multiobjective optimization of simulated moving bed by tissue p system

Chin. J. Chem. Eng., 15(5) 683-690 (2007)

Multiobjective Optimization of Simulated Moving Bed by Tissue P System*

HUANG Liang($$i%), SUN Lei (aG) , WANG Ning(3 ?)** and JIN Xiaoming(&% 8) National Laboratory of Industrial Control Technology, Institute of Advanced Process Control, Zhejiang University, Hangzhou 310027, China

Abstract The binaphthol enantiomers separation process using simulation moving bed technology is simulated with the true moving bed approach (TMB). In order to systematically optimize the process with multiple productive objectives, this article develops a variant of tissue P system (TPS). Inspired by general tissue P systems, the special TPS has a tissue-like structure with several membranes. The key rules of each membrane are the communication rule and mutation rule. These characteristics contribute to the diversity of the population, the conquest of the mul- timodal of objective function, and the convergence of algorithm. The results of comparison with a popular algorithm- the non-dominated sorting genetic algorithm 2(NSGA-2) illustrate that the new algorithm has satisfactory performance. Using the algorithm, this study maximizes synchronously several conflicting objectives, purities of different products, and productivity. Keywords simulated moving bed, tissue P systems, multiobjective optimization, Pareto optimality, evolutionary algorithm, binaphthol enantiomers separation process

1 INTRODUCTION The simulated moving bed (SMB) technology is

receiving more and more attention as a convenient technique for the large-scale continuous chroma- tographic separation in areas such as biotechnology, pharmacy, and fine chemistry[ 1-41. SMB chromatography have the advantages in getting high purity products and achieving higher productivity with lower solvent consumption. The methods for SMB modeling have been studied and the factors affecting its performance have being analyzed[5,6]. The selection of the operating conditions to achieve high separation performances is acknowledged to be the major problem in running an SMB unit. Because of the complexity of the process and the coupling between the operating variables, it is hard to take a systemic analysis. Many studies has been carried out, and an explicit criterion for the choice of the operating conditions to achieve the prescribed separation was proposed, which is called the “triangle method”[7,8]. However, it is hold true only in the ideal state.

In practice, the process of SMB has to be optimized simultaneously on several incommensurable and conflicting objectives, such as the purities of different products and the productivity. There is no single optimal solution but rather a set of alternative solutions. The problem is too complex to be solved by exact methods, such as gradient search and linear pro- gramming, since the output of classical search and optimization methods is a single optimized solution. Recently, many studies on optimizing the process have been camed out with the evolution algorithm[9]. A new powerful algorithm is suggested based upon other researchers’ current studies. The new algorithm adopts the idea of tissue P systems and evolution algorithm[ 1 @-131.

As a powerful computing device, P systems have

significant potential to be applied to various problems of biology as well as to computer science[ 14,151. However, P systems are difficult to directly solve multiobjective optimization problems. This article explores the basic frame of a kind of tissue P system to implement the multiobjective optimization of SMB. The new algorithm is called TPS.

TPS has two unique characteristics. Firstly, it has the frame of general tissue P systems[ 10,111. Secondly, several subsystems exist in an algorithm. Some are single objective optimization subsystems, Others are multiobjective optimization subsystems. These fea- tures assure that the algorithm works well. The new algorithm is applied in the optimization of the binaph- tho1 enantiomers separation process using simulation moving bed technology. The following simulation shows that TPS outperforms a popular current algorithm, the non-dominated sorting genetic algorithm 2 (NSGA-2)[ 161.

2 SIMULATED MOVING-BED SYSTEM AND VARICOL PROCESS

The SMB process is illustrated in Fig.1. The inlets and outlets divide the system into four sections, with the configuration 2-2-2-2 (two columns in each section). These inlets and outlets are periodically switched in the direction of fluid phase flow, so the countercurrent movement of the solid phase and liquid phase is simulated. The countercurrent movement leads the more retained component B to move to the extract and the less retained component A to move to the raffinate. The feed is injected between sections I1 and 111. The component B is collected as the extract, which is between sections I and 11. The component A is collected as raffinate, which is between section I11 and IV. After being regenerated in section IV, the

Received 2006-09-06, accepted 2007-03- 14. * Supported by the National Natural Science Foundation of China (No.60421002).

** To whom correspondence should be addressed. E-mail: [email protected],edu.cn

684 Chin. J. Ch. E. (Vol. 15, No.5)

extract section I1 feed

direction of fluid flow and port switching

J L 7 - T Y eluent section IV raffinate

Figure 1 Four-section SMB

liquid phase is recycled to section I together with fresh desorbent. which is fed between sections I and IV.

3 MATHEMATICAL MODEL OF PROCESS

Different models to characterize the performance of a SMB process have been proposed[l7]. There are two main strategies of modeling a SMB process: one representing the real SMB, taking into account the periodic switch of the inlets and outlets, and the other by assuming the equivalence with the TMB.

Although transient evolution of the SMB and TMB approaches are different, they have similar steady-state performances in terms of average property. The primary objective is to characterize steady-state performance of the SMB process, without considering the transient process. And it has been tested that the TMB model has adequate accuracy[ 171. The optimization of the SMB operation conditions can be safely carried out on the basis of TMB model[l8]. In addi- tion, because of a large number of simulations, it is very important that a simulation method that is less time-consuming should be used. So the TMB model is selected as the simulation model in this work.

In the TMB model, the solid phase is assumed to move with plug flow in the opposite direction of the fluid phase, while the inlet and outlet lines remain fixed. As a consequence, each column plays a different function, depending on its location.

OF SIMULATED MOVING-BED

(1) Model equations for the TMB Mass balance in a volume element of the bedj

Mass balance in the particle aq.. aq.. rJ = us rJ + k ( 4,; - q; j ) at az

Initial conditions t = O ; c.. U =q.. 1J = o (3)

Boundary conditions for section j

DL, dqj z = 0 ; CIj - -~ = c;j,o

v j dz (4)

where,

sectionj

is the inlet concentration of species i in

For extract and raffinate nodes c.. = c . .

l j IJ+1,0

For the eluent node

1/+1,0 VI

VlV c.. =-c..

I/

For the feed node

(7)

And 4.. IJ = 4.. lJ+1,0 (9)

Global balances Eluent node

VI =vlV (10)

vw = v l - v x (11)

(12)

(13) Multi-component adsorption equilibrium iso-

Extract node

Feed node

Raffinate node v, = vn - VF

VN = vm - VR

therm

where, i = A , B refers to the species in the mix-

(2)Model parameters Peclet number

ture, and j = I , I1 ,111,IV is the section number.

VjLj

DLl

Pej =-

(3) Simulation results The separation of enantiomers is an important

issue in various areas and particularly in the health-related field. The most recent SMB application is related with chiral technology. In this work, the chromatography resolution of binaphthol enantiomers was considered for the simulation purpose. The adsorption equilibrium isotherms are of bi-Langmuir type:

(16)

(17)

The operation conditions and model parameters used in the simulation for the TMB approach are pre- sented in Table 1.

* 2 . 6 9 ~ ~ 0.1OCA + = 1 + 0 . 0 3 3 6 ~ ~ + 0.0466cB 1 + CA + 3cB

* 3 .73CB 0 . 3 0 ~ ~ + qB = 1 + 0 . 0 3 3 6 ~ ~ + 0 . 0 4 6 6 ~ ~ 1 + CA + 3cB

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Multiobjective Optimization of Simulated Moving Bed by Tissue P System 685

Table 1 Operating conditions and model parameters for the TMB approach[l7]

TMB operation conditions Model parameters and unit geometry Feed con- Solid flow Recycling Eluent flow Feed flow Extract flow Raffhate Column Section centration rate flow rate rate rate rate flow rate Pe diameter length

7.11 0.4 0.1s-' 2000 2.6cm 21.0cm 2.9 11.15 27.95 21.45 3.64 17.98

EL' m1.min-l m1.min-l ml.min-' mlmin-' m1.min-I m1.rnin-I

4 MULTIOBJECTIVE OPTIMIZATION OF THE SMB

The SMB performance is characterized by four performance parameters: purity, recovery, eluent consumption, and adsorbent productivity. These are de- fined in Table 2.

Figure 2 shows the steady-state internal concentration profiles of TMB model. The simulation predictions are 97.72% for extract and 99.25% for raffinate, while the simulation predictions obtained in the lit- erature[ 171 were 97.7% and 99.3%, respectively. It can be seen that the TMB model is correct and precise, so it can be applied for the operating optimization of the SMB process.

g 1.2

5 0.8 0.6

0 0.4 0.2

0 1 2 3 4 5 6 7 8 column

A; -l3 Figure 2 The concentration profiles of TMB

- - - -

There are two strategies in the optimization of SMB. The first one is to seek the better size of length and diameter, and the particle size distribution of the adsorbent solid, in order that the facility is more effi- cient.'The second issue arises when a SMB unit is al- ready available and the objective is to use it dealing with a new separation. In the latter case the operation conditions must be chosen so as to achieve the prescribed separation performances.

In this work the second issue addressed is, assuming that the SMB unit and its geometric and packing

parameters are fixed. Therefore, the operating parameters to be selecEed are the internal flow rate Qj, the switching time t (corresponding to the solid flow rate Qs), and the feed concentration. But in most cases, the feed concentration is fixed, and cannot be operated freely, so that it is constant is assuming.

It is obvious that if the minimum amount of solid is used to separate a given amount of feed or the maximum amount of the feed is separated by a given SMB unit, great economic profit would be obtained. So the interests is in determining the optimal operating conditions to maximize the total adsorbent productivity of two components. Meanwhile, the high product purity is also a key factor, sometimes de- mands are very rigorous. Considering the relationship between the internal flow rate (10)-(13), it can be seen that only four variables are free. Eluent flow rate QD, extract flow rate QE, feed flow rate QF, recycle flow rate QRF and the switch time t* are selected as the decision variables.

The objective functions can be formulized as follows:

(18) ~aximize: J, = P, ( t* , Q,, QE, QF, eRE )

Maximize: J, = PR (t* , QD , QE , QF, QRF ) (20)

G Q E I ',"" is the total adsorbent where, P, =- "S VS

productivity of extract and raffinate. PE, and PR represent the purity of the extract and raffinate, respectively.

Equations (18)-(20) are subjected to the con- straints on the decision variables of:

2 m i n G t t G 4 m i n (21) 1 0 m l ~ m i n ~ ' ~ Q ~ G 3 0 m l ~ m i n ~ ' (22) 10ml.min-' G QEG 30mi.min-' (23) 2ml.min-' GQFG5ml.min-' (24)

Table 2 Definition of process performance parameters

Performance Parameter -~

Purity, % Productivity, g.min-'.L-' Recovery, % Eluent consumption, L.g-'

extract

raffinate

1ooc; c," + c," lOOC," c," + c,"

c,"e, "S

100C,"QR

C,"QF

Chin. J. Ch. E. 15(5) 683 (2007)

686 Chin. J. Ch. E. (Vol. 15, NOS)

lOrnlmin-' d Q&40mlmin-' (25) The multi-objective optimization problem chases

two goals: (1) To find a set of solutions as close as possible

to the Pareto-optimal front. (2) To find a set of solutions as diverse as possi-

ble. These must also be sparsely spaced in the Pareto-optimal region.

5 TPS ALGORITHM In order to optimize the three objective functions

effectively and simultaneously, TPS incorporates the idea of tissue P systems and evolution algorithms. TPS have a special structure which consists of several membranes. Each membrane has its own subpopulation (set of objects). Each subpopulation evolves in the respective membrane which works as an effective tube. These subpopulations communicate through channels among them according to the communication rule. The communication rule, mutation rule and other rules are similar to their counterparts in P systems. Moreover, the algorithm pays attention to the ends of the Pareto front. The ends constitute the skeleton of the Pareto front, so it is never paid any more attention than necessary. In other words, TPS emphasize the optimization of each single objective. Therefore, a complete set of solutions will approach the Pareto front faster. In fact, many real-world problems require that one special objective is optimized and other objectives do not turn for the worse. The emphasis on the single objective makes the set of solutions faster and more effective an approximation to the Pareto front.

5.1 The tissue-like structure As a general tissue P system, the structure of TPS

is a tissue-like. The number of membranes is different according to the number of objectives and the diffi- culty of the optimization problem. As is shown in Fig.3, membranes mo, ml, .-, m6 form the structure of the system. Membranes ml, m2, m3 are subsystems for different single objective optimizations respectively. Membranes m4, m5, m6 are different subsystems for three objective optimizations together. Membrane mo collects the nondominated solutions from m6 as the systemic result.

population

Figure 3 Tissue-structure

5.2 Flowchart of describing the algorithm Similar to standard P systems, each membrane

evolves respectively with maximal parallel. Therefore, the algorithm should be implemented on a cluster of computers or the evolutions of all membranes are implemented one by one on a computer. One of the strategies is described by following flowchart as Fig.4.

5.3 The rules The Objects in each membrane evolve as the as-

sociated rules. The evolution of the subpopulation can incorporate ideas from most popular evolutionary algorithms. However, one key rule must be adopted from P systems, i.e., the communication rule. Each membrane exchanges its several best objects with the membrane, which links by a channel. However, the copies of the solutions remain in it.

The mutation rule of TPS is also unique. It is described as follows:

S + S ' (26)

where, s = X1X2 . . . x, { S ' = YlY, ... Yl '

pi = rand yi = x

i = 1,2;. ., 1 (28) 1 is the length of the strings or the number of variables of a problem; xi, yi are the variables of a problem; S is an approximate solution of the problem; xi m a I Xi min is the upper limit and lower limit of variable xi; P, is the probability of mutation; < E (0,l) is a random number; qi is a random number with normal distribution. The range of distribution of qi is different in different membranes. The selection of the ranges is crucial for the convergence of the TPS.

At each generation, membrane m6 sends its front into the membrane mo. Therefore, there are a mass of solutions, some of which are dominated by the other. In the membrane mo, the dominated solutions are deleted and only the non-dominated solutions are stored at each generation. Many seminal works report that the incorporation of elitism improves the performance of algorithms. However, the elitists in mo do not evolve. The function of such strategy is not obvious in the early stages of evolution. When non-dominated solutions exceed the size of the offspring population, some of them will be deleted. If the Pareto front of the problem is wide and its shape is complex, the front with little solutions coming from the systems makes it difficult to represent the real front. The store of the elitists will increase the number of the Pareto solutions and improve their distribution.

5.4 The analysis of algorithm NSGA has b5en criticized for its computational

complexity O(MN ), where, M represents the objectives and N represents the population size. The NSGA-2 and the strength Pareto evolutionary algorithm (SPEA)

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1 ( I ) Initialize: specify parameters of operation and I create random chromosomes for each subsystem

1 (2) Each subsystem evolves specified times

c

/.-----

I

c

I (4) Exchange chromosomes according their communication rules I 1

Y

(6) The membrane m,, output the Pareto

Figure 4 The flowchart of the new algorithm

needs at most O(MN2) computations in each generation[l2]. The TPS only require at most O( M,,,N; ), where the Msub is the number of objectives oP b e maximum subsystem, and NSub is the size of the maximum subpopulation. Because the whole population is divided into several subpopulations and there are some single objective optimization subsystems, the computational complexity deceases and therefore the algorithm runs faster.

5.5 Test of algorithm

for the test of algorithm[l3]. ZDT 4:

The famous difficult test function ZDT4 is used

where m = 63, x1 E [0,1] , and %;. .,xm E [-5,5]. It contains 2162 local Pareto-optimal fronts. The

sheer number of multiple local Pareto-optimal fronts produces a large number of hurdles for an algorithm to converge to the global Pareto-optimal front. It tests the algorithms, NSGA, NSGA-2, SPEA, SPEA-2 and an intelligent multiobjective optimization evolutionary algorithms for large parameter optimization problems (IMOEA). Thirty independent runs were performed using the same stopping criterion which is Neva, =25000. The parameters of NSGA, NSGA-2, and SPEA, SPEA-2 are summarized as follows: P,= 0.8, Pm=0.1, tdom=10, 6,h,=0.48862, and Npop= 100. The parameter settings of IMOEA are Npop=30, NEma = 30, P, = 0.2, P, = 0.6, P, = 0.01 [ 191. The results of test are shown in Fig.5. The comparison illustrates that TPS obtained the best Pareto front, which is much closer to the Pareto-optimal front when compared with the other algorithms.

Chin. J. Ch. E. 15(5) 683 (2007)

688 Chin. J. Ch. E. (Vol. 15, NOS)

0 0.1 0.2 0.3 0.4 0.5 0.6

Figure 5 Comparison of algorithms o TPS; 4 TMOEA; NSGA; 0 NSGA 2; & SPEA;

% SPEA 2: __ Pareto frontier

.A

6 EXPERIMENTS 6.1 Design for the experiment

The optimization of the binaphthol enantiomers separation process using SMB with the TMB model [Eq.( 18)-(25)] is implemented by TPS and NSGA-2, respectively. The same parameter settings are adopted. The population size is 50 and the stopping criterion is 50 generations. In the following experiments, the TPS has several membranes. These membranes share the population with 50 objects as shown in Table 3. The ranges of mutation which belong to different membrane should be different according to the different decision spaces. Each membrane exchanges its 4 best solutions with another membrane through the corresponding channel.

Table 3 The parameters of each membrane

Membrane Size of subpopulation Ranges of mutation (qi)

ml 5 0.01 m2 5 0.01 m3 5 0.01 m4 5 0.2 m5 10 0.05 ms 20 0.01

6.2 Results and discussions As is shown in Fig.6, NSGA-2 obtains 50

non-dominated solutions and TPS provides 41 1 non-dominated solutions. Each solution represents an optimal operating point of SMB. The performance comparison of algorithms a and p often uses the cover metric of their nondominated solution sets. The cover metric is expressed as follows:

the number of individuals in fl weakly dominated by a CM (a, P> =

the number of individuals in p (30)

Figure 6 Three-objective optimization of SMB by TPS and NSGA-2

O NSGA-2; oTPS

The value CM (a, p) = 1 means that all individuals in are weakly dominated by a. On the contrary, C M ( a , P ) = 0 denotes that none of individuals in p is weakly dominated by a. The comparison of the solution sets obtained by NSGA-2 and TPS is listed in Table 4. The comparison of the results shows the result of TPS is closer to true Pareto surface. Solutions of NSGA-2 are dominated by the solutions of TPS.

Without considering the productivity, the relationship of purity is illustrated by Fig.7. Obviously, the solutions of TPS are superior to the solutions of NSGA-2. The same figure also shows the effect of higher productivity on purity. The solutions have higher productivity while their purity is lower. The relationship between the productivity and the purity of extract is shown clearly by Fig.8. Fig.9 shows the relationship between the productivity and the purity of raffinate. Considering the three objectives and practi- cal product, several optimal solutions from each algorithm are selected and listed in Table 4. The optimal solution of TPS dominates the optimal solution of NGAS-2. In fact, TPS only evolves less than 5 generations while obtaining the same level solutions which obtained by NSGA-2 run 50 generations.

O r p

1 I , I 0

95 96 97 98 99 100 P L

Figure 7 The relationship of purities O TPS; O NSGA-2

Table 4 The optimal solutions and their comparison of TPS and NSGA-2

CM(NSGA-2, CM(TPS, QD QE QF QRF TPS)=O NSGA-2)=1 Algorithms P E PR Pt t

99.47 99.33 95.4 3.4934 21.1658 15.9887 3.9438 21.2602 99.70 99.61 65.18 3.2266 31.9438 12.4095 2.7225 10.8884

TPS

NSGA-2 99.47 98.60 59.3 2.3228 24.5712 11.5721 2.6355 31.0858

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98.5 99.0 99.5 100 p,

Figure 8 Productivity and purity of extract 0 TPS; V NSGA-2

1101

Figure 9 Productivity and purity of raffinate V NSGA-2; o TPS

Three-objective optimizations have been performed in this study. Then, the relationships of two objectives are analyzed and illustrated by Figs.7+. At the same time, the effect of another objective is illustrated by a single figure. While several objectives are optimized together, the algorithm can overcome local min- ima easily because the population distributes uniformly and will not converge at a local minimum. Moreover, the computing is time-consuming. Three-objective optimization decreases the time of simulations because it illustrates the problems that are illustrated by several problems with two-objective optimization.

7 CONCLUSIONS The binaphthol enantiomers separation using

SMB technology is investigated for many years. It is intractable that SMB is systematically optimized with many operating parameters, including switching time interval and liquid flow rates. In most cases, many conflicting optimal objectives are required by factors for economical and product quality. In this article, three objectives are optimized simultaneously.

In order to implement the optimization of SMB, this article develops a strong-performing multiobjective optimizer that is called TPS inspired by the tissue P systems. On one hand, it extends the application of P systems. On the other hand, it is a new strategy for the multiobjective optimization problems. As shown by experiments above, the quality of non-dominated solutions obtained by TPS is superior to those of NSGA-2. The relative performance of TPS demon- strated that TPS may be well qualified to join in the current set of “best performers” in the multi-objective optimization.

NSGA-2 and TPS both provide a set of nondominated solutions. Engineers may make a trade-off according to the result. Compared with NSGA-2, TPS provides better operating points in term of the purity of the extract, the purity of the raffinate, and the total adsorbent productivity of extract and raffinate.

ACKNOWLEDGEMENTS We are grateful to Professor Gheorghe Pa’un for

direction and introduction of the exciting field of P systems.

NOMENCLATURE less retained component more retained component average concentration of component i in the inlets and outlets during one switch period cover metric feed concentration of component i concentration of component i in sectionj desorbent axial dispersion coefficient extract feed corresponding to F, D, E, R mass transfer coefficient capacity to maintain the best nondominated individuals the number of evaluation the number of individuals in a population probability of crossover probability of mutation set of objects at t generation set of objects evolves from P, volumetric liquid flow rate in the inlets and outlets average adsorbed phase concentration adsorbed concentration in equilibrium with fluid phase concentration raffinate set of objects is merged from P, and P 1 ~ time the size of the comparison set solid velocity fluid velocity in sectionj axial coordinate niche count bed porosity

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