann and Marquardt 62 and are introduced in Table 10. Previous investigations have already shown the potential of the rigorous optimization with conceptual design models to improve the previously reported shortcut-based process designs 64 . Table 10: Specifications for ethanol dehydration by means of entrainer-enhanced pressureswing distillation (adopted from Br¨ uggemann and Marquardt 62 ). flowrate [ kmol ] s

mol acetone [ mol ]

1

0.000

feed ethanol product water product

ethanol [ mol ] water [ mol ] mol mol 0.042 ≥ 0.999

0.948 ≥ 0.999

Compared to the previous optimization problem, the optimization of the process becomes more complex due to the additional entrainer, such that the optimization runs would exceed the specified time limit of 3h. However, a slight adjustment of the parameters of the EA, increasing the number of individuals per generation and the diameter of the tabu zones, is sufficient to reach convergence within the time limit for most runs. The increase of the tabu-zones is reasonable due to the previously demonstrated good performance of the local optimization approach. Table 11: Specifications of the EA for the entrainer-enhanced pressure-swing distillation. λinit 30

λ 15


dlo i 60 40 10 80 15

dup i 115 115 110 135 110

dtabu i 3 5 5 3 5

col2 Nutility hx Nutility

dlo i 1 1

dup i 3 3

dtabu i 0 0

general

µ κ ∆fmin 3 6 10 ea pi 1 1 1 1 1

ng,max 4

nf,max 15

[0.03 [0.03 [0.03 [0.03 [0.03

si 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2

pi [0.5 0.3 0.1 0.01] [0.5 0.3 0.1 0.01]


tmax 10800 s 0.5] 0.5] 0.5] 0.5] 0.5] si − −


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The adjusted parameters for the EA are listed in Table 11. Some additional adjustments have to be made for the optimization of the entrainer-enhanced pressure-swing distillation process. Due to the critical pressure of acetone, the maximum pressure has to be decreased to 45 bar. Fresh entrainer can be fed into the recycle stream and the initialization strategy is adjusted to account for a certain amount of acetone in the recycle stream, by first solving the overall mass balances, before the single distillation columns are initialized. The remaining solution strategy is similar to the process without the entrainer. Again, 15 consecutive optimization runs were performed utilizing up to ten instances of GAMS in parallel. The runs terminated after 5-8 generations. An overview of the performance data is provided in Table 12. The table lists the minimum (min), average (∅) and maximum value (max) for the number of generations (ng ), the total time of the optimization run (ttot ) and the convergence ratio for all 15 consecutive runs. In addition the minimum, average and maximum values obtained col1 col1 for the TAC, the time for the local optimization (topt ) and the DDoF [Ntrays , Nfcol1 eed , Nrecycle , col2 col1 col2 Ntrays , Nfcol2 eed , QB , QB ] are listed for the best individuals of the 15 consecutive runs.

Table 12: Performance evaluation of the 15 runs for the entrainer-enhanced pressure-swing distillation.

ng min ∅ max

min ∅ max

ttot


5 5.87 8

7379 8887 10800

col1 Ntrays

Nfcol1 eed

col1 Nrecycle

79 82 85

75 77 81

70 72 76

best[individual per run ] Me TAC a topt [s]

75.56 78.96 85.56

3.7638 3.7643 3.7653

239 370 536

best individual per run col2 Ntrays Nfcol2 Qcol1 [MW] Qcol2 [MW] B B eed 127 132 135

66 70 74

8.81 8.84 8.85

7.81 7.83 7.84

While similar to the process without entrainer all optimized processes utilize the HP steam in the reboiler of the second column and operate at the maximum pressure, there is much less variance in the remaining DDoF. The heat duty of the intermediate HX is not listed in 32 ACS Paragon Plus Environment

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Table 12, since it is not used by any of the optimized processes. The overall best solution has a TAC of 3.7638 Me and is characterized by a total number of 82 trays in the first column, a with feed tray 78, recycle tray 73 and a total number of 132 trays in the second column with tray 71 as feed tray. However, the design of all other optimized processes varies by less than 5%, emphasizing the constant performance of the hybrid approach. Figure 11 illustrates the composition profiles of the distillation columns.

Figure 11: Composition profiles for the best determined design of the entrainer-enhanced pressure-swing distillation process. In order to illustrate the performance of a single run Figure 12 (left) illustrates the evolution of the fitness over the generations and the cost distribution for the evaluated individuals (right). The results are similar to the process without the additional entrainer, highlighting the excellent quality of the local optimization approach.



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Figure 12: Evolution of the fitness (TAC) over the generations (left) and cost distribution (right) for a single run. The distribution of the operating and investment cost shares for those individuals with HP steam in column two are illustrated in more detail in Figure 13 (left). The different configurations represent an approximation of the Pareto front, while there is no influence of the utility selection for the intermediate HX, since it is not used in the local optimization. However, in contrast to the pressure-swing process without entrainer, the optimum design does not operate at the minimum recycle flowrate, as illustrated in Figure 13 (right). There is a distinct tradeoff, which relates to the tangential pinch in the first column and the restrictions on the reflux ratio of the second column (cf. Figure 11).

4.3

Extractive distillation

The last case study further increases the utilization of the EA by combining the selection of a suitable entrainer and the optimization of the process design for the separation of an acetone/methanol mixture by means of an extractive distillation process. This separation was previously investigated by Kossack et al. 65 , applying the first three steps of the process synthesis framework in series. Different entrainer candidates were first screened by means of a literature study and the application of a CAMD method 66 . Ten of these candidates were considered for a first screening according to the MED of the process, performing short-


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Figure 13: Cost distribution of individuals with HP steam in column two (left) and TAC over the recycle (top product of column 2) flowrate (right). cut calculations by means of the RBM 67,68 . Finally, an MINLP optimization of the MESH models for the extractive distillation processes was performed for the best five solvents. The superstructure for these processes is depicted in Figure 14.

Figure 14: Superstructure for the extractive distillation process. The specifications of the feed and the names and approximated costs for the six entrainer candidates considered in this study are listed in Table 13. Cost estimates are based on commodity prices listed on alibaba.com (October, 2012). The six entrainer candidates include all the components considered by Kossack et al. 65 in the final optimization, as well as ethanol, 35 ACS Paragon Plus Environment


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which was discarded in the shortcut screening due to low solubilities. The same property models and parameters as used by Kossack et al. 65 are used for the current study. Table 13: Specifications for extractive distillation process. flowrate [ mol ] acetone [ mol ] s mol feed acetone product methanol product

10

mol methanol [ mol ] entrainer [ mol ] mol

0.7774 ≥ 0.995

0.2226

0.000

≥ 0.995

entrainer candidates chlorobenzene mesitylene p-xylene

1000 1800 1250

$ ton $ ton $ ton

DMSO ethanol water

1400 800 0.03

$ ton $ ton $ ton

Although product purities of 99.5 mol% are desirable, the lower bound on the methanol product purities is decreased to 98.9 mol% for DMSO and 98.2 mol% for water as entrainer, since higher purities cannot be reached according to Kossack et al. 65 . Although other authors like Luyben 69 achieve the higher purities for these solvents, making use of different thermodynamic models, this adaption is reasonable for a comparison of the results with those of Kossack et al. 65 . Additional cost approximations are introduced to account for the entrainer losses. In case methanol represents a valuable product, the lower quality should also be considered for a fair comparison of the different entrainer candidates. While the operating pressure could be specified as DDoF similar to the previous case study, this is not necessary for the current process and the choice of utility is consequently discarded. Instead, the EA determines the optimal entrainer (Nentrainer ) as a nominal integer DDoF. The entrainer choice effects all thermodynamic property calculations and has therefore a tremendous effect on the resulting model, which is used in local optimization. However, the general process configuration is very similar to the previously investigated entrainerenhanced pressure swing distillation, such that no significant changes to the settings of the EA are required. The parameters are listed in Table 14. Besides the specified lower and upper bounds, an additional constraint limits the initial number of trays for the first column


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to a maximum of 70 trays, for all entrainer candidates other than water. This constraint was added to decrease the search space, since only few equilibrium trays are utilized in the final designs. Table 14: Specifications of the EA for the extractive distillation process. general


Nentrainer

λinit 30

λ 15

µ κ ∆fmin 4 6 10 ea

dlo dup i i 30 120 3 110 3 110 150 70 3 45

dtabu i 3 5 5 3 5

dup i 6

dtabu i 0

dlo i 1

pi 1 1 1 1 1

ng,max 4

nf,max 100

[0.03 [0.03 [0.03 [0.03 [0.03

si 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2

pi [0.5 0.4 0.1 0.01]

tmax 10800 s 0.5] 0.5] 0.5] 0.5] 0.5] si −

Depending on polarity, the entrainer either associates with methanol, such that acetone is obtained as top product of the first column, or the other way around. However, instead of specifying the dipole moment of the entrainer candidates a simple flash calculation with 50 mol% of the entrainer and 25 mol% of acetone and methanol is performed as one of the first initialization steps in the local optimization approach. The component which accumulates most in the equilibrium vapor composition is specified as the top product of the first column. Based on an initial estimate of an entrainer-to-feed ratio of 1.5, which is rather pessimistic, initial specifications for the recycle stream and the bottom products are determined based on the overall mass balances. Afterwards the single unit operations are further initialized and aggregated. The utilities (cf. Table 6) are selected according to the temperature in the reboiler just before the aggregated process model is optimized for TAC. As in the previous case studies, 15 consecutive optimization runs were performed utilizing up to ten instances of GAMS in parallel. The runs terminated after 4-6 generations. An overview of the performance data is provided in Table 15.



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Table 15: Performance evaluation of the 15 runs for the extractive distillation process.

ng min ∅ max

min ∅ max

ttot


4 5.33 6

5201 7608 9386

col1 Ntrays

Nfcol1 eed

col1 Nrecycle

36 42 46

25 31 35

4 4 4

best [individual per run ] ke TAC a topt [s]

85.71 89.88 97.78

194.521 195.020 195.565

102 579 1097

best individual per run col2 [kW] Qcol2 [kW] Qcol1 Ntrays Nfcol2 B B eed 12 15 18

6 7 9

546 550 556

185 188 194

While there is some degree of variance in the DDoF for the distillation columns, all of the best individuals of each run utilize DMSO as entrainer and the deviation in the objective function is smaller than 0.6%. The average convergence ratio is approximately 90% and thus 10% higher than for the entrainer-enhanced pressure-swing distillation. Although the separation processes are of similar complexity, the optimization of the pressure seems to complicate the local optimization problem.

Figure 15: Evolution of the fitness (TAC) over the generations (left) and the variance of the fitness for the different entrainer candidates (right) for a single run. In order to illustrate the performance of a single run Figure 15 (left) illustrates the evolution of the fitness over the generations and the variance of the fitness for the different entrainer


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candidates (right). While a uniform distribution of the entrainer candidates is enforced in the initial generation, ethanol is not considered further on due to its significantly inferior performance. The other candidates are considered in the subsequent 2-3 generations. The most prominent and best performing entrainer is DMSO, which is solely considered in the last two generations. However, in contrast to the previous case studies, there is a distinct variance in the solution quality for each of the entrainer candidates. This is further illustrated in Figure 15 (right). Obviously the worst DMSO individual is outperformed by several individuals with chlorobenzene and mesitylene as entrainer. Thus, this time the hybrid optimization approach does not only generate different configurations with approximately equal fitness, but truly improves the quality of the final solution.

Figure 16: Cost distribution for all individuals (left) and for those with TAC below 220 (right).

ke a

This is further illustrated by the distribution of the operating and investment cost shares which are illustrated in Figure 16 (left) and with a focus on the best solutions in more detail in Figure 16 (right). The figures underscore the high concentration of the individuals in the region of the best performance and Figure 16 (right) illustrates that the solutions rather cluster in the region of minimum operating cost, than presenting a larger approximation of the Pareto front as in the previous case studies.



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5

Conclusions

This paper introduced a novel hybrid evolutionary-deterministic optimization approach for conceptual design, which builds on the efficient solution of SR-MINLP model formulations, as introduced by Kraemer et al. 18 and further extended by Skiborowski et al. 3 ,4 . The approach is illustrated for three case studies, demonstrating not only the reliability of the deterministic local optimization approach, but also the capability of the hybrid approach to improve the solution quality and provide additional information on design variability. The hybrid approach is also capable of optimizing complex design decisions like the choice for utilities and entrainer candidates, which tremendously complicate a gradient-based optimization. While there is little potential for improvement of the local optimization approach for simpler problems, like the optimization of the single distillation column, the hybrid approach reduces the necessary user input for the initialization and strengthens the confidence in the quality of the results. In addition, the variety of different and approximately equivalent solutions, generated by the hybrid approach, can further be evaluated by means of additional criteria, including process controllability metrics. The extractive distillation case study confirms the potential of the hybrid approach to improve the solution quality in case of complex optimization problems, for which the results of the local optimization approach show higher variance for different initial DDoF values. It also demonstrates the potential of the hybrid optimization approach to directly tackle a extremely complex optimization problem, characterized by the manipulation of the rigorous thermodynamic models for equilibrium and enthalpy calculation, according to the entrainer selection, in combination with the closed-recycle process design for the extractive distillation process. The application to even more complex case studies should well be feasible, as the computationally efficient local optimization in combination with the parallelization facilitated an optimization of the current case studies in less than 3 h CPU time. A possible case study with further increased combinatorial complexity would be the simultaneous optimization of the process design as well as the choice of membrane material and utilities for each membrane 40 ACS Paragon Plus Environment

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stage in a multistage membrane-assisted distillation process. But also, the optimization of a process combining reaction and separation, as e.g. the synthesis of MTBE in a reactive distillation as investigated by Urselmann et al. 49 is of further interest. This case-study could then also be used for a direct comparison of the two hybrid optimization approaches. Based on the current results, it is recommended to utilize the potential of the local optimization approach to the greatest extent possible, by optimizing all continuous as well as those discrete design decisions which can efficiently be reformulated by means of NCP-functions. Further benefits could be achieved by means of a reformulation of the local optimization problem as a GDP and a logic-based optimization 20 in combination with the decoupling of the complex property and equilibrium calculations 3,4 .

Acknowledgement The authors would like to acknowledge financial support of ”Deutsche Forschungsgemeinschaft” for the project ”Optimisation-based framework for the synthesis of membrane-assisted hybrid processes” (MA 1188/32-1).

Supporting Information Available The supplementary information contains information on the economic models and thermodynamic property data for the three case studies. This material is available free of charge via the Internet at http://pubs.acs.org/.

Nomenclature Abbreviations DMSO dimethyl sulfoxide EA evolutionary algorithm ES evolutionary strategy



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GA genetic algorithm HEN heat exchanger network HP high pressure HX heat exchanger LP low pressure MA memetic algorithm MP medium pressure NCP nonlinear complementary problem SR-MINLP successively relaxed MINLP TAC total annualized cost

Symbols $ [ unit ]

c specific cost d DDoF for EA

[-]

∆fmin minimum improvement in fitness function

[-]

κ maximum age of parents

[-]

λ number of individuals in a generation

[-]

µ number of parents for a generation

[-]

N integer number

[-]

Ntrays number of trays

[-]

Nf eed tray number for feed tray

[-]

Nrecycle tray number for recycle tray

[-]

nf number of failed attempts for generation of new individuals

[-]

ng number of generation without significant improvement

[-]

p mutation probability of DDoF in EA

[-]

s mutation scale of DDoF in EA

[-]


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t time

[-]

ttot computational time for a hybrid optimization run

[s]

topt computational time for the local optimization

[s]

QB heat duty for the reboiler

[kW/MW]

Superscripts col column HX heat exchanger

Subscripts entrainer entrainer choice init initial value lo lower bound min minimum max maximum up upper bound utility utility choice

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