Mnemonic Enhancement Optimization (MEO) for Real-Time

Ind. Eng. Chem. Res. 2009, 48, 499–509

499

Mnemonic Enhancement Optimization (MEO) for Real-Time Optimization of Industrial Processes Xueyi Fang, Zhijiang Shao,* Zhiqiang Wang, Weifeng Chen, Kexin Wang, Zhengjiang Zhang, Zhou Zhou, Xi Chen, and Jixin Qian State Key Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang UniVersity, Hangzhou 310027, People’s Republic of China

In this paper, the model-based real-time optimization (RTO) is viewed as a kind of nonlinear parametric optimization problem which is solved repeatedly when parameter values change. A novel RTO strategysmnemonic enhancement optimization (MEO)sis proposed. The method preserves the past optimal solutions and corresponding parameter values as experience and approximates the optimum based on the experience. The approximation is used by the optimization algorithm as a starting point to find the real optimum. The optimum is proved to be a continuous function of the parameter. This ensures that the distance between the optimum and the initial point tends to decrease as RTO continues to run. Thus MEO can improve the performance of RTO continually. Numerical experiments illustrate the continuity of the optimal set mapping, and the MEO method is compared with the traditional method. The results show that MEO outperforms the traditional method concerning the solution time, the number of iterations, and the percentage of successful optimizations. 1. Introduction The chemical process industry has been increasingly propelled to run profitably in a very dynamic and worldwide market with external and internal uncertainties.1 The increasing competition results in decreasing profit margins. This emphasizes the importance of online adjustment of operating plants to maximize profit dynamically. Real-time optimization (RTO) with rigorous models has attracted increasing attention of many researchers during the past decades and has become an important part of enterprise-wide optimization.2 For many processes, fluctuating system load and changing product specifications result in frequent variations of the optimal operating points. RTO refers to a sequence of online economic optimizations of the operating point when these operating conditions change or the process is disturbed. The optimization problems are solved repeatedly and must be solved quickly, or else the answer is out-of-date for the continually moving problem conditions. A lot of effort has been devoted to high performance optimization algorithms such as the reduced space sequential quadratic programming and interior point methods.3-7 However, with the rapid increase of process scale and complexity, optimization still remains a challenge. An initial guess sufficiently close to the optimum is important to both local and global optimization algorithms.8 The traditional method in RTO to generate initial guesses is to take the latest optimal solution as the starting point of current optimization. Some industrial practitioners also keep a set of base cases as initial conditions in case of convergence difficulty. However, this method is prescribed and fixed. Its implementation needs human interaction, which may debase the real-time performance of optimization. This paper views the RTO problem as a kind of parametric optimization problem and suggests a novel method, mnemonic enhancement optimization (MEO), which memorizes the past solving experience to approximate the optimum and searches * To whom correspondence should be addressed. Fax: +86-57187951068. E-mail address: [email protected].

for the real optimum using the approximations as starting points. From the viewpoint of bionics, it is similar to the long-term memory mechanism of the human brain. According to the experience accumulated, MEO can make wise decisions on where to start the optimization so as to get quickly to the optimum. The experience (or “mnemon”) could be “enhanced” for RTO calculation in various ways, such as the Lagrange interpolation, the curve fitting methods, and even the artificial neural network. The optimum is proved to be a continuous function of the parameter. This ensures that the initial point provided by the MEO method tends to go to the optimum as experience accumulates. In other words, MEO can improve the quality of the initial point continually. The rest of the paper is organized as follows. The next section introduces the problem formulation of parametric RTO and suggests the idea of MEO based on which the MEO algorithm is developed. The continuity analysis is given in the third section. Section four illustrates the continuity of the optimal set mapping and compares the MEO method with the traditional method, based on the steady-state optimization problems of two chemical process models. Section five concludes the paper. 2. MEO Frame In this section, we first introduce the parametric RTO problem formulation. Then we demonstrate the idea of MEO with an example of a parametric optimization problem. Finally, we summarize the MEO algorithm. 2.1. Problem Formulation of Parametric RTO. For typical closed-loop RTO with a rigorous steady-state model, the external and internal uncertainties such as the economic data, the operating conditions, and the disturbances are used to update the process model through data measurement and reconciliation.9,10 Every RTO run is driven by the change of the values of the uncertainties. The optimization subproblems solved in all RTO runs have the same structure and differ only in the values of these uncertainties. In other words, RTO can be seen as solving a sequence of similar parametric optimization subproblems. The objective functions as well as the models of

10.1021/ie800166p CCC: $40.75  2009 American Chemical Society Published on Web 12/01/2008

500 Ind. Eng. Chem. Res., Vol. 48, No. 1, 2009

Figure 1. Landscapes of f(x,R) with nine different parameter values of R. The nine pentacle stars on the thick line represent the minimizers x* of the nine curves. The collection of all such minimizers forms the thick line which represents f(x*,R) as a function of x*.

different subproblems are the same except that their parameters are different. The parametric form of RTO subproblem is given as follows. min{f(x, R)|x ∈ M(R)}

(1)

where R ∈ D (D ⊂ R ) is the parameter and the constraint set mapping M(R) is defined as d

M(R) ) {x ∈ Rn|ci(x, R) g 0, i ∈ I}

(2)

f and ci, i ∈ I in (1) and (2) are real-valued functions on a subset of Rn × Rd and are assumed to be bounded and twice continuously differentiable in both arguments x and R. It should be noted that the form of the constraints in (2) is convenient for the theoretical analysis of parametric optimization problem.11 This form is compatible with equality constraints, because any equality constraint ci(x,R) ) 0 is equivalent to two inequality constraints ci(x,R) e 0 and ci(x,R) g 0. Each time the value of R varies, the parametric optimization problem (1) with the new parameter value needs to be solved to find the new optimum x*. The optimum x* of the parametric problem (1) is a function of the parameter R. The function is called the optimal set mapping11 and is defined as x*(R) ) {x|f(x, R) ) φ(R)} where φ(R) is the extreme value function11 defined as φ(R) ) inf f(x, R) x∈M(R)

(3) (4)

2.2. Idea of MEO. MEO learns from past solving experience to construct the graph of the optimal set mapping in multidimensional space and improves RTO performance consecutively according to the graph. A simple example is given to demonstrate the idea. Example. The following parametric optimization example has one variable and one parameter.

{

min f(x, R) ) 0.03(x - R2)2 + R s.t. 0 e R e 10

(5)

f(x*, R) ) φ(R) ) R (7) Change the value of the parameter R nine times from integer 1 to 9. Figure 1 shows the nine corresponding curves of the objective function f(x,Ri), x ∈ R, i ) 1, 2, ..., 9. Suppose that R varies randomly nine times. The parameter variation and the optimal set mapping for one possible realization are shown in Figure 2. According to the traditional method, when the value of R changes from R5 to R6, for example, x*(R5) will be taken as the initial point for the sixth optimization. However, we can see in Figure 2 that x*(R1) is much closer to x*(R6) than x*(R5). It is better to take x*(R1) as the starting point than x*(R5) in order to find out x*(R6). As another instance, when the value of R changes from R7 to R8, a simple approximation method such as the first-order Lagrange interpolation using x*(R4), R4 and x*(R5), R5 can provide an initial point very close to the solution x*(R8), compared with choosing x*(R7) as in the traditional method. The two instances show that if we preserve the past optimal solutions, we can make initial guesses which may be much closer to the optimum than the latest optimal solution does. And, an initial point sufficiently close to the optimum is important to optimization algorithms.8 The idea of the MEO method can be summarized as (1) storing the past solving experience to learn the graph of the optimal set mapping, (2) approximating the optimum according to the experience, and (3) finally sending the approximation to the optimization algorithm as initial point. There are many methods for MEO to approximate the optimum, such as Lagrange interpolation, curve fitting, and artificial neural network. In this paper, the simplest zero-order approximation method is used and is defined as follows. In the (k + 1)th RTO interval, the zero-order approximation method chooses from the accumulated experience set {(x*(R1),R1), (x*(R2),R2), ..., (x*(Rk),Rk)} a previous solution (x*(Ri),Ri) satisfying i(k) ) arg min |Rk+1 - Rj | j)1,...,k

Then, the optimal set mapping is x*(R) ) R

The extreme value function is

2

(6)

(8)

where | · | is L2 norm. Then x*(Ri(k)) is taken as the initial point to find x*(Rk+1).

Ind. Eng. Chem. Res., Vol. 48, No. 1, 2009 501

Figure 2. Optimal set mapping x*(R) and parameter changes of nine RTO runs. The left-hand side graph in this figure records the nine values R1,R2, ..., R9. The x-axis denotes time, and the y-axis denotes the parameter R. The right-hand side picture is the optimal set mapping with x* and R being the x-axis and y-axis, respectively.

Figure 3. Flowchart of RTO with MEO. The flowchart describes the RTO system with MEO method briefly. It contains the control and sensor layer, the blocks of data reconciliation, the parametric model, the optimization algorithm, and the MEO method.

The MEO flowchart and algorithm are shown as follows. 2.3. Flowchart of RTO with the MEO Method. The brief flowchart of RTO with the MEO method is given in Figure 3. It contains the systems of RTO and MEO. According to the flowchart, an RTO interval begins with the disturbance of the external and internal uncertainties, such as the economic data or the operating conditions.1 The external uncertainties are sent directly to the MEO block. The changes of the internal uncertainties are first detected by the control and sensor layer. Then they are sent to the data reconciliation block. The data from the data reconciliation block is transferred to the MEO block, instead of the block of parametric model. Then the MEO block approximates the optimum and sends the approximation to the optimization algorithms block as a starting point. Once the optimization converges, the solution is implemented by the control and sensor layer. Simultaneously, the optimal solution is returned to the MEO block to be stored as experience. And the RTO interval terminates.

2.4. MEO Algorithm. Step 1. Set k ) 1. Use the point supplied by the user as the starting point for the first optimization. Store the solution and go to step 2. Step 2. When the parameter Value changes, set k ) k + 1. Use zero-order approximation (8) to select a previous solution. Send the selection to the optimization algorithm to find the real optimum x*(Rk) and go to the next step. Step 3. Implement the solution x*(Rk) through control system. Add (x*(Rk),Rk) to the experience set {(x*(R1),R1), (x*(R2),R2), ..., (x*(Rk-1),Rk-1)}. Go to step 2. It should be noted that the method the industrial practitioners use keeps a fixed set of base cases as the initial conditions in case of convergence difficulty. The decision about which base case should be used does not depend on the parametric feature of RTO, but rather on human experience. This is not consistent with the goal of RTO. MEO automatically accumulates past solving experience and provides the initial point based on the


parametric feature of RTO. There are various methods enhancing MEO approximation, such as Lagrange interpolation, curve fitting, and neural network, while the method of industrial practitioners does not have such approximation methods. The experience database of MEO is not prescribed and fixed. It is formed and enriched during RTO runs. And MEO constructs the initial point automatically.

Γ(R) ∈ Ω ⊂ Vε(Γ(R0)) This means that

(13) |Γ(R) - Γ(R0)| < ε According to this lemma, the optimal set mapping is continuous if it is a point to point mapping and is usc-B. To prove the optimal set mapping is usc-B, the closedness of

3. Theoretical Analysis of MEO Method

Γi(R) ){x ∈ Rn|ci(x, R) g 0}, i ∈ I

A good initial guess should be sufficiently close to the optimum. If the optimal set mapping x*(R) is continuous and the region of the parameter R is closed and bounded, then MEO guarantees that lim(x*(Ri(k)) - x*(Rk+1)) ) 0

kf∞

(9)

where Rk+1 represents the current parameter value and Ri(k) is selected according to the criterion (8). In order to show the continuity of the optimal set mapping, three assumptions are made. Assumptions. (i) The optimal set mapping x*(R) is point to point mapping, or injective function. (ii) The region D ⊂ Rd of the parameter R is closed and bounded. (iii) The objective function f and the constraint functions ci, i ∈ I, are bounded and differentiable in both arguments x and R which has already been assumed in the formulation of parametric RTO problem. It should be noted that there were reports on the multiple solutions of process simulation, such as the work of Lin et al.12 In RTO however, all variables are constrained in certain range. And most of the research work on optimization algorithms for RTO does not take the situation of multiple solutions into consideration. So in this paper, we focus on the parametric RTO problem which has injective optimal set mapping. The second assumption is reasonable in industrial practice because the regions of meaningful parameters in chemical processes are closed and bounded. In many optimization algorithms, it is assumed that the objective function and the constraint functions are bounded and differentiable. And all the parameters in parametric optimization problems derive from the variables of nonparametric optimization problems. So the third assumption holds. To prove the continuity of x*(R), the concepts of closedness, upper semincontinuous and lower semincontinuous of the point to set mapping are needed.11,16 The definitions are given in the Appendix. Through the rest of this paper, the upper semincontinuous and lower semincontinuous abbreviated to usc-B and lsc-B, respectively.11 Define Vε(x0) as Vε(x0) ) {x||x - x0 | < ε}

(10)

where ε > 0. And define Γ to be a mapping of D into the power set (set of all subset) 2X of X where D ⊂ Rd is the domain of R and X ⊂ Rn the domain of x. Let us observe the continuity of Γ. Lemma 1. Suppose that Γ is a point to point mapping and is usc-B, then Γ is continuous. Proof. According to the definition of usc-B, for any ε > 0, there exists δ > 0 such that when |R - R0 | < δ there is an open set Ω satisfying

(11)

(12)

(14)

should be checked. Lemma 2. The point to set mapping Γi(R) is closed if assumption iii holds. Proof. If x0 ∈e Γi(R0) for a pair of sequences {Rt} ⊂ Rd and {xt} ⊂ Rn, t ) 1,2, ... with the properties Rt f R0, xt ∈ Γi(Rt), xt f x0 there exists a constant ε > 0 so that ci(x0, R0) < -ε < 0

(15)

According to the definition of Γi(R), we have ci(xt, Rt) g 0

(16)

|ci(xt, Rt) - ci(x0, R0)| > ε

(17)

i.e., ∀ δ > 0, there is

when ||

[][]

xt x0 || > δ Rt R0

This contradicts the continuity of c(x,R) according to assumption iii. So Γi is closed at R0. And according to the arbitrariness of R0, we have that Γi(R) is closed. Now according to the Theorem 4.2.2 in the work of Bank et al.,11 the continuity of the optimal set mapping can be proved if the constraint set mapping M is lsc-B. Lemma 3. The constraint set mapping M is lsc-B, if assumption iii holds. Proof. Let Ω be any open set satisfying Ω ∩ M(a0) * φ

(18)

0

and let x satisfys x0 ∈ Ω ∩ M(R0) (19) Suppose that there does not exist such δ ) δ(Ω) > 0 that for any a ∈ Vδ(a0), M(a) ∩ Ω * φ (20) holds, i.e. for any δ > 0 there exists an a ∈ Vδ(a0) satisfying M(a) ∩ Ω ) φ (21) 0 Then, there is x ∈e M(a), because x ∈ Ω. This means there exists an index j such that 0

cj(x0, a) < -ε < 0

(22)

where ε > 0. But because cj(x0,a) is continuous at (x0,a0), so for cj(x0,a0) g 0 there is a δ > 0 such that cj(x0, a) > -ε/2

(23)

This contradicts (22). So there exists a δ ) δ(Ω) > 0 such that Ω ∩ M(a) * φ for any R ∈ Vδ(R0), which means the constraint set mapping M is lsc-B.


Figure 4. Function of the optimal reflux flow of the depropanizer with respect to the feedstocks of S502 and S538. The x-axis is S538, the y-axis is S502, and the z-axis the optimal reflux flow of the depropanizer. The parameters and the variable are all scaled.

Theorem 1. The optimal set mapping x*(R) of the parametric optimization problem (1) is continuous if assumptions i, ii, and iii hold. Proof. The objective function f is upper semicontinuous and lower semicontinuous according to assumption iii. The constraint set mapping M is closed according to Lemma 2 in this paper and Lemma 2.2.4 in the work of Bank et al.11 Because the bounded set in the finite dimensional space is compact, there exists a nonempty compact set K satisfying x*(R) ∈ K for any R ∈ D where D is the domain of R. M is the lsc-B according to the Lemma 3. And because f is upper semicontinuous, the extreme value function φ is upper semicontinuous. On the basis of the Theorems 4.2.1 and 4.2.2 in the work of Bank et al.,11 it can be concluded that the optimal set mapping x*(R) is usc-B. According to assumption i and Lemma 1, we know that x*(R) is continuous. According to assumption ii, this theorem implies that the optimal set mapping is uniformly continuous in the closed and bounded region of the parameter. Thus when the experience density grows continuously in the closed and bounded region, the initial guess given by MEO method tends to go to the real optimum as RTO continues to run. This is what (9) means. So, MEO can provide an initial guess that is sufficiently close to the real optimum when the experience is adequate. And the optimization algorithm can benefit from it. As the experience accumulates, the initial point provided by MEO method tends to be closer to the real optimum. This leads to a continual improvement of the optimization performance. 4. Numerical Results

Figure 5. Graph between the extreme value function and the parameters S502 and S538. The x-axis is S538, the y-axis is S502, and the z-axis the extreme value function of the depropanizer. The parameters are scaled.

Now we are at the position of proving the continuity of the optimal set mapping x*(R).

The numerical experiments in this section are based on the optimizations of two models. One model is a depropanizer and debutanizer distillation sequence; the other is the ethylene separation process. The experiments contain three parts. The first part illustrates the continuity of the optimal set mapping of a parametric optimization problem based on the depropanizer and debutanizer distillation sequence. In the second part, the MEO method is compared with the traditional method based on the parametric optimization problem of the depropanizer and debutanizer distillation sequence. On the basis of the optimiza-

Figure 6. Demonstrations of the changes to some parameters of the depropanizer and debutanizer distillation sequence. All parameters are scaled.

504 Ind. Eng. Chem. Res., Vol. 48, No. 1, 2009 Table 1. Statistical Results of Optimizations Based on the Depropanizer and Debutanizer Distillation Sequence

traditional method MEO method

average solution time (s)

average iterations

116.2 102.2

5.6 4.7

tions of the ethylene separation process, the third part compares the MEO method with the traditional method. 4.1. Continuity of the Optimal Set Mapping. In this experiment, the continuity of the optimal set mapping of a parametric optimization problem for the depropanizer and debutanizer distillation sequence is illustrated. The distillation sequence model built in Matlab environment is the same as that in the work of Wang et al.13 and Jiang et al.14 The RSQP Toolbox for Matlab is used as the nonlinear programming (NLP) optimization solver, which was developed by the authors. The convergence tolerance is set as 10-6.

The parametric optimization problem in this experiment has the same objective function and the same constraints as those in the work of Wang et al.13 except that the flow rate of the two feedstocks, S502 and S538, are taken as parameters. We select a rectangle region whose center represents the standard loads of these two parameters. Then we partition the region into 21 × 21 mesh with 441 nodes. The parametric optimization problem is solved for 441 times based on the 441 different parameter values of these nodes. The function between the two parameters and the optimal reflux flow of the depropanizer is shown in Figure 4. It is observed in this figure that the surface is continuous and smooth. This is consistent with the continuity analysis. Figure 5 shows the graph of the extreme value function f(x*(R),R). It is also observed the function is smooth and continuous in the region.

Figure 7. Comparison between MEO and traditional method concerning the norm of ∆x*. The figure records |∆x*| of all 800 RTO runs for the both methods. The variables are all scaled. The asterisks represent |∆x*| for the traditional method, and the circles represent |∆x*| for the MEO method.

Figure 8. Histogram of the solution time and the number of iterations based on the optimizations of the depropanizer and debutanizer distillation sequence.


Figure 9. Flowsheet of ethylene separation process.

4.2. Optimizations Based on the Depropanizer and Debutanizer Distillation Sequence. In this experiment, the MEO method is compared with the traditional method which takes the latest optimal solution as the initial condition of current optimization. The parametric optimization problem is the same as in subsection 4.1 except that the parameters are selected to be the flow rates, the temperatures, the pressures, and the percentages of the components of the two feedstocks, S538 and S502. In this

experiment, there are a total of 32 parameters. These parameters are changed for 800 times, corresponding to 800 RTO runs. Some of the parameters are shown in Figure 6 (scaled). In order to control the experience density and the space complexity, we suggest a simple method to manage the experience data set. When the number of iterations of current optimization is less than a certain value, such as 3, the past optimal solution chosen to be the initial point may be close to

506 Ind. Eng. Chem. Res., Vol. 48, No. 1, 2009 Table 2. Unit Description of the Ethylene Separation Process unit

description

E-DA-301 E-FA-309 E-DA-401 E-DA-404 E-DA-405 E-DC-402 E-DC-40X

unit

description

demethanizer E-DC-401 acetylene converter demethanizer reflux drum E-DA-408 C2 green oil absorber de-ethanizer E-DA-402 ethylene column depropanizer E-DA-407 methane gas stripper debutanizer E-DA-406 propylene fractionator allene converter E-DA-409 C3 refractionator flash drum

Table 3. Purity Control of the Products of the Ethylene Separation Process product

purity

ethylene discharged at S460 ethane discharged at S452 butane discharged at S592 propylene discharged at S594 propylene discharged at S550

G99.9% G97.9% G98.0% G99.6% G96.0%

the optimal solution of current optimization. MEO stores the optimal solution of current optimization and deletes from the experience set the past optimal solution corresponding to the initial point. In this experiment, all the optimizations converge successfully. The statistical data in Table 1 show that MEO leads to a reduction of 12.0% for the solution time and 16.1% for the number of the iterations. Denote the scaled distance between the initial point and the optimum by |∆x*| ) |x* - x0|, where x* is the optimum and x0 stands for the initial point to find out x*. The MEO algorithm tends to reduce |∆x*| as RTO continues to run, while the traditional method has no such tendency. This is reflected in Figure 7. It is also observed that in most cases, the initial points provided by MEO method are closer to their optimums than those provided by the traditional method. Figure 8 records the statistical data of the solution time and the number of iterations in four histograms. It is observed that the MEO method outperforms the traditional method in the solution time and the number of iterations. For each histogram, all elements in the statistical data are grouped according to their numeric range. Each group is shown as one bin. The x-axis of the histogram represents the range of values. The y-axis shows the number of elements that fall within the groups. 4.3. Optimizations Based on the Ethylene Separation Process. In this experiment, the MEO method is compared with the traditional method based on the ethylene separation process.

Figure 10. Changes of the parameters of the ethylene separation process.

The process has more than 30 000 variables. The flowsheet is established in the Aspen environment15 and is shown in Figure 9. The four mixtures from the condensate system first feed into the demethanizer, E-DA-301. Then they go through the demethanizer, the acetylene hydrogenation reactor, the ethylene column, the depropanizer, the debutanizer, the allene and methylacetylene hydrogenation reactor, and the propylene fractionator. There are four main products, the ethylene, the ethane, the butane, and the propylene. The products of ethylene and ethane are discharged at the ports S460 and S452 of the ethylene column E-DA-402, respectively. The butane product is discharged at S592 of the debutanizer E-DA-405. Propylene is discharged at S594 and S550 of the propylene fractionator E-DA-406 with two different grades. The process consists of 13 units including distillation columns, an absorber, reactors, and a flash drum. The detailed description of each unit is listed Table 2. The main products concerned in this model are the ethylene, the ethane, the butane, and the propylene. The objective is to optimize the profit of these products. The optimization problem has the objective function 5

J)

∑wS

i i

(24)

i)1

where Si, i ) 1, 2, and 3 are the molar flow rates (kmol/h) of the products ethylene, ethane, and butane, S4 and S5 are the molar flow rates (kmol/h) of the product propylene at ports S594 and S550, respectively. The term wi, i ) 1, ..., 5 are the weighting coefficients (prices/h) of the product flowrates. The purity control of these products is given in Table 3. The parametric optimization problem is formulated as

{

min - J s.t. Parametric functional constraints: mesh equations(mass equation, energy equation, etc.); connection equations of the distillation columns; Purity control. (25)

The problem has 33 839 variables and 12 degrees of freedom which are several reflux ratios and heat duties. There are a total of 21 parameters including the loads, temperatures, and pressures of seven feedstockssS319, S316, S378, S312, S415, S384P, and S267. Let the 21 parameters be changed 500 times, corresponding to 500 RTO runs; we compare the MEO method with the


Figure 11. Number of iterations of 500 optimizations about the ethylene separation process. The stars represent the numbers of iterations for the 500 optimizations with the traditional method. The circles represent those with the MEO method.

Figure 12. Solution time of 500 optimizations about the ethylene separation process. The stars represent the solution time for the 500 optimizations with the traditional method. The circles represent the solution time with the MEO method.

traditional method based on the parametric optimization problem (25). The parameters are illustrated in Figure 10. The optimization algorithm used is DMO. In Figure 10, the x-axes of the 21 graphs represent RTO runs. The first column records the flow rates (kmol/h) of the seven feedstocks. The second and the third columns record the temperatures (°C) and the pressures (kg/cm2), respectively. Figures 11 and 12 show the results of 500 optimizations. In Figure 11, the numbers of iterations are compared. It is observed that most circles congregate below the stars. The similar result is shown in Figure 12 which compares the solution time of optimizations. Figure 13 compares the histograms of the number of iterations and the solution time between the traditional method and the MEO method. Compared with the traditional

method, the numbers of iterations concerning the MEO method are highly centralized in a small valued region. This guarantees that the solution time is mainly distributed within a region of short time. Table 4 records the statistical data of the optimizations of the ethylene separation process. MEO has improved the percentage of successful optimization by 8.4%. Only 8 optimizations with MEO failed, while 50 optimizations with the traditional method failed to converge. The total number of iterations concerning MEO method is less than a half of that concerning the traditional method. MEO has reduced the average solution time and the number of iterations by 37% and 52%, respectively. The first experiment illustrates the continuity of the optimal set mapping as well as the extreme value function. The result


Figure 13. Histograms of the solution time and the number of iterations based on the optimizations of the ethylene separation process. Table 4. Statistical Results of Optimizations Based on the Ethylene Separation Process traditional method MEO method total solution time (s) total number of iterations percentages of successful optimization

11221.97 4635 90%

7068.73 2206 98.4%

is consistent with the theoretical analysis in this paper. The second experiment validates the MEO method and compares it with the traditional method. The results show that MEO leads to a reduction of 12.0% for the solution time and 16.1% for the number of the iterations. The MEO algorithm tends to reduce the distance between the initial point and the optimum as RTO continues to run, while the traditional method has no such tendency. The third experiment is based on the ethylene separation process. It validates the MEO method and compares it with the traditional method. The results show that the MEO method outperforms the traditional method and leads to a reduction of 37% for the solution time and 52% for the number of the iterations. At the same time, the percentage of successful optimizations is increased by 8.4%. 5. Conclusions In this paper, the real-time optimization problem for continuous processes is viewed as a kind of parametric optimization problem and the MEO method is developed accordingly. The method stores the past optimal solutions and corresponding parameter values as experience and approximates the optimum based on the experience. Then the approximation is used by the optimization algorithm as starting point to find the real optimum. There are various approximation methods for MEO. From a bionic viewpoint, MEO is similar to the long-term memory mechanism of human brain. It memorizes the past solving experience in order to better finish the future optimization task. The optimum is proved to be a continuous function of the parameter under reasonable assumptions. As RTO continues to run, the initial point provided by the MEO method tends to go to the real optimum. In the numerical experiments, the continuity of the optimal set mapping is illustrated. And the MEO method is compared with the traditional method. The

results show that the MEO method outperforms the traditional method with respect to the solution time, the number of iterations and the percentage of successful optimizations. The paper can be viewed as a preliminary study of the MEO method. Future research work is needed to develop the method. For instance, other approximation methods, such as Lagrange interpolation and curve fitting, should be used. The solving experience of Lagrange multiplier can also be stored and used in MEO method. Acknowledgment This research was supported by the 973 Program (2009CB320603), the 863 Program (2007AA04Z192), and the National Key Technology R&D Program (2007BAF22B05) of China. We would like to thank Professor Michael Fu of the University of Maryland and Professor Arthur W. Westerburg of Carnegie Mellon University, for their kind suggestions and corrections. Appendix Let D ⊆ Rm and X ⊆ Rn. A point to set mapping Γ:D f 2X is (i) closed at a point R0 ∈ D if for each pair of sequences {Rt} ⊂ D and {xt} ⊂ X, t ) 1, 2, ... with the properties Rt f R0, xt ∈ Γ(Rt), xt f x0 it follows that x ∈ Γ(R0); (ii) upper semicontinuous (according to Berge, or simply: B) at a point R0 ∈ D, if for each open set Ω containing Γ(R0) there exists a δ ) δ(Ω) > 0 such that 0

Γ(R) ∩ Ω * φ ∀ R ∈ Vδ{R0} holds; (iii) lower semicontinuous (according to Berge, or simply: B) at a point R0 ∈ D, if for each open set Ω satisfying Ω ∩ Γ(R0) * φ, there exists a δ ) δ(Ω) > 0 such that Γ(R) ∩ Ω * φ ∀ R ∈ Vδ{R0} holds. Literature Cited (1) Varma, V. A.; Reklaitis, G. V.; Blau, G. E.; Pekny, J. F. Enterprisewide modeling & optimizationsAn overview of emerging research challenges and opportunities. Comput. Chem. Eng. 2007, 31, 692–711.

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(10) Sequeira, S. E.; Graells, M.; Puigjaner, L. Real-Time Evolution for On-line Optimization of Continuous Processes. Ind. Eng. Chem. Res. 2002, 41, 1815–1825. (11) Bank, B.; Guddat, J.; Klatte, D.; Kummer, B.; Tammer, K. NonLinear Parametric Optimization; Birkha¨user: Basel, 1983. (12) Lin, W.; Seader, J. D.; Wayburn, T. L. Computing Multiple Solutions to Systems of Interlinked Separation Columns. AIChE J. 1987, 33 (6), 886–897. (13) Wang, K.; Shao, Z.; Zhang, Z.; Chen, Z.; Fang, X.; Zhou, Z.; Chen, X.; Qian, J. Convergence Depth Control for Process System Optimization. Ind. Eng. Chem. Res. 2007, 46, 7729–7738. (14) Jiang, A.; Shao, Z.; Chen, X.; Fang, X.; Geng, D.; Zheng, X.; Qian, J. Simulation and Optimization of Distillation Column Sequence in Large-Scale Ethylene Production. J. Chem. Ind. Eng. China 2006, 57, 2128. (15) Geng, D. Simulation and Optimization on Ethylene Separation Process. M.S. Thesis, Zhejiang University, China, 2006. (16) Berge, C. Topological spaces; Oliver and Boyd: London, 1963.

ReceiVed for reView January 29, 2008 ReVised manuscript receiVed July 14, 2008 Accepted October 6, 2008 IE800166P

Mnemonic Enhancement Optimization (MEO) for Real-Time

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