Differential Evolution: Method, Developments and Chemical Engineering Applications
2.4 Chemical Engineering Applications
With its ease of use, DE is a good choice for a general purpose optimizer.
It has been used in many Chemical Engineering applications, which can be classified into two types:
(1) Model building and parameter estimation, where the aim is to reduce model error and then to control a process or to determine unknown coefficients. This includes neural network training, regression analysis and other modeling methods.
(2) Process optimization, where the goal is to maximize yield or maximize profit by determining the best operating conditions. Decision variables are mainly process variables (flow rate, temperature etc.)
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Tables 2.7 and 2.8 summarize the Chemical Engineering problems where DE has been applied successfully. In addition, Srinivas and Rangaiah (2007c) evaluated DE with tabu list (DETL) on many application prob-lems taken from the literature, which are either NLPs or MINLPs. Their results show that DETL requires, on average, about 40% fewer objective function evaluations compared to both DE and modified DE of Babu and Angira (2006).
2.5 Conclusions
Since its inception in 1995, DE has undergone many changes and devel-opments, and is now one of the effective global optimization techniques.
This direct search algorithm, based on nature’s evolution, has the benefit of being simple and instinctive for beginners to understand, and yet power-ful and flexible for researchers to customize for their applications. DE code and concepts are readily available on the internet (http://www.icsi.berkeley.
edu/∼storn/code.html), and it is good as a general, global optimizer with a few parameters. With these advantages, it is not surprising that DE has been used in many applications in Chemical Engineering, where the bulk of the problems involve continuous variables, which is what DE is adept at. Fur-thermore, developments in DE, summarized in this chapter, have improved its capabilities to tackle a variety of problems (namely, constrained, mixed integer, multi-objective and large dimensioned), provided modifications which can be customized to suit the problem, enhanced the speed and reliability of the method, and made tuning of its parameters easier. Given these developments and merits, DE is attractive and useful both as a simple general optimizer and as a sophisticated tool to solve Chemical Engineering problems.
Exercises
For the following exercises, use the code Rundeopt.m in the Supplementary Material available on the book website for solving the modified Himmelblau function (equation 2.2). This and other m-files in the folder are based on the DeMAT software of Price et al. (2005). MATLAB software is required for running these m-files. Suggested changes in the tuning parameters will have to be made in Rundeopt.m file and saved before running. The animated
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56S.Chen,G.P.RangaiahandM.Srinivas
Table 2.7 Applications of DE in modeling and parameter estimation
Application (Reference) Objective
Selected/Independent decision
variables Remarks
Estimation of heat transfer parameters in a trickle bed reactor (Babu and Sastry, 1999)
Sum of squared errors Thermal conductivity of bed, wall heat transfer coefficient, effective diameter of packing, liquid and gas flow rates
DE is faster and more reliable compared to radial temperature
Mean squared errors in a weighted min-max
Uses hybrid DE which includes acceleration and migration operations
Modeling (1) true boiling point curve, and (2) effect of pressure on entropy, of crude oil (Chen et al., 2002)
Sum of squared errors Weights in the feedforward neural network models
Uses an improved DE which includes Levenberg-Marquardt
Weights of fuzzy cognitive maps DE is found to be robust, effective and efficient for the problems studied
Five kinetic and other parameters Model simplifications could lead to erroneous results. DE is faster than GA for this application Differential Evolution in Chemical Engineering Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 06/03/17. For personal use only.
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DE:Method,DevelopmentsandApplications57
biodegradation of phenol case study (Bhat et al., 2006)
and actual ones other parameters in the biofilm and kinetic models
improves the efficiency of DE
Macro-kinetic model for oxidation of p-xylene (Yan, 2007)
Mean of squared, relative errors
Weights and thresholds in the neural network model
Four rate constants are modeled by an artificial neural network First order liquid phase
reaction and catalytic gas oil reaction (Angira and Santosh, 2007)
Fractional difference variation
A few rate constants Trigonometric DE was found to be faster than DE
Estimation of biomass and intracellular protein in E . Coli (Ko and Wang, 2007)
Mean squared errors Tunable parameters in the model Hybrid DE of Chiou and Wang (1999) was used
Fuzzy inference systems for modeling four-effect evaporator and continuous stirred tank reactor (Eftekhari et al., 2008)
Mean squared errors Membership function width and position
DE was superior to ANFIS, a well-known technique for fuzzy technique
(Continued )
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Three to four parameters in the expression for thermal diffusivity as a function of temperature and moisture content
DE was successful in this application
Controller design for the ALSTOM gasifier
(Nobakhti and Wang, 2008)
Integral of absolute error 20 tuning parameters in the proportional-integral controllers
DE with an adaptive mutation factor (F) was used
Rate expression for sulfur dioxide oxidation, and diesel catalytic cracking reaction (Wu et al., 2008)
Error between predicted and actual values
Six parameters in sulfur dioxide rate expression and three kinetic parameters in cracking reaction
Immune DE for these problems was faster and more reliable than the classic DE
Soft sensor for naphtha 95%
cut point in crude
distillation unit (Yan, 2008)
Mean of squared errors Parameters in the ridge regression estimator
DE with modified mutation operator was employed
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DE:Method,DevelopmentsandApplications59
Fed-batch fermentation process (Chiou and Wang, 1999)
Maximize ethanol production Glucose feed concentration and flow rate, initial glucose concentration, initial volume and fermentation time
Using hybrid DE (with migration and acceleration operations)
Number and size of parallel equipment at each stage, and location and size of storage
The multi-objective optimization (MOO) problem was converted into a fuzzy goal optimization problem, which was solved using mixed integer hybrid DE Optimal control problems in
(1) continuous stirred tank reactor and
(2) bi-functional catalyst blending (Cruz et al., 2003)
(1) Minimize squared
Trajectory of cooling fluid flow rate in problem 1, and trajectory of mass fraction of the hydrogenation catalyst in problem 2
DE was concluded to be reliable and relatively efficient after the first stage, for lactic acid production (Wang and Lin, 2003)
Maximize lactic acid productivity and glucose conversion simultaneously
Biomass, glucose and lactic acid concentrations, feed glucose concentration, dilution rate, flow rate and bleed ratio and volume ratios
The MOO problem was converted into a fuzzy goal optimization problem, which was solved using hybrid DE.
Inclusion of extractor improves the process
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April20,201717:3DifferentialEvolutioninChemicalEngineering9inx6inb2817-ch02page60 converted into a fuzzy goal optimization problem, which was solved using hybrid DE Fermentation process for
Trajectory of feed rate (in both applications), feed
concentration of glucose, initial concentration of glucose and initial working volume (in ethanol process only)
Non-uniform control vector parameterization was used.
Case of two feeds was also considered
Adiabatic styrene reactor (Babu et al., 2005)
Maximize yield, selectivity, and productivity of styrene
Ethyl benzene feed flow rate and temperature, inlet pressure and steam to reactant ratio
Multi-objective DE was proposed and used for solving the MOO problem units, in flow, out flow, temperature and composition
MINLP problems solved by the modified DE
Alkylation process operation (Babu and Angira, 2006)
Maximize profit Olefin feed rate, acid addition rate, alkylate yield, acid strength, motor octane number, isobutane to olefin ratio
Solved by the modified DE
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2006)
(1) Isothermal CSTR, and (2) batch reactor for A→B→C (Babu and Angira, 2006)
Solved by the modified DE
Fed-batch fermentor for protein production using E. Coli (Ko and Wang, 2006)
Maximize protein production rate
Feed rate trajectory, feed glucose concentration and
fermentation time
Hybrid DE was used, and two additional experiments were conducted for validation (1) Plug flow reactor for A→B
and A→C, and (2) Batch reactor for A→B→C (Angira and Santosh, 2007)
Maximize yield of product B
Trajectory of (1) the control variable, and (2) reaction temperature
Used trigonometric mutation in DE
Plug flow reactor catalyst blend (Angira and Santosh, 2007)
Maximize yield Composition of catalyst, mole fraction of components
Maximize yield Composition of catalyst, mole fraction of components
Multi-modal problem, solved by trigonometric mutation in DE
(Continued )
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April20,201717:3DifferentialEvolutioninChemicalEngineering9inx6inb2817-ch02page62 (for PE), and tangent plane distance function (for PS)
Moles of each component in each phase (for PE), and mole fraction of each component (for PS)
DE was shown to be more reliable but computationally less efficient compared to tabu search
Shell and tube heat exchanger design (Babu and
Munawar, 2007)
Minimize cost of exchanger Tube diameter, length, pitch, passes, shell head type, baffle spacing and cut
DE faster than GA for the design problem studied
Solvent mass flow rate and binary variables for appearance and position of a structural group in the molecule
The mixed integer nonlinear programming problem was solved by mixed integer hybrid DE
Lactic acid recovery by esterification in a CSTR and hydrolysis in a reactive distillation (Rahman et al., 2008)
Minimize total annual cost CSTR temperature, number of stages, reflux ratio, feed
Minimize total annual cost String vector (for arrangement of exchangers), heat load, heat capacity flow rates of split streams
Solved five case studies taken from the literature, and found better HENs in some cases Differential Evolution in Chemical Engineering Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 06/03/17. For personal use only.
plot produced by this code shows how the population in each generation varies after the mutation and crossover operations. The red circles represent the population vectors. Run the code several times and observe how DE searches the terrain. Record the success rate of the algorithm in finding the global minimum and the average number of iterations it takes.
2.1. Try Np = 5, 10, 20, 25 and 50, and run the code several times for each setting. What happens to the success rate and the speed of convergence?
Does the search gets trapped in a local minimum? What is the best set-ting, which achieves a balance between the success rate and speed of convergence? Is the recommended tuning guideline of Np = 10N rea-sonable?
2.2. To understand the effect of F , try a very low value of F (e.g. 0.1).
Use the best setting for Np found in the previous exercise. Does convergence happens? Also, try very large values of F (e.g. 1.5 and 2). Is convergence fast or slow? Is the recommended F ≈ 0.8 a good choice?
2.3. Choose a low value of Cr, say 0.1 along with Np = 20 and F = 0.8.
Note the movement of the points (red circles) in the search domain.
Do they move a lot, or is it much less than the case when Cr is high? Is convergence fast or painfully slow? What does this tell about the use of Cr in increasing variation?
2.4. Try initializing in an area which does not include the global minimum, for example from 0 to−6 for both variables. Is it still possible for con-vergence to the global minimum or does it happen at a local minimum?
Is DE able to search outside its initial space? What is the success rate of locating the global minimum as compared to the original case where initialization is done over the whole feasible space?
2.5. Change the value to reach (F_VTR in the code) to a bigger number such as 0.01 instead of 0.000001, and solve the problem several times.
Does DE algorithm converge significantly faster? What does this show about DE? Is the algorithm built to find an accurate answer? If not, suggest ways to find accurate answer efficiently.
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