Design of Graph-Based Evolutionary Algorithms: A Case Study for Chemical Process Networks Michael Emmerich
[email protected] Center for Applied Systems Analysis, Informatik Centrum Dortmund (ICD/CASA), Joseph von Fraunhofer Straße 20, 44227 Dortmund, Germany
Monika Grotzner ¨
[email protected] Department of Technical Thermodynamics, Technical University Aachen (RWTH), Schinkelstrasse 8, 52062 Aachen, Germany
Martin Schutz ¨
[email protected] Center for Applied Systems Analysis, Informatik Centrum Dortmund (ICD/CASA), Joseph von Fraunhofer Straße 20, 44227 Dortmund, Germany
Abstract This paper describes the adaptation of evolutionary algorithms (EAs) to the structural optimization of chemical engineering plants, using rigorous process simulation combined with realistic costing procedures to calculate target function values. To represent chemical engineering plants, a network representation with typed vertices and variable structure will be introduced. For this representation, we introduce a technique on how to create problem specic search operators and apply them in stochastic optimization procedures. The applicability of the approach is demonstrated by a reference example. The design of the algorithms will be oriented at the systematic framework of metricbased evolutionary algorithms (MBEAs). MBEAs are a special class of evolutionary algorithms, fullling certain guidelines for the design of search operators, whose benets have been proven in theory and practice. MBEAs rely upon a suitable denition of a metric on the search space. The denition of a metric for the graph representation will be one of the main issues discussed in this paper. Although this article deals with the problem domain of chemical plant optimization, the algorithmic design can be easily transferred to similar network optimization problems. A useful distance measure for variable dimensionality search spaces is suggested. Keywords Evolutionary algorithms, genetic programming, chemical process optimization, network representations, metric-based evolutionary algorithms, minimal moves.
1
Introduction
Chemical plants contain multiple chemical engineering devices (e.g., pumps, distillation columns, and chemical reactors) that spread a complex net of connecting material, heat, and information streams. They serve to perform a chemical process during which raw materials are converted into desired products. c 2001 by the Massachusetts Institute of Technology
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The availability of high-speed parallel computers and process simulators, which are capable of rigorous owsheet simulations, has given way to new methodologies for the optimization of chemical processes that no longer need to deal with strong simplied mathematical models of the given tasks. The use of evolutionary algorithms (EAs) seems to be a promising approach here, because they have proven to be robust parallel search techniques for difcult highdimensional and non-linear search spaces. A special advantage of EAs in this context is that they provide intuitive concepts to search on natural representations of solutions, like in this case, parameterized graphs. In this paper, we present an approach where a chemical plant is represented as a special kind of parameterized network (graph), and we design genetic operators working on both the structure and the parameters of this graph. It shall be emphasized that the choice of the problem representation and genetic operators is crucial in this context. There exist only a few systematic approaches for the design of EAs on nonstandard representations. The evolution of discrete structures has been mainly stressed in the eld of Genetic Programming (Ba¨ ck et al., 1997), dealing with the evolution of computer programs. With the memetic algorithms, Radcliffe and Surry (1994) present EAs that are based on non-orthogonal representations and stress working principles of genetic algorithms, like principles of inheritance of features. Metric-based evolutionary algorithms (MBEAs) (Droste and Wiesmann, 2001) are a class of evolutionary algorithms that fulll certain requirements for the design of the representation, mutation, and recombination. They explicitly address design principles for operator design that are common in many evolutionary algorithms. MBEAs for the optimization of ordered binary decision diagrams and nite automata turned out to be superior to classical approaches like evolutionary programming and genetic programming (see Droste and Wiesmann (2001)). The representations addressed here are similar to the representation of chemical plant networks. In Section 2 we will discuss guidelines for the design of EAs. Taking these guidelines into consideration, in Section 4 a graph representation for chemical process networks together with genetic operators will be derived. The successful application of these guidelines relies upon the denition of a suitable metric on the search space. Hence, the denition of a distance measure on graphs will be a main issue discussed in Section 4. This is done after an introduction into the problem class and a reference example, given in Section 3. In Section 4.5 we introduce different algorithmic variants and describe their performance on the test case. These experiments demonstrate the applicability of our approach.
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Evolutionary Algorithms for General Representations
In this article, chemical process networks will be represented as graphs with variable connectivity and size. Evolutionary algorithms usually deal with sequential representations like real or integer vectors. Therefore we will have to dene evolutionary algorithms in a more general way. 2.1 General Concepts for Search Algorithms The terms introduced in this section are fundamental for every search algorithm, not only for EAs. The formal description of these concepts is similar to that introduced by Jones (1997). 330
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2.1.1 Search Space, Representation, and Fitness The search space is the set of solution objects to be considered by the search algorithm. The set of objects that represents solutions and on which the search operators of the algorithm operate will be called the representation space . A representation is a function assigning objects from the search space to objects in the representation space. In evolutionary algorithms, is called the genotype space, whereas is called the phenotype space. Elements of the genotype space are called genotypes. An objective function assigns a value out of the tness space to every object of the search space . On the tness space , a partial order has to be declared. In the algorithms presented in this article, the real numbers will serve as tness space. Optimization algorithms are special kinds of search algorithms, trying to nd optimal points in the search space, i.e., points with . In the following, we introduce some important terms that will be needed in the discussion of guidelines for EAs. Feasibility is an important concept, whenever constraints (implicit or explicit) are dened on the search space. The representation space can be partitioned into an infeasible subspace and a feasible subspace , where the constraints are fullled. By overrepresentation or degeneracy, it is meant that one element in the search space is assigned to more than one element in the representation space , i.e.,
A similar concept is described by underrepresentation. Underrepresentation means that not all elements in the search space are represented in the representation space, i.e.,
This is almost inevitable when dealing with innite search spaces. Radcliffe (1994, 1995) distinguishes between an allelic representation and a genetic representation. The rst is dened as sets of features, whereas the second is dened as a sequence of features, i.e., each feature has a xed position in the genotype. On so-called metric representation spaces, a distance measure is declared, i.e., a function with
A global minimum (maximum) is dened as an element condition holds:
for which the following
2.1.2 Search Operators In EAs we use various kinds of search operators like mutation and recombination in order to generate new solutions. They are often implemented as procedures that operate on one or several genotypes and generate randomly variations of them. A useful formalization of probabilistic search procedures has been suggested by Jones (1997). It is especially well suited for reasoning about general properties of search operators, since it allows omission of the details of their implementation. Here, search operators describe procedures that transform an input multiset into an output multiset. Multisets are dened as follows (Jones, 1997): A multiset is a mathematical entity that is like a set, but it is allowed to contain repeated elements, e.g., is a multiset, which is equivalent to . The number of objects (cardinality) in a multiset Evolutionary Computation Volume 9, Number 3
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Algorithm: Standard (
) EA
Initialization: Evaluation: while not terminate do Recombination: Mutation: Evaluation: Replacement: t := t + 1 end while Figure 1: Main loop of an EA. will be denoted by , e.g., = 3. A multiset whose elements are drawn from all elements of a set will be denoted by , and the set of multisets with elements drawn from with xed cardinality is described by . D EFINITIO N 1: Search operators are dened as functions assign a probability value to each pair of elements from .
that
The probability value is interpreted as the probability that the second multiset results from the rst multiset if the procedure of this search operator is applied. Whenever deterministic operators are used, this probability takes either the value or . A series of applications of search operators , on multisets will be denoted by:
A symmetric search operator is an operator with the property: for a given cardinality . Mutation operators in EAs are often symmetrical search operators. We have now introduced a general notion of search operators and some general terms and operators related to them. The following section deals with special search operators of EAs, like recombination and mutation. 2.2 Evolutionary Algorithms Evolutionary Algorithms are inspired by simple models of biological evolution. They are known as robust optimization algorithms based on the collective learning process within a population of individuals. Each individual represents a search point in the representation space . By the iterative processing of an evolution loop, consisting of selection, mutation, and recombination, a randomly initialized population evolves towards better regions of the search space. The tness function delivers the quality information necessary for the selection process to favor individuals with higher tness for reproduction. The reproduction process consists of the recombination mechanism, responsible for the mixing of parental information, and mutation introducing undi332
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rected innovation into the population. In Figure 1 the main loop of an EA is described. It covers only the basics of EAs necessary for understanding the article. For a detailed description of all variants of EAs, the reader is referred to the literature (B¨ack et al., 1997; B¨ack, 1996; Schwefel, 1995; Rechenberg, 1994). In this algorithm, denotes a population of initial individuals that might be given by the user as starting points. This multiset may also be empty. In this case, the initialization procedure may generate individuals of randomly or by heuristics. The evaluation procedure assigns a tness value to each element of a population. If we are not dealing with noisy tness functions, this is done in a deterministic way. The generational loop consists of the following steps: Genotypes taken from the population are selected (depending on their tness) and combined to new genotypes by the recombination operator . The individuals of are then randomly modied by the mutation operator and evaluated. From these new so-called offspring individuals and the parental population , the replacement operator selects individuals that build the new parental generation . The evaluation procedure may contain a local search procedure starting from the given individual to obtain a tness value. In one of the algorithms we present, the hierarchical structure evolution, this local search seeks for continuous or discrete parameters for a given network conguration described by the individual. The algorithms presented here are based on a -ES as introduced by Schwefel (1995). Only the search operators and representation have been adapted to the given task. The recombination operator of evolution strategies (ESs) differs from other EAs in the way that it chooses mating partners for recombination uniformly distributed from the parental generation, neglecting their tness values. Furthermore, a surplus of offspring is usually generated by , i.e., . The replacement operator selects only individuals that exist less than iterations. In analogy to biology, is called the maximal life span. In the case , we speak of a -ES, and in the case ,a -ES. In the following, we will characterize the two main variation operators in evolutionary computation: mutation and recombination. We focus on these search operators since the initialization and replacement operator have been applied in a standard way in this article. 2.2.1 Mutation Operators A mutation operator is a search operator with that generates a variation of a given individual. Mutation is an asexual operator, therefore, the process of mutation can be described by the reduced mutation operator with . In the following, we will omit the multiset notation whenever we talk of operators working on multisets of cardinality . Every mutation operator generates variations of the parental chromosomes, i.e., . Some characteristics can be found in many mutation operators. Some of them are listed in the following. Mutation is often applied as a local search operator. Any small variation of the genotype is generated with a higher probability than any large variation. Assuming that is a distance measure on , this property can be formalized as follows: Often symmetry is one of the design principles for a mutation operator in order to avoid a drift. A bias-free mutation operator has the following property:
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Another important design principle for mutation operators is ergodicity. Every point in the search space should be reachable within one application of the mutation operator from any start point, i.e.,
Ergodicity is an important condition for the global convergence of many mutationbased EA systems (see for example Droste and Wiesmann (2000)). The usefulness of design principles for mutation operators depends strongly on the specic problem. Nevertheless, in evolution strategies principles like local search, symmetry, and ergodicity turned out to be good guidelines in the design of mutation operators (cf. Rudolph (1994)). In an MBEA, exactly these three conditions need to be fullled. 2.2.2 Recombination Operators The recombination operator is an operator with xed cardinality. Without loss of generality, we will look at the reduced recombination operator. This operator generates one individual from two given individuals. In our MBEA system, it will be applied times, whereby the parent individuals are chosen randomly from the given individuals. In an MBEA system, the recombination operator should fulll the following guidelines. First, similarity should be kept. That means that the offspring individual should not differ more from one of its parents than the parents differ from each other. It may be formalized as follows:
The second criterion for a recombination operator in a MBEA system is that it should not introduce an additional bias into the search process, i.e.,
Whenever problem-specic knowledge is given, the set of guidelines may be extended by further guidelines. Some guidelines that are especially useful for nonstandard representations have been described by Radcliffe and Surry (1994) and Radcliffe (1995) . They have been introduced for allelic representations, which are sets of features describing the individual. They are: Assortment: All combinations of features of the parent individuals are possible for the offspring individual Respect: Features that can be found in both parents are also features of the offspring individual Transmission: Each feature of the offspring individual is taken from at least one of the parents Linkage: Features that semantically belong to each other are transferred to the offspring individual in a coupled fashion 334
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The guidelines of assortment, transmission, respect, and linkage do not contradict the guidelines of the MBEA, so we will take them into consideration when designing the the graph-based EA in Section 4. Before this, the problem domain for the case study will be introduced.
3
Computer Aided Chemical Plant Design
In this chapter, the reader is introduced to the main aspects of computer aided design of chemical plants. It provides basic knowledge needed to understand the approach for an EA design presented in the next section. A chemical plant can be described as a complex network of chemical engineering devices connected by material, energy, and information streams. The graphical representation of a chemical process structure is called a owsheet (see Figure 3). The process optimization comprises the modication of a given process structure, as well as the adjustment of the unit operation parameters described above in order to nd the global optimum to perform a certain chemical task. A rigorous approach is followed (see Grossmann (1990)). Mixed-integer, non-linear programming techniques are applied to equation-based models of chemical process superstructures to nd the global optimum of the respective target value in a mathematical sense. These approaches require a predened superstructure that restricts the scope of variations. In addition, the target function denition has to accomplish mathematical conditions like convexity to guarantee global convergence. The process synthesis comprises the generation of an initial process structure as well. Unfortunately, some authors also denote process synthesis as an optimization process during which new structures are evolved. Nevertheless, the development of a feasible owsheet, starting with a rough idea concerning the desired product, consists usually of an iterative so-called evolutionary design process, which successively goes into higher levels of details. The denition of an initial owsheet can be supported by expert systems that contain relevant heuristics as well as thermodynamic laws and strategies of owsheet synthesis. Their operation relies in many cases on a rule-based algorithm that successively assimilates the raw materials and the products of a process with reference to their thermodynamic state, their composition, and their amounts (Siirola and Rudd, 1971; Mahalec and Motard, 1977). Apart from hierarchical decomposition approaches (e.g., Douglas (1995)), heuristic branch and bound techniques exist (e.g., Schembecker and Simmrock (1996)), and both require the continuous interaction with the supervising engineer. The engineering knowledge on which these expert systems rely may be regarded as a vast set of heuristic rules being ascertained by repetitive successful applications in former design processes, although without a pretension to be regarded as infallible. These methods will not guarantee nding the global optimum but strongly enhance the quality and speed of nding very good solutions to the problem in question. In order to evaluate process proposals, a detailed modeling and simulation of the process has to be performed. Therefore, the progress of simulation technique enhances the possibilities of process optimization. Commercial simulators, like ASPEN PLUSTM used in this paper, offer the features of calculating thermodynamic states, designing technical devices, and evaluating economic prot. The interaction between the optimizer based on EAs and the commercial simulator is shown in Figure 2. The owsheet represented in an optimizer-specic way has to be translated into an inputle for the simulator. The simulation performed with ASPEN PLUSTM determines all material streams comprising all variables of state and other process data that serves Evolutionary Computation Volume 9, Number 3
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Process definition
Translation into the inputfile of the simulator ASPEN PLUS TM Simulation
OPTIMIZER (EA)
Calculation of thermodynamical states
Cost calculations
Examination of simulation errors, Constraint Control Flowsheet Evaluation
Figure 2: Interaction between simulator and optimizer.
as a base for cost calculations. As many simulation errors are likely to occur, a detailed analysis of the severeness of simulation errors and a constraint control takes place. The results serve to modify the tness value (the product price in this paper) (Groß, 1999) and to evaluate the chemical process proposals generated by the EA. This is given back to the optimizer. 3.1 Modeling and Simulation of Chemical Engineering Processes The thermodynamic and chemical changes of state induced by the chemical engineering devices are described by different unit operations. The modeling of a chemical plant consists of developing a net of unit operations and specifying as many variables as demanded by an analysis concerning the degrees-of-freedom. As a result, we obtain a graph where vertices reecting unit operations are connected by edges symbolizing heat, material, and information streams. In addition, in most cases, raw materials, products, and the reaction mechanisms applied are concluded from experiments. These preliminary studies often reveal that certain conditions have to be maintained to carry out the desired reactions. The corresponding unit operations can be regarded as xed vertices during the optimization process. These xed vertices mostly imply constraints for some of the thermodynamic state variables. The simulation of a given owsheet demands the solution of a non-linear equation system, whereby the restricted areas for many variables with physical or technical meaning have to be considered. The equation system is given by the relationship between the input and output variables and parameters of the unit operations. ASPEN PLUSTM performs the steady-state simulation of continuous chemical processes applying a sequential-modular solution algorithm. It comprises net decomposition and tear stream denition as a owsheet often contains cyclic substructures. 336
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Split Ratio {0.0, 1.0}
Hydrogen Recycle
Material stream
Purge
Splitter Compressor
Pump
Temperature {895.2, 978.2} K
Toluene
Hydrogen (Methane)
Conversion {0.3, 0.98} Temperature {183.2,373.2} K Pressure {1.0, 25.0} bar
Mixer Heater
Pump
Furnace
Reactor
Pressure {5.0, 15.0} bar
Flash Cooler
Valve
Hydrogen, Methane
Condenser Utilities [BRINE, WATER]
Toluene Recycle
Valve
Benzene Pressures {1.0, 5.0} bar Numbers of trays {10,100} Numbers of feed tray {5,50}
Diphenyl Reboiler Utilities [D15, D45, D150, D250, D600]
Stabilizer-Column Toluene-Column
Benzene-Column
Figure 3: Flowsheet of the HDA process (taken from Douglas (1988)) with the different chemical devices connected by material streams. Fixed vertices are marked in grey. Associated to the devices are continuous, integer, and discrete parameters. 3.2 The Reference Process To illustrate the functioning of a chemical plant, the well-documented problem of a hydrodealkylation of toluene (HDA) (Douglas, 1988) serves as our reference process. We present some of the details of this process in order to convey the various constraints that might occur with this problem class. The aim of the HDA process is the production of benzene (Equation 1) with the formation of diphenyl (cf. Equation 2) being an undesired side reaction. (1) (2) The homogeneous gas-phase reactions (1) and (2) take place in the temperature range of 900-980 K at a pressure of about 35 bar. At higher temperatures, cracking might occur, and at lower temperatures, the conversion rate decreases rapidly. The reaction can be carried out in either an adiabatic or an isothermal reactor. A molar ratio of at least 5:1 hydrogen to aromatics has to be maintained to prevent coking. These chemical facts impose constraints to the thermodynamical state variables entering the reactor. Figure 3 shows a possible owsheet for the HDA process that serves as base conguration in the following. Fixed components of the owsheet are the toluene and hydrogen feed, the benzene product, and the reactor. As mentioned before, they imply constraints to their input state variables and should be left untouched during optimization. Furthermore, the xed structural parts divide the owsheet into three sections: 1) pretreatment (purication and heating) of the feeds, 2) reaction, and 3) separation, which Evolutionary Computation Volume 9, Number 3
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can be subdivided into phase splitting and vapor and liquid recovery (Douglas, 1988). Nevertheless, the choice of the relevant subsystems depends on the specic process. The toluene and hydrogen raw material streams are heated and combined with the recycled toluene and hydrogen streams before they are fed to the reactor. The product stream leaving the reactor contains hydrogen, methane, benzene, toluene, and the undesired diphenyl. We attempt to separate most of the hydrogen and methane from the aromatics1 by using a partial condenser to condense the aromatics, and then the light gases are ashed away. In order to prevent the accumulation of methane, which enters as an impurity in the hydrogen feed stream and is also produced by reaction (1), a purge stream is required in the gas recycle loop. Not all the hydrogen and methane can be separated from the aromatics in the ash drum, and therefore, we remove most of the remaining amount in a distillation column (the stabilizer) to prevent them from contaminating the benzene product. The benzene product is then recovered in a second distillation column, and nally, the recycled toluene is separated from the unwanted diphenyl. The cost of this process depends mainly on the size and choice of apparatus as well as on the energy and material demands of the process. Another important factor for the costs are the products, i.e., the purity of the main product and the mass-ow of the waste products. 3.3 Difculties Induced by This Problem Class for Optimization There are different classes of parameters involved in chemical process optimization. The most common classes of parameters, besides the structural modications of the owsheet, are: metric real parameters (e.g., pressure and temperature level, conversion rate of the reactor, splitting factors), integer parameters, which denote, for example, the number of trays (models for thermal separation steps) in a distillation column or dene where the feed stream enters a distillation column, nominal discrete parameters from a nite set of alternatives (e.g., the kind of fuel used in a furnace or the pressure level for heating steam). All parameters are taken from a limited domain of values. Sometimes, even the limits depend on each other (e.g., the number of the feed trays has to be lower than the total number of trays). The optimization of a chemical process turns out to be a challenging task as the modication of the process structure and its parameters’ details entail strongly correlated physical and economic effects. Therefore, it is not sufcient to deal with single devices or functional parts of the structure to obtain the global optimum, but one has to examine the entire owsheet. Apart from constraints to the network structure, many parameter settings entail strongly correlated effects as the change of one state variable inuences the values of other state variables. This may change the suppositions for the setting of a unit operation which is not in the neighborhood to the one with the modied parameters. For example, a small change in the pressure or temperature level may impose changes 1
Aromatics are toluene, benzene, and diphenyl.
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to the state of aggregation. As many unit operations demand a dened state of aggregation, the choice of an appropriate one may have to be re-taken or the owsheet becomes infeasible. We can conclude that the assessment of the impact of certain parameter modications on the process performance during the optimization process is not obvious.
4
Evolutionary Algorithms for Process Optimization
This section deals with the problem-specic individual representation and the genetic operators that work on it. The choice of these components has been based on the guidelines given in Section 2. Additionally, a distance measure for variable dimensional search spaces will be introduced. 4.1 Graph Representation As mentioned in Section 2, the choice of a representation is a crucial step in the design of an EA. Most of the data structures for the process representation that can be found in the literature are matrix or string based (Grossmann and Yeomans, 1999). These representations are useful for the analysis of static chemical process structures, but applying them to structure optimization often entails further restrictions. Additionally, they require a high amount of memory. Up to now, there are only a few approaches based on graph representations reecting chemical process structures for optimization tasks. For instance, Friedler et al. (1992) suggested using so-called P-Graphs for structure synthesis. Unfortunately, these graphs are not suited for the representation of detailed structures, because they only take nite domains of states for the material streams under consideration. With this paper, we will introduce another type of directed graph as a suitable representation, the so-called -Graphs (Structure-Graphs). The advantage of this type of graph is its applicability to the formulation of algorithms that work on graphs with variable structure. Furthermore, when using -Graphs as data-structures, no redundant information about the process structure has to be stored as it does in many matrix representations. Each vertex of an -Graph represents a unit operation. An edge represents a stream from an exit of a unit operation to an entry of another unit operation. Exits and entries are represented by the so-called outlet- and inlet-connectors, and an edge is a pair of one inlet- and one outlet-connector. Each vertex has a dened number of inlet- and outlet-connectors. Each inlet- and outlet-connector is dened as a tuple of an index and a vertex. The index is needed to distinguish between different inlet (or outlet) streams of the same vertex. This is important because each of the entries and exits of the unit operations may have an individual function. For example, we need to distinguish between the gas outlet stream and the liquid outlet stream describing a ash. This leads to the following denition of -Graphs: D EFINITIO N 2: An -Graph is a tuple , where is a set of vertices, is the subset of xed vertices of , and are functions of the type , is a set of edges with , and is a function that assigns an element of a nite set of operation-types to each vertex.
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The set can be derived from the problem denition and contains vertices that have to be integrated in all feasible process structures (cf. Section 3.1), e.g., the vertices that represent the main product, the reaction, or certain raw material feeds. The following denition describes the set of inlet- and outlet-connectors of the graph, which are pairs of indices and vertices: D EFINITIO N 3: Let inlet-connectors for
denote an -Graph. Then the set of all is given by
Similarly, the set of all outlet-connectors for
is given by
Furthermore, the indegree of a vertex denotes the number of incoming edges, whereas the outdegree denotes the number of outgoing edges of this vertex. If we represent the unit operations of the chemical process as vertices and its streams as edges, then almost every process structure can be represented as an -Graph. E XAMPLE 1: The as
-Graph ,
shown in Figure 4 is dened ,
,
On the other hand, not every -Graph leads to a reasonable process structure. A signicant number of infeasible process structures can be excluded from the search space by forcing -Graphs to respect the axiom in the following denition: D EFINITIO N 4: A complete connected -Graph ( -Graph) is dened as an -Graph that fullls the following axiom:
Under this axiom, represents a one-to-one mapping from the set of outletconnectors to the set of inlet-connectors . This axiom is reasonable because it assures that every outgoing stream of a unit operation should lead to exactly one entry of another unit operation or to one of the products, and every incoming stream should have a source in exactly one exit of a unit operation or in one of the feeds. To get a consistent model, we represent feeds (products) of the whole process as vertices with in-degree (out-degree) 0 and out-degree (in-degree) 1. Whenever optional 340
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Figure 4: The gure shows an example for a small -Graph. The black circles denote inlet-connectors and the white circles outlet-connectors. Connector indices are printed next to the connector if these are more than one.
inlet- or outlet-connectors appear, we represent the unit operation by different operation types. For instance, one mixer type might have two inlet streams and another type three inlet streams. By comparing the search space size for feasible -Graphs that can be combined from a given nite sets of possible inlet- and outlet- connectors, we will demonstrate how much the search space complexity decreases if we apply this apparently simple restriction (cf. Emmerich (1999)). Let and denote maximal nite sets of possible outlet- and inlet-connectors, i.e., all connectors of vertices that might be integrated into the graph. For reasons of simplicity, we assume that both sets contain the same number of elements, i.e., . Then the number of all possible one-to-one mappings from subsets of to subsets of is given by . This determines an upper bound for the number of valid -Graphs , with and if we take no graphs with isolated vertices2 under consideration. In comparison to this, we would get more than possible -Graphs. This number equals the number of possible sets of connector pairs comprising connectors out of at the rst and connectors out of at the second position. Thus, we achieve a signicant reduction of the search space by the application of the axiom given in Denition 4. Unfortunately, we still have to cope with a search space that, at least in the worst case, grows exponentially with the number of its possible connectors. Therefore, the application of heuristic algorithms for the graph optimization is still justied. An -Graph is a very intuitive representation for process structures, with a simple graphical illustration (cf. Figure 4). From this representation, the connectivity of the process can easily be derived. In the following, -Graphs will represent structures of chemical processes. The individual representation for the EA will then consist of an extension of -Graphs with parameterized edges and vertices and is introduced next. 2
Isolated vertices are vertices with in-degree and out-degree of zero.
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4.2 Parameter Functions for Vertices and Edges In order to represent unit operations and streams with all the information needed for heuristic analysis, owsheet simulation, and optimization, the -Graphs have to be extended to parameterized graphs. For this reason, each vertex of the graph has been assigned a tuple consisting of integer, discrete, and real-valued parameters that are optimization variables or serve as variables in heuristic calculations. Furthermore continuous parameters are assigned to the edges to manage thermodynamic states like pressure, concentration and temperature, whenever heuristic calculations need to be performed. The parameterized -Graphs will be denoted as graphs. We have designed a heuristic procedure for the reduction of the parameter search space of a suggested conguration in order to accelerate the local search procedure. This heuristic reduces the ranges of the optimization parameters without excluding feasible variables from the search space. The rules are specic, interval-based descriptions of the function of each unit operation, estimating the range of possible values for the physical state variables at each outlet depending on intervals for physical state variables at the inlets and specications for the apparatus parameters. A detailed description is given in Emmerich (1999) that focuses on structural variations and not on the search for a good parameterization for a given conguration. 4.3 Distance Measures between Graphs The question has not yet been answered whether a metric on the structure search space can be dened. While distance measures for vectors of constant dimensions are well known, little is known about distance measures between graph structures. We introduce a general concept for a distance measure, which can easily serve to dene the distance between two -Graphs. The distance measure will be related to sets of simple, symmetric search operators called minimal moves that dene minimal transitions between elements of the representation space. The weight of a transition will be determined by the expected change of the tness value caused by the transition. The main advantage of this distance measure will be that it is is easy to adapt to different problems on discrete variable dimensional representations. The knowledge about smooth transitions between structures can be expressed as a set of minimal moves. These minimal moves then serve as the basis for the denition of a problem-specic metric as well as for the formulation of the mutation operator that respects this metric in the sense of the MBEA guidelines. After some preliminary denitions, we will state a theorem on how a distance measure can be formulated whenever a set of minimal moves is given. Since the set of multisets is isomorphic to the set , for reasons of simplicity we will just write when declaring the scope of search operators. D EFINITIO N 5: A set of search operators (minimal moves) will be called a complete set of minimal moves if and only if it satises the following conditions: 1.
, i.e., all search operators are symmetric
2.
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Whenever the expected impact of the transition a minimal move describes is known, this knowledge can be represented by a weight assigned to that operator: D EFINITIO N 6: Let denote a complete set of minimal moves and a function assigning a positive weight to each operator. Then the pair will be called a complete set of weighted minimal moves. It shall be emphasized that assigns a strictly positive number to each operator. This will be essential for the denition of a metric related to such a set of operators. Given a complete set of (weighted) minimal moves, a distance measure can be dened. Before introducing this measure, we make some useful denitions. Let be a set of weighted minimal moves: D EFINITIO N 7: Let denote the set of all tuples (sequences) of elements from . The function assigns the sum of weights of all minimal moves to each (non-empty) sequence of minimal moves, i.e.,
D EFINITIO N 8: To each pair , the function assigns the set of sequences of minimal moves by which successive application a transition from element to element is possible, i.e.,
T HEOREM 1: Distance measure for discrete search spaces: Let denote a discrete representation space and a complete set of weighted minimal moves. Then with if otherwise and
denes a distance measure.
P ROOF : The proof shows that is a metric on in the mathematical sense. This means the identity on zero, the symmetry axiom, and the triangular inequality are fullled. In this proof, will be abbreviated by . 1. Identity on zero ( Case Case
),
. 2. Symmetry Case Case
:
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3. Triangular Equation: The proof for is trivial, since all distances are sums of positive numbers or equal to zero. For the case , we show that the assumption leads to a contradiction. Let ,
and
describe transitions with and
describes a transition from
to
, . Then
with
. Now, it is obvious to see that the assumption would lead to contradiction. It shall be remarked that it is essential in this proof that all weights are strictly positive and that the distance measure is dened by the sum of weights (not, e.g., by their product). Furthermore, symmetry is of importance, i.e., the fact that for each procedure, a reverse procedure is dened to which the same weight is assigned. To apply this concept to a search space of -Graphs, a complete set of weighted functions has to be dened. This is done in Section 4.4. As one can see, this method provides a exible way to dene a problem-specic metric on graph-based search spaces. Given a set of minimal moves that leads to smooth transitions in the search space, a problem-specic formal metric can be derived. This metric can provide the basis for the formulation of search operators of an MBEA system. Furthermore, it enables us to quantify characteristics of the tness landscape induced by the minimal moves, like its tness distance correlation (Jones, 1997). 4.4 Genetic Operators Up to now, we have introduced a graph representation together with a suitable concept of a distance between objects in that representation. In the following, we look at problem-specic genetic operators. Mutation The algorithm for the mutation operator can be subdivided into the parameter mutation and the structure mutation, i.e., the mutation of the process network connectivity. For the parameter mutation, standard mutation procedures for real, integer, and discrete parameters, as described in Rudolph (1994), Schwefel (1995), and Schutz ¨ (1996), are combined. A detailed, formal description of procedures for the employed mixed integer parameter mutations is given in Emmerich (1999). Besides the parameter mutation, the network structure of the process has to be mutated, too. In this study, a minimal moves mutation operator is derived from a given set of minimal moves. The idea to employ minimal moves for the mutation of discrete structures is not new and can be found elsewhere (Radcliffe and Surry, 1994). For the mutation of the -Graphs, these minimal moves take only information about the type of vertices and their connectivity into account. They recognize existing subgraph patterns and replace them by other, similar patterns. Furthermore, we assume that for a 344
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Algorithm: Minimal Moves Mutation Input: : original individual, : mutation strength, minimal moves and function Output: : mutated individual, mutated and mutate and with logarithmic normal distribution (Emmerich et al., 2000) choose with a geometrical distribution with mean (Emmerich et al., 2000) , while do choose randomly with ; ; for all do if then ; else if then end if end for end while Figure 5: Sketch of the search procedure minimal moves mutation neighboring structures. The neighborhood search is based on Dijkstra’s algorithms for shortest paths in a graph (Guting, ¨ 1992; Dijkstra, 1959). The operation tests if there is a structure isomorphic to in . A fast graph isomorphism test for labeled graphs is given by Ullmann (1976).
given , the minimal moves assign the same probability to all neighboring structures. The minimal moves mutation operator described in Figure 5 is designed to meet the local search property, as it is required for MBEA systems. A geometrically distributed value is assigned to an integer variable, determining the strength of the mutation. The distance measure introduced in Theorem 1 will be used in this study, thereby considering a set of minimal moves that is also used to generate neighboring structures in the mutation and a positive weighting function on this set. The randomized search procedure for neighboring structures is designed similarly to the algorithm of Dijkstra for shortest paths in networks. The working principles of this algorithm are described in Guting ¨ (1992) and Dijkstra (1959). It collects neighboring structures in the order of their distance to the original individual due to the distance measure for discrete search spaces. Structures having equal distance to are visited in random order. The algorithm stops after the th nearest neighbor has been collected and returns this structure as the mutated solution. We will now analyze whether the guidelines of an MBEA (similarity, ergodicity, and symmetry) are met by the proposed algorithm. Since the probability of the geometric distribution (Emmerich et al., 2000; Rudolph, 1994) decreases for increasing , it is plain to see that the probability of a solution to be obtained by mutation decreases with its distance to the original solution, as it is required for the MBEA. Since the support of the geometric distribution is innite, every could occur, and therefore, every Evolutionary Computation Volume 9, Number 3
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structure in is a possible result of the MBEA. Thereby, the ergodicity guideline of MBEAs is met if the (problem-specic) set of minimal moves allows transitions from any element of the search space to any other. Last, the symmetry guideline is met because structures with equal distance to the original individual appear with the same probability. The reason for this is that such structures are always visited in random order, each order with the same probability, by the mutation procedure. It seems to be advantageous to adapt the weights of the minimal moves during the evolution. This has been benecial in the case study. Furthermore, the width of the geometrical distribution is adapted by a mutative self-adaptation schedule. For both adaptations, a logarithmic normal distribution, described in Schwefel (1995) and Rudolph (1994), has been applied. Nevertheless, it might be of practical importance to limit in order to achieve moderate computation times. Ergodicity, the other guideline of an MBEA, says that it is always possible to get from one structure in the representation space to every other structure. Whether this guideline is met or not depends on the choice of problem-specic minimal moves. Instead of employing a general set of minimal moves on -Graphs, we dened a problem-specic set for the test case by which each relevant structure could be reached. This is due to the fact that only a very low ratio of possible -Graphs leads to feasible chemical process networks. In the following, we will sketch the minimal moves employed for the test case of a HDA-plant (cf. Section 3.2) (Emmerich, 1999). Each of these moves follows the principle of replacing a given subgraph pattern with another, semantically similar, subgraph pattern. To keep the symmetry condition, every replacement procedure is reversible: 1. Insert (delete) a heater parallel to an existing heater 2. Insert (delete) a pump (valve) parallel to an existing pump (valve) 3. Insert (delete) a sequence compressor-cooler followed by a compressor 4. Insert (delete) a heater in sequence to an existing heater 5. Swap a stabilizer column with a cooler (valve) and ash combination 6. Swap two columns in a sequence of two liquid separation columns 7. Swap a by-product stream with a recycle stream Figure 6 illustrates the pair of modication operators ins comp seq and their inverse procedure del comp seq, whereas Figure 7 describes the pair of modication operators exc column flash and their inverse procedure exc flash column. The graphical notion is lent from chemical reaction diagrams. Since it is very intuitive, we omitted a further formal description of the minimal moves. The upper parts of the diagrams show the original pattern in the graphs and new vertices that shall be inserted in the graphs. The lower parts show the resulting graph patterns and the vertices that left the graph. The inverse operators (like the reversible reaction) would be presented if the drawings were ipped upside down. Rectangular boxes dene vertices of the graph that are connected with the graph but for which the type of unit operation is of no relevance for the modication operator. By an application of ins comp seq, two vertices with types cooler and compressor are inserted as a sequence into the graph, and del comp seq deletes such a sequence from the graph. The alternative subgraphs represent quite similar physical functions, 346
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Compressor
Cooler
+
+
f1-1
f1
Figure 6: Illustration of the minimal moves ins comp seq and del comp seq. A further explanation of this diagram can be found in Section 4. 2 2
Stabilizer Column
1
+
+ Cooler
f2
f2-1
Flash 1
2
2
+ 1
1
Figure 7: Illustration of minimal moves exc flash column and exc column flash. but the compression with an intercooler has a lower energy consumption than the alternative without an intercooler. On the other hand, it has a higher equipment costs. Through the application of exc column flash and exc flash column, a stabilizer column is replaced by a combination of a cooler and a ash. Both subgraphs represent a part of the chemical process that is removing gas (light ends) from a liquid stream. Initialization Another important evolutionary operator is the initialization of chemical process structures. The generation of a feasible start solution requires a lot of engineering knowledge and experience. Since the design of an expert system for the initial generation of chemical process structures would be no simple task and would require a large knowledge base (cf. Schembecker and Simmrock (1996)), an automatic initialization has not yet been designed. Here we plan to integrate a probabilistic expert system for the generation of initial chemical process structures that will be developed in another research project (cf. Gro¨ tzner and Roosen (1999) and Grotzner ¨ (2001)). In this article, we start with a user-dened initial process. The diversity of the start population should be high. Therefore, the structure mutation procedure is applied several times on the initial structure to produce the initial population. Recombination Last, a simple recombination operator for chemical process networks will be introduced. The application of a standard recombination operator to process Evolutionary Computation Volume 9, Number 3
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parameters is quite difcult if the population consists of differing process structures. The reason for this is the different length and physical meaning of the parameter lists of the individuals in the population. In a hierarchical structure evolution, no such problem occurs. This is due to the fact that the structure is kept constant in each of the subpopulations. For the simultaneous structure evolution, we have designed a recombination operator that obtains one Graph from two parental -Graphs, operating on both structure and parameters. The so-called subsystem recombination utilizes the heuristics that many processstructures can be partitioned into a few subgraphs, which represent functional units of the process. For a suitable partitioning of a chemical process the reader is referred to Section 3.2, where we introduced several subsystems that build up a typical process. Generally, such subsystems should be chosen in such a way that they have a uniform interface and perform similar functions. The subsystem recombination algorithm builds an offspring individual by choosing each subgraph randomly from one of the parental individuals, thereby building up a new -Graph. Subgraphs that perform a similar function are exchanged. Therefore, it can be expected that we get a high ratio of feasible graphs. The subsystem recombination is based on the guidelines for representation independent EAs. Assortment, respect, and transmission are fullled if we consider subsystems as features of a genetic representation. Assortment is present because every feasible combination of parental subsystems is a possible output of the recombination procedure. Transmission and respect are present because every subsystem in the offspring individual has its origin in one of the parental subsystems. Since subsystems sum up semantical units of the process that are inherited in a coupled fashion, linkage is present, too. The guidelines of a MBEA system can be ensured under the condition that one minimal move considered for the metric never changes the structure of more than one subsystem at the same time. In this case, the distance measure can be decomposed as the sum of distances between the subsystem graphs. The guideline of similarity is kept since the offspring individual will never have a lower number of subsystems in common with one of its parents than the parents have in common with each other. P ROOF : Let denote two arbitrary parents graphs and denote an arbitrary offspring graph. Furthermore, let be the distances between the subsystems and be a binary vector determining from which parent each subsystem is inherited by the offspring. Then
The second guideline, that the recombination is bias free, is met since the subsystems are chosen with equal probability from one of the parents.
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P ROOF : Let be two arbitrary parental graphs. Let between parental subsystems. Let
be the distances , then:
4.5 Hierarchical and Simultaneous Evolution One main characteristic of the given problems is that a network structure as well as its parameterization needs to be found. The underlying problem may be classied as a mixed integer optimization problem, when we code the decisions on the network structure as discrete variables. But it needs to be emphasized that the meaning of such discrete parameters has to be distinguished from the meaning of discrete parameters in the parameterization of a given structure, because their value entails the existence of other, continuous and discrete variables. Similar problems have been described by Lohmann (1991), who investigated the evolution of neural networks with variable topology. He terms this “structure optimization” and the corresponding algorithm “structure evolution.” Whenever the optimization of parameters has a signicant inuence on the order of tness values for different congurations compared to the order of tness values with initial parameterizations, we talk of an hierarchical problem. In this case, it is advantageous to detect the parameterization of a given conguration in each iteration by a parameter sub-optimization. The corresponding EA will be termed a “hierarchical structure evolution.” On the other hand, if the optimization of the parameters has only a small inuence on the order of tness values for different congurations compared to the order obtained by their initial parameterization, it is recommended to omit the local search and optimize both parameterization and congurations in the same loop. In this case, the resulting EA will be called “simultaneous structure evolution.” For the given application, the latter one is the only feasible approach, since the rst approach needs too much function evaluation. A summarizing description of this evolutionary algorithm for process optimization has been given in Figure 9. The results of the simultaneous structure evolution on the reference example are summarized in Table 1. The adaptation of mutation rates for the structural genetic operators is important, mainly to enable a ne tuning for the parameterization of the optimal conguration at the end of the search. This enables the use of smaller population sizes. The solution that we have obtained corresponds to the solution that Gross (1999) found with a mixed integer approach using a superstructure and a -ES. As shown in Table 1, the evolutionary algorithms with constant numbers of structural mutations perform better in the beginning, but soon a stagnation of the tness value can be observed. The strategies with adaptive numbers of structural mutations are able to nd much better solutions. This is due to the fact that in the beginning of the evolution, structural mutations are likely to lead to improvements. Therefore the search process benets from a high mutation rate. In the later stages of evolution, it is more important to conserve the found structure of the owsheet in order to allow a ne tuning of the design. Here a small mutation rate is advantageous. Evolutionary Computation Volume 9, Number 3
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Hydrogen Recycle
Material stream
Split Ratio {0.16}
Purge
Splitter Compressor
Pump Toluene
Conversion {0.98} Temperature {23 °C}
Hydrogen (Methane)
Pressure {25.0 bar}
Mixer Heater
Furnace
Reactor
Pressure {5.0, 15.0}
Pump
Flash Cooler
Valve
Cooler
Condenser Utilities {WATER}
Biphenyl,Toluene Recycle
Hydrogen, Methane Benzene
Valve
Pressure {1 bar} Number of trays {22} Number of feed tray {5,50}
Pressure {25.0 bar}
Reboiler Utility {D45}
Stabilizer-Column
Figure 8: Flowsheet of the optimized HDA process. In the optimal solution (Figure 8), the main improvements are the replacement of the rst distillation column by a ash and a cooler, the introduction of a recycling stream for biphenyl byproduct (in this case, a column has been removed but the size of nearly all apparatus have been enlarged), and the choice of optimal utilities for the distillation column. Besides, about 18 continuous and integer parameters for the conguration have been adopted. Computational efforts of simulations have been very large, so that by now we have only a few test runs of the algorithms for the HDA test case. For more empirical studies, the reader is referred to extensive investigations on a similar problem, the optimization of heat exchanger networks, where a similar graph-based approach has been successfully applied on several benchmark problems (B¨ack et al., 1998). Furthermore, the algorithm has been applied successfully for the optimization of feed water strings of thermal power plants (Hillermeier et al., 2000).
5
Summary and Outlook
In this paper, we presented a graph representation and concepts for the design of genetic operators that can be applied for the optimization of chemical plants. A special set of genetic operators has been dened for the reference problem – the HDA process optimization. A problem representation with typed vertices and variable dimension has been developed, which keeps essential syntactic constraints for the conguration of owsheets. A technique for the development of problem-specic search operators has been introduced and applied to the reference example of the HDA plant optimization. The technique adheres to guidelines known from EA literature. A suggestion for the denition of a metric on variable dimensional discrete search spaces has been made based on weighted sets of problem-specic minimal moves. 350
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Algorithm: - Evolution strategy on -Graphs t := 0 initialize Set (Population) of -Graphs Extend Graphs to by generating intervals with parameter estimation algorithm evaluate all -Graphs while Termination criteria not fullled do recombine Graphs from by exchanging subsystems (see Section 4) mutate the Graphs applying minimal moves (see Section 4) Estimate parameter domains for each new Graph mutate parameters evaluate the Graphs (after parameter optimization, if hierarchical structure evolution is used) select the best Graphs for from new Graphs and old Graphs from with age lower than end while Figure 9: Main loop of a
-ES applied for
-Graph evolution.
Table 1: Comparison of different evolutionary algorithms on the test case. For each strategy and different numbers of generations (GEN), the difference of the average tness for 5 runs to the best known tness of the HDA plant is given. The (5, 5, 20) EA and the (15, 5, 100) EA have been compared, both with adaptive and constant number of discrete mutations. Gen 10 25 50 100 150 200
(5, 5, 20) adaptive 0.09 0.05 0.05 0.04 0.02 0.007
(5, 5, 20) constant 0.05 0.05 0.05 0.05 0.05 0.05
(5, 5, 100) adaptive 0.02 0.015 0.01 0.008 0.007 0.002
(5, 5, 100) constant 0.008 0.004 0.004 0.003 0.003 0.003
Finally, different algorithmic variants – hierarchical and simultaneous structure evolution – have been introduced. Moreover, a mutative adaption of individual mutation probabilities for mutation operators has been described and successfully applied on the test case. There have been other publications that focus on further aspects of this algorithm. The treatment of continuous, integer, and discrete parameters, which has been a combination of standard techniques, is described in detail and investigated in Emmerich et al. (2000). The knowledge-based estimation of the parameter domains for chemical process networks (ICPN-algorithm) is described in Emmerich (1998). For further details on the problem and a superstructure approach for the test case, we refer to Groß(1999) and Douglas (1988). For an overview of our work on related chemical engineering Evolutionary Computation Volume 9, Number 3
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problems, the reader is referred to Emmerich et al. (2000). Finally, a comparison of the hierarchical and simultaneous EA approaches can be found in Olschewski (1999) and Schutz ¨ (1996). The proposed technique is an alternative to the common MINLP algorithms. Since it requires no denition of a superstructure, it is especially well suited to design problems with a large variety of possible structures, like the design of heat exchanger networks and power plants. The modular representation of the plant enables the convenient integration of knowledge into the operators and algorithm. Further research must focus on this knowledge integration in order to achieve a fast exploration of complex search spaces. Acknowledgments The authors would like to thank Thomas B¨ack and Dirk Wiesmann for their valuable comments. Monika Grotzner ¨ gratefully acknowledges support by the German Society of Research (Deutsche Forschungsgemeinschaft). Michael Emmerich and Martin Schutz ¨ gratefully acknowledge nancial support by the German Bundesministerium fur ¨ Bildung, Wissenschaft, Forschung und Technologie, grant 01IB802B. The authors are responsible for the contents of this publication.
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