Programming Strategies of Sequential Incremental-Scale

Article pubs.acs.org/IECR

Programming Strategies of Sequential Incremental-Scale Subproblems for Large Scale Data Reconciliation and Parameter Estimation with Multi-Operational Conditions Zhengjiang Zhang,†,‡ Zhijiang Shao,§ and Junghui Chen*,‡ †

College of Physics and Electronic Information Engineering, Wenzhou University, Wenzhou 325035, People’s Republic of China Department of Chemical Engineering, Chung Yuan Christian University, Chungli, Taiwan 320, Republic of China § State Key Laboratory of Industrial Control Technology, Institute of Industrial Control, Zhejiang University, Hang Zhou 310027, People’s Republic of China ‡

ABSTRACT: Data reconciliation and parameter estimation (DRPE) is a crucial issue in model-based applications, such as realtime optimization and process control. In order to obtain more reliable parameter estimates, a series of measurement data sets from different operational conditions will be used for DRPE problems. However, the dimensionality of DRPE problems increases directly with the number of measurement data sets. The number of degrees of freedom in DRPE problems is usually very large. Therefore, it is very difficult to solve the DRPE problem with multioperational conditions. On the basis of the characteristics of the DRPE problem, two directions, including the direction of incremental objectives of the DRPE problem and the direction of incremental parameters of the DRPE problem, are considered to decompose the original DRPE optimization problem into a series of incremental-scale subproblems. Three programming strategies are proposed to solve a series of incremental-scale subproblems one by one. The solutions to the current subproblem are used as a set of good initial guesses of the next optimum subproblem after including a new subproblem. By solving a series of subproblems, the optimum values of the original large scale DRPE optimization problem can be derived efficiently. The effectiveness of the proposed strategies can be demonstrated through two industrial processes, including free radical polymerization of styrene and the purified terephthalic acid oxidation process system.

1. INTRODUCTION

values of the model parameters and the unmeasured variables. It reconciles the measured variables simultaneously as well. On the basis of the process model, DRPE is combined with optimization techniques to reconcile the measurement data and estimate the model parameters. In order to obtain more reliable parameter estimates, a series of measurement data sets from different operational conditions will be used for DRPE. Yip and Marlin used multiple measurement data sets to improve the DRPE problem.9 They pointed out that the expected advantage of using multiple measurement data sets is a reduction in the effects of stationary noise, but the number of parameters estimated increases. The most straightforward approach to the DRPE problem is to use nonlinear programming to estimate the measured and unmeasured variables, and the model parameters simultaneously. However, the dimensionality of the DRPE problem increases directly with the number of measurement data sets. Since the process model is usually large-scale and nonlinear, and the number of degrees of freedom in the DRPE problem is very large, a feasible path optimization approach can be very expensive because an iterative calculation is required to solve the undetermined model. Therefore, special attention was paid to the computation strategies in the DRPE problem.

The model-based applications, such as real-time optimization and process control, are often used in industrial processes. In these applications, the actual state of the process is determined based on a detailed process model.1 Although great efforts have been made to develop rigorous models for many processes, most of these models contain some undetermined model parameters which have to be estimated based on the process measurement data. However, the measurement data gathered from the real plant usually contain random and gross errors, and they should be reconciled to improve the accuracy of the estimated model parameters. Therefore, data reconciliation and parameter estimation (DRPE) is considered as the key to the successful improvement of real-time optimization and process control. The inaccurate measurement data violate conservation laws and other process constraints. Data reconciliation resolves the contradictions between the measurements and their constraints. It also transforms the contaminated data into consistent information. After data reconciliation, parameter estimation is carried out. The reconciled values of the measured variables are used to estimate the values of the model parameters.2,3 The computing procedures are complex. Without fully using the redundancy in the models, the results of the two-step approach are not accurate. This inefficient two-step approach has led to the development of simultaneous strategies for DRPE.4−8 Simultaneous DRPE minimizes the least-squares error in measurements and subjects to model constraints and bounds. It can estimate the © 2015 American Chemical Society

Received: Revised: Accepted: Published: 5697

December 23, 2014 April 14, 2015 April 22, 2015 April 22, 2015 DOI: 10.1021/ie504977k Ind. Eng. Chem. Res. 2015, 54, 5697−5709

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Industrial & Engineering Chemistry Research A more efficient way is to use an infeasible path approach to solve the NLP problem. The infeasible path approaches, such as the successive quadratic programming method (SQP) and the interior point method, are only efficient in solving large-scale NLP problems with few degrees of freedom.10 They are inefficient in solving large-scale NLP methods sometimes as the DRPE problem contains large scale and nonlinear constraints and many degrees of freedom. The initial guess sufficiently close to the optimum values is important to optimization algorithms.11 If the initial value of the optimization variables is far away from the optimum, the large-scale NLP methods usually cannot give convergent results or may converge to a (sub) local optimum for the large scale DRPE problem. Many computational strategies were proposed, such as the reduced successive quadratic programming strategy,12 the nested three-stage computation framework,6 particle swarm optimization,7 parallel calculation methods,13,14 iterative algorithms,15 and pervasive knowledge discovery strategies by just-in-time learning,16 etc. When the production of multigrade chemical and polymer products is changed to meet the market demand, it requires the change of the operating conditions as well. However, most papers were concentrated on how to improve the effectiveness of the largescale NLP algorithms. The decomposition strategies for the large scale DRPE problem with multiple-operational conditions were seldom considered in literature. This paper focuses on the large scale DRPE problem with multiple-operational conditions, which is a large scale optimization problem with many degrees of freedom. On the basis of the characteristics of the DRPE problem, the direction of incremental objectives of the DRPE problem, and the direction of incremental parameters of the DRPE problem are considered to decompose the original DRPE optimization problem into a series of incremental-scale subproblems. Then an integration routine is easily used to expand the subproblems and to determine the current optimal subproblems until returning to an original large scale problem. The solutions to the current subproblem are used as initial guesses for the next optimal subproblem after including a new subproblem. By solving the series of incremental-scale subproblems, the optimum values of the original DRPE optimization problem can be obtained. In this work, three programming strategies are proposed to solve the large scale DRPE problem with multi-operational conditions. The rest of the paper is organized as follows. The characteristics of the large scale DRPE problem with multi-operational conditions are described and formulated in the next section. Three sequential incremental-scale subproblems programming strategies are then introduced in section 3. The effectiveness of the proposed strategies is demonstrated by the numerical results of two case studies in chemical processes in section 4. Finally, conclusions are drawn in section 5.

C

J= min

x c , uc , θ

P(C , K ):

c = 1, ⋯ , C

s. t .

∑ Jc (xc , yc) c=1

f(xc , uc , θ) = 0 x L ≤ xc ≤ x U

c = 1, 2, ..., C c = 1, 2, ..., C c = 1, 2, ..., C

u L ≤ uc ≤ uU θ L ≤ θ ≤ θU (1)

where C is the total number of operational conditions; Jc(xc,yc) is the objective function depending on the difference between the measured and the reconciled data of the measured variables in the c th operational condition; yc = [yc,1, yc,2,...,yc,M ] is the set of the measured data in the cth operational condition and M is the total number of the measured variables; xc = [xc,1,xc,2,...,xc,M ] is the set of the reconciled data in the cth operational condition; uc = [uc,1,uc,2,...,uc,N ] is the set of the unmeasured variables in the c th operational condition and N is the total number of the unmeasured variables; xL, xU, uL, and uU are the lower and upper bounds of the measured and unmeasured variables respectively; θ = [θ1,θ2, ..., θK] is the set of parameters in the process model. It also includes the locations of the estimated variable biases which are known a priori, and K is the total number of model parameters; θL and θU are the lower and upper bounds of the parameters, respectively; and f is the set of equality constraints. When the residuals (xc,m − yc,m, m = 1,...,M) of the mth measurement variable in the cth operational condition are normally distributed with zero mean and known deviation, weighted least-squares can be directly applied.17 The formulation of Jc(xc,yc) in eq 1 can be written as M

Jc (xc , yc) =

∑ (xc ,m − yc ,m )2 /σc ,m2 m=1

(2)

where σc,m is the standard deviation of the mth measured variable in the cth operational condition. In the large scale DRPE problem with multi-operational conditions, there are C(M + N) + K optimization variables. The equality constraints are usually considered as the rigorous process model equations. Therefore, there are a huge number of equality constraints. If f is a G-vector of functions, the number of degrees of freedom is C(M + N − G) + K. This indicates that the number of degrees of freedom increases linearly with the number of operational conditions. Solving this kind of optimization problem requires high-performance optimization algorithms and good initial guesses of the optimum values. A lot of efforts have been devoted to high-performance optimization algorithms, such as the reduced space sequential quadratic programming and interior point methods.12,13,18−20 However, these large-scale NLP algorithms are sometimes inefficient in solving this optimization problem, because it is usually difficult to obtain good initial guesses for so many optimization variables. If the initial guesses of the optimization variables are far from the optimum values, these large-scale NLP algorithms usually cannot yield convergent results for the large-scale DRPE problem. The next section proposes an incremental method to construct a series of incremental-scale subproblems for programming step by step so that good initial guesses of the optimum can be generated.

2. FORMULATION OF A LARGE SCALE DRPE PROBLEM WITH MULTI-OPERATIONAL CONDITIONS AND ITS CHARACTERISTICS The general formulation of a large scale DRPE problem with multi-operational conditions can be described as 5698

DOI: 10.1021/ie504977k Ind. Eng. Chem. Res. 2015, 54, 5697−5709

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Industrial & Engineering Chemistry Research

3.1. Incremental Objective Based Large Scale DRPE Programming. In the objective function (eq 1), the original DPRE problem (P(C,K)) can be partitioned into a series of incremental-scale subproblems (P(1,K), P(2,K), ..., P(C,K)) based on the operational conditions. The first subproblem P(1,K) uses the measured data with one operational condition only; the second subproblemP(2,K) contains the measured data of the previous operational condition in P(1,K) and the measured data of an additional operational condition. Finally, the last subproblem, P(C,K), uses the measured data of all the operational conditions. Thus, the problem can be formulated as follows,

3. PROGRAMMING STRATEGIES FOR SEQUENTIAL INCREMENTAL-SCALE SUBPROBLEMS Since the DPRE problem with multi-operational conditions is a large-scale and nonlinear optimization problem with a large number of degrees of freedom, it is inefficient to solve this problem directly by the large-scale NLP algorithms. Moreover, the initial guesses should be provided when the optimization routine is conducted. Then the optimization routines will iteratively improve the initial guesses for the convergences to optimal solutions. Traditionally, solutions of a special DRPE problem P0 = P(c = 0,K) in the normal operational condition are used first as the initial guesses. The problem P0 is defined as P0: f (x 0, u 0, θ0) = 0

1

(3)

J=

where x0,u0, and θ0 are the measured variables, unmeasured variables, and parameters in the normal operational condition, respectively. Then the original DRPE problem P(C,K) is solved using the solutions of problem P0 as initial guesses (Figure 1). In

min

x c , uc , θ

P(1, K ): s . t .

∑ Jc (xc , yc ) c=1

c=1 c=1 c=1

f(xc , uc , θ) = 0 x L ≤ xc ≤ x U u L ≤ uc ≤ uU θ L ≤ θ ≤ θU

(4)

2

min

x c , uc , θ

J=

∑ Jc (xc , yc) c=1

s . t . f(xc , uc , θ) = 0 c = 1, 2

x L ≤ xc ≤ x U P(2, K ):

c = 1, 2

u L ≤ uc ≤ uU

Figure 1. Procedure of solving the original DRPE problem P(C,K) by using the solutions to problem P0.

c = 1, 2

θ L ≤ θ ≤ θU ⋮

Figure 1, z0* = [x0 u0 θ0] is the solutions to problem P0, z0c = [x0c u0c θ0 ], c = 1,2,...,C are the initial guesses of problem P(C,K). In practical industrial processes, the operational conditions usually switch frequently in a wide range, the initial values z0c may be generally far away from the optimum zc* of the DRPE problem P(C,K). With the bad initial guesses, it is difficult to solve this DPRE problem using the large-scale NLP algorithms. The search space in this large scale problem is very large; this means that the initial guesses of optimization variables in the DRPE problem are difficult to derive. Therefore, the optimum points of the parameters and the estimated variables cannot be guaranteed. To reduce the search space, based on the characteristics of the original DRPE problem (P(C,K), the whole optimal problem is decomposed into the series of incremental-scale subproblems. Then the approach that finds out the optimal solutions of the large scale problem is an incremental strategy. That is, the initial subproblem for optimization is a small one and the optimal problem sequentially grows by adding new variables and constraints. The incrementalscale subproblems are solved iteratively; the solutions to the current subproblem are used as initial values of the next subproblem. This incremental strategy not only can gradually increase the dimension of the search space explored by the NLP algorithms but also may substantially reach convergence, because good initial guesses for part of the variables are provided by the optimal solution to the previous subproblem. In this section, three different incremental methods, including incremental objectives, incremental parameters, and hybrid increments of objectives and parameters, are proposed.

⋮ ⋮

(5)

C

J= min

x c , uc , θ

P(C , K ): s . t .

∑ Jc (xc , yc ) c=1

f(xc , uc , θ) = 0 x L ≤ xc ≤ x U

c = 1.2, ..., C c = 1.2, ..., C c = 1.2, ..., C

u L ≤ uc ≤ uU θ L ≤ θ ≤ θU

(6)

The series of subproblems increase the number of optimization variables step by step. As the objective in each subproblem is different, these sequential subproblems can be called as incremental objective based DRPE problems (IODRPE problems). In the IO-DRPE problems, starting from the subproblem P(1,K) that contains one condition only, it is easiest to be solved; the last subproblem P(C,K) is the most difficult one to be solved because the number of variables in P(C,K) is significantly larger than that in P(1,K). If the solutions to the current subproblem are used as initial guesses of the next subproblem, the optimum values of the original DRPE problem can finally be obtained because the good initial guess points are provided in the large search space. The procedure of solving IODRPE problems can be depicted in Figure 2. First, the solutions 5699


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subproblem P(c + 1, K), the next operational condition which is close to the previous conditions (c) can be determined by (ic , jc ) = arg min SIMi , j i,j

i = 0, 1, ..., c ; j = 1, ..., C − c (8)

With the similarity operation, the IO-DRPE problems are presented with a data set containing yc, c = 1,...,C; the size of the subproblem is sequentially enlarged after the most similar operational condition is added using the following steps iteratively: Step 1: Set the solutions to problem P0 as the initial guess (z0*) and c ← 0. Step 2: Calculate similarity SIMi,j, i = 0,1,...,c;j = 1,...,C − c using eq 8 and select the most similar condition (jc). Step 3: Set c ← c + 1. Add the measured data in the jcth operational condition to construct subproblem P(c,K). Set z0i = zi*, i = 1,...,c − 1 and z0c = zic* Step 4: Use the large-scale NLP algorithms to solve the subproblem P(c, K) with the initial guesses (z0i ,i = 1,2,...,c). If the solutions to the subproblem are converged, the convergent solutions to this subproblem are used as initial guesses for the next subproblem P(c + 1,K) and go to Step 5; otherwise, go to Step 6. Step 5: If c ≤ C, go to Step 2; otherwise, the solutions to the large scale problem (P(C,K)) are completed. Step 6: IO-DRPE programming fails; try other sequential incremental-scale subproblems strategies. 3.2. Incremental Parameter Based Large Scale DRPE Programming. As many parameter values of the nonlinear rigorous models derived from basic principles of physics and chemistry cannot be accurately predicted by theory, the unknown parameters must be determined by process measured data. On the basis of model parameters, the original DPRE problem can also be partitioned into a series of incremental-scale subproblems (P(C, 1), P(C, 2), ..., P(C, K)). The first subproblem P(C,1) tunes one parameter only while the values of the other parameters are fixed. The second subproblem P(C,2) contains two tuned parameters, including the previous parameter in P(C, 1) as well as a new parameter, whereas the rest of the parameter values are fixed. Finally, the last subproblem P(C,K), which is the original DPRE problem, tunes all of the model parameters. Thus, sequential subproblems can be called incremental parameter based large-scale DRPE problems (IPDRPE problems), which are formulated as follows,

Figure 2. Procedure of solving IO-DRPE problems. Note: ic is derived from eq 8

z*0 to problem P0 are used as initial guesses of subproblemP(1,K). And then the first subproblemP(1,K) is solved. The dimension of the search space to be explored by the NLP algorithms is not large and subproblem P(1,K) can be solved easily. Sequentially, the solutions z*1 to problem P(1,K) are used as initial guesses of subproblemP(2,K) and the second subproblem P(2,K) is solved. Therefore, to solve the current subproblem, the previous solution is used as the initial guesses, starting from P0. The subproblem size is gradually increased per operational condition until the full problem is recovered. The subproblems of IO-DRPE problems are based on the decomposition of the operational conditions. In order to guarantee that each subproblem with good initial guesses is solved, the next subproblem (P(c + 1,K)) to be solved should be similar to the previous subproblem (P(c,K)), which has been solved successfully. Also, P(c + 1,K) contains the same operational conditions of P(c,K) and an additional operational condition, assuming that similar solutions should be used to solve similar subproblems. Similarity can be applied to finding out which operational condition is the best to be added into P(c, K). When the Euclidean distance is chosen as the similarity measure between measurement vector yi in condition i and measurement vector yj in condition j, the similarity function can be defined by SIMi , j = 1 −

|| yi − yj || max

c1= 0,1,..., c

C

J=

|| yc − yc || 1

2

min

c 2 = 1,..,. C − c

i ≠ j ; i = 0, 1, ..., c ; j = 1, ..., C − c

x c , uc , θ

P(C , 1): s . t .

(7)

where y0 = x0, c is the total number of operational conditions that have been added to the current subproblem (P(c, K)), and C − c is the total number of operational conditions waiting to be added. On the basis of the similarity criterion, to build the next

∑ Jc (xc , yc) c=1

f(xc , uc , θ1) = 0 x L ≤ xc ≤ x U

c = 1, 2, ..., C c = 1, 2, ..., C c = 1, 2, ..., C

u L ≤ uc ≤ uU θ1L ≤ θ1 ≤ θ1U 5700

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Industrial & Engineering Chemistry Research C

min

x c , uc , θ

J=

∑ Jc (xc , yc) c=1

s . t . f(xc , uc , θ1 , θ2) = 0 x L ≤ xc ≤ x U L

U

L

θ1U

u ≤ uc ≤ u

P(C , 2):

θ1 ≤ θ1 ≤

c = 1, 2, ..., C c = 1, 2, ..., C c = 1, 2, ..., C

θ2 L ≤ θ2 ≤ θ2U ⋮ ⋮ ⋮ (10) C

J= min

x c , uc , θ

P(C , K ): s . t .

∑ Jc (xc , yc) c=1

f(xc , uc , θ ) = 0 x L ≤ xc ≤ x U

c = 1, 2, ..., C c = 1, 2, ..., C c = 1, 2, ..., C

u L ≤ uc ≤ uU θ L ≤ θ ≤ θU

(11) Figure 3. Procedure of solving IP-DRPE problems (θ0 = [θ0,1 θ0,2... θ0,k] is the vector of parameters in problem P0).

In the IP-DRPE problems, starting from the subproblem P(C, 1) that contains one tuned parameter only, it is easiest to be solved; the last subproblem P(C, K) is the most difficult one to be solved because the number of tuned parameters in P(C, K) is significantly larger than that in P(C, 1). If the solutions to the current subproblem are used as initial guesses of the next subproblem, the optimum of the original DRPE problem can finally be obtained because the solutions to the current subproblem provide a good starting point for the next subproblem in the large search space. The procedure of solving IP-DRPE problems is depicted in Figure 3. First, the solutions x*0 , u*0 , and θ01 = θ*1,0 to problem P0 are used as initial guesses of subproblem P(C,1). And then the first subproblem P(C,1) is solved. The dimension of the search space to be explored by the NLP algorithms is not large and the subproblem P(C,1) can be solved easily. Sequentially, the solution θ02 = θ*2,0 to problem P0 and the solutions x*c , u*c , c = 1,...,C and θ01 = θ1* to problem P(C,1) are used as initial guesses of subproblem P(C,2) and then the second subproblem P(C,2) is solved. Therefore, the basic concept in solving the solutions to the current subproblem is to use the solutions as the initial guesses of the next subproblem starting from P0 and gradually increase the subproblem size one by one parameter until the original problem is constructed. The subproblems of IP-DRPE problems are formed based on the decomposition of the parameters. In order to guarantee that each subproblem with good initial guesses is solved, the next subproblem (P(C,k + 1)) should be solved using the previous subproblem (P(C, k)) and a new parameter. This assumes that the sensitive parameters should be solved first because the parameter search space is smaller in the initial stages with fewer parameters. That is, the parameters which have more impact on the measured variables should be selected first in the initial stage. The sensitivity analysis is used in mathematical modeling to determine the influence of the parameter values on the equality

models, providing a means for finding out which parameter should be added to P(C,k). In the normal operating condition, the sensitivity coefficient S of the parameters can be defined as

∂f (12) ∂θ To find out the variance contribution of the individual parameter, the parameter ranking by orthogonalization is applied here.21 First, through QR factorization with the column permutation for sensitivity coefficient matrix S = {Sm,k}, as SE = QR, where the permutation matrix E provides a reference parameter ranking, and the matrix R offers a ranking to examine individual parameter variances. SE can be further refined by extracting the diagonal components with the cross product (SE) T (SE). The information matrix of the parameters can be expressed as,8,21 S=

[(SE)T (SE)]−1 = UD−2 UT

(13) −1

where D = diag[d1...dK] and U = R D is a unit upper triangular matrix. The total information matrix is defined as tr(ETSTSE) =

∑ i

uTi u i di2

(14)

where ui is the ith column in U. The norm of the individual variance contribution of each parameter is given by σθi ≡

|| u i ||2 di2

i = 1, ···, K (15)

where (∥ui∥ measures the cost in terms of additional variance to the whole parameter set. On the basis of the sensitivity criterion, for building the next subproblem P(C, k + 1), 2

/di2)

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Industrial & Engineering Chemistry Research the additional tuned parameter, which should be strongly related to the models of the subproblem P(C, k + 1), can be determined by ik = arg max σθi i

i = 1, ..., K − k

(16)

where k is the total number of parameters added to the current subproblem (P(C, k)), and K − k is the total number of parameters to be added. With the sensitivity analysis, the IPDRPE problem is presented with a tuned parameter set containing θk, k = 1,...,K; the size of the subproblem is sequentially enlarged after the most sensitive parameter is added iteratively using the following steps: Step 1: Set the solutions to problem P0 as the initial guess (xc*, * , i = 1,...,K) and k ← 0. uc*, c = 1,...,C and θ0,i Step 2: Calculate sensitivity σθi, i = 1,...K − k using eq 16 and select the most sensitivity parameter (ik). Step 3: Set k ← k + 1. Add the ikth parameter to construct subproblem P(C,k). Set x0c = x*c , u0c = u*c c = 1,...,C; θ0i = θ*i , i = 1,...,k − 1 and θ0k = θ*0,k. Step 4: Use the large-scale NLP algorithms to solve the subproblem P(C,k) with the initial guesses (x0c , u0c , c = 1,...,C and θ0i , i = 1,...,k). If the solutions to the subproblem are converged, the convergent solutions (x*c , u*c , c = 1,...,C and θ*i , i = 1,...,k) to this subproblem are used as initial guesses of the next subproblem P(C, k + 1) and go to Step 5; otherwise, go to Step 6. Step 5: If k ≤ K, go to Step 2; otherwise, the solutions to the large scale problem (P(C, K)) are completed. Step 6: The IP-DRPE programming is not converged; try other sequential incremental-scale subproblem strategies. 3.3. Hybrid Incremental Objective and Parameter Based Large Scale DRPE Programming. Although the incremental objective approach and the incremental parameter approach can be applied to solving the large scale DRPE problems, it usually takes a long time for the simple optimization methods to converge the solution. One cannot tell for certain whether IO-DRPE problems or IP-DRPE problems are better, but there is still room for further improvement. One may observe if the tuned parameters or the variables are easier to search and converge in the incremental search space, and then the incremental objective approach and the incremental parameter approach can be used interchangeably. This approach can deal with a hybrid incremental parameter and objective based largescale DRPE problem (HIPO-DRPE problem) as it combines the incremental objective method and the incremental parameter method for fast optimization of all the parameters and variables. Now, the original DRPE problem is decomposed via two directions (i.e., the operational condition and the parameter) shown in Figure 4. The horizontal direction denotes the number of incremental operational conditions while the vertical direction represents the number of incremental tuning parameters. Before discussing the optimization in the HIPO-DRPE problem, some basic definitions and properties are set forth concerning the accessible path from one subproblem to another in HIPO-DRPE problems. Definition of accessible path: A path from a subproblem P(k,c) to a subproblem P(i,j) is accessible if the solutions to the subproblem P(i,j) can be obtained by solving a series of subproblems from P(k,c) to P(i,j). The accessible path from the subproblem P(k,c) to the subproblem P(i,j) is labeled as ACC(P(k,c), P(i,j)).

Figure 4. Procedure of solving HIPO-DRPE problems.

For example, if the solutions to subproblem P(3,3) can be determined by solving a series of subproblems, such as P(1, 1) → P(2, 1) → P(3, 1) → P(3, 2) → P(3, 3) P(1, 1) → P(1, 2) → P(1, 3) → P(2, 3) → P(3, 3), or P(1, 1) → P(1, 2) → P(2, 2) → P(2, 3) → P(3, 3) etc.

the path from the subproblem P(1,1) to the subproblem P(3,3) is accessible. That is, the optimum of the original DRPE optimization problem P(C,K) can be solved if an accessible path from either P(c,1), c = 1,...,C or P(C, k), k = 1,...,K to P(C, K) exits in the series of subproblems. Like the IO-DRPE approach or the IP-DRPE approach, the starting point in the HIPO-DRPE problem from either the subproblem P(C,1) (that contains one tuned parameter) or the subproblem P(1,K) (that contains one operational condition) is solved first; the last subproblem P(C,K) is the most difficult one to be solved because the number of tuned parameters and the number of operating conditions in P(C,K) are significantly larger than those in P(1,K) or P(C,1). The search strategy of the HIPODRPE problem is to find an accessible path from either P(C,1) to P(C,K) or P(1,K) to P(C,K), and to solve the series of HIPODRPE subproblems from either P(1,K) P(C, 1) to P(C,K) on the accessible path. If the solutions to the current subproblem are used as initial guesses of the next subproblem, the optimum of the original DRPE problem can finally be obtained because the solutions to the current subproblem provide a good starting point for the next subproblem in the large search space. Considering the properties of HIPO-DRPE subproblems, two search algorithms are proposed, including the objective-first search algorithm and the parameter-first search algorithm. 3.3.1. Objective-First Search Algorithm. The objective-first search algorithm considers the horizontal (operating condition) direction as the primary search direction and the vertical (tuned parameter) direction as the secondary search direction. The procedure of using the objective-first search algorithm for the HIPO-DRPEE subproblems is shown in Figure 5. First, the solutions to the DRPE problem P0 in the normal condition are used as initial guesses to solve subproblem P(C,1). If the 5702


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are used as initial guesses for the next subproblem P(C, k + 1). If the subproblem cannot be solved, go to Step 7. End Step 6: If c ≤ C, go to Step 2; otherwise, the solutions to the large scale problem (P(C,K)) are obtained. Step 7: The HIPO-DRPE programming with the objectivefirst search algorithm is not converged; try other sequential incremental-scale subproblems strategies. 3.3.2. Parameter-First Search Algorithm. The parameterfirst search algorithm is contrary to the objective-first search algorithm. It considers the vertical direction as the primary search direction, and the horizontal direction as the secondary search direction. The procedure of using the parameter-first search algorithm for HIPO-DRPEE subproblems is shown in Figure 6. Figure 5. Procedure of solving the HIPO-DRPE problem using the objective-first search algorithm.

subproblem P(C,1) is solved successfully, the incremental objective is kept along with the horizontal direction to form the subproblem P(2,K). The solutions to problem P(1,K) are used as initial guesses of the subproblem P(2,K) and the second subproblem P(2,K) is solved. (In Figure 5, the green arrow represents the subproblems solved successfully and the red arrow represents the subproblems solved unsuccessfully). When the solution to any subproblem P(c,K) in the horizontal direction fails to converge, change the search direction to the vertical one and solve the subproblem starting from P(c,1); then solve the subproblems in the vertical direction one by one tuned parameter (P(c,1) → P(c,2) → ... → P(c,K)). After P(c,K) is obtained, keep solving the rest of the subproblems in the horizontal direction until the original problem P(C,K) is constructed. To sum up, the HIPO-DRPE problem with the objective-first search algorithm is presented with a set a data containing yc, c = 1,...,C and a tuned parameter set containing θk, k = 1,...,K; the size of the subproblem is sequentially enlarged after the parameter strongly related to the previous conditions or the most sensitive parameter is added iteratively using the following steps: Step 1: Set the solutions to the problem P0 as the initial guesses and c ← 0, k → K, A = {P0}. Step 2: Calculate similarity SIMi,j, i = 0,1,...,c; j = 1,...,C − c using eq 8 and select the most similar condition (jc). Step 3: Set c ← c + 1. Add the measured data in the jcth operational condition to construct the subproblem P(c,K). Set z0i = zi*. i = 1,...,c − 1 and z0c = zic*. Step 4: Use the large-scale NLP algorithms to solve the subproblem P(c, K) with the initial guesses (z0i .i = 1,2,...,c). If the solutions to the subproblem are converged, set A = A ∪ {P(c,k)}; otherwise, go to Step 5. The convergent solutions to this subproblem are used as initial guesses of the next subproblem P(c + 1,K) and go to Step 6. Step 5: For k = 1:K. (i) Calculate sensitivity σθi, i = 1,...,K − k + 1 using eq 15 and select the most sensitive parameter (ik). Add the ikth parameter to construct the subproblem P(c,k). The procedure of selecting initial guesses is the same as that of the IP-DRPE programming. (ii) Use the large-scale NLP algorithm to solve the subproblem P(c, k). (iii) When the solutions to the subproblem are converged, set A = A ∪ {P(c, k)}. The convergent solutions to this subproblem

Figure 6. Procedure of solving the HIOP-DRPE problem using the parameter-first search algorithm.

First, the solutions to the DRPE problem P0 in the normal condition are used as initial guesses to solve the subproblem P(C,1). If the subproblem P(C,1) is solved successfully, the incremental parameter is kept in the vertical direction to form the subproblem P(C,2). The solutions to the problem P(C,1) are used as initial guesses of the subproblem P(C,2) and the second subproblem P(C,2) is solved (in Figure 6, the green arrow represents the subproblems to be solved successfully and the red arrow represents the subproblems to be solved unsuccessfully). When the solution to any subproblem P(C,k) in the vertical direction fails to converge, change the search direction to the horizontal one and solve the subproblem starting from P(1,k); then solve the subproblems in the horizontal direction one by one operational condition (P(1,k) → P(2,k) → ... → P(C,k)). After P(C,k) is obtained, keep solving the rest of the subproblems in the vertical direction until the original problem P(C,K) is constructed. To sum up, the HIPO-DRPE problem with the parameter-first search algorithm is presented with a set of data containing yc, c = 1, ... ,C and a tuned parameter set containing θk, k = 1,...,K; the size of the subproblem is sequentially enlarged after the parameters strongly related to the previous conditions 5703


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product quality and process efficiency. The mathematical model of Schmidt and Ray is used to describe the process of polymerization in a jacketed continuous stirred tank reactor.22 The material balance equations for the concentration of polymers of different molecular weights are presented and shown to be zero, the first, and the second moment balances. The reaction mechanisms of a free radical polymerization of styrene would start with the decomposition reaction of an initiator, and then initiator reaction, propagation until the termination reaction. From the simulation runs, the concentration of polymers with the chain length of more than 8 000 units of polymers has the concentration of 10−12 or smaller, so all the resulting discussions involve MWD with the degree of polymerization, and the number of the chain length is set to be 8 000. Thus, the total number of equations in the process model is 8 014. The process model is large scale and nonlinear. The simulation environment for this example is AMPL.23 The NLP solver is IPOPT. A finite difference method is used to derive sensitivity information in this test. The other efficient methods for sensitivity analysis can also be used. Pirnay et al. introduced an efficient strategy to obtain the sensitivity information.24 The sensitivities to parameters can be determined with the minimal computational cost by reusing matrix factorizations from the IPOPT solver.24 A state-of-the-art nonlinear programming (NLP) solver proposed by Zavala and Biegler can also deal with the problem of calculating sensitivity information.25 Although the distribution output process has been discussed in the field of process control design, there is not much investigation into the DRPE problem to enhance distribution output in the literature. The present study is the first one to investigate the performance of the programming strategy of sequential incremental-scale subproblems. Case 1: The measured variables are assumed to be the monomer concentration in the inlet monomer feed stream (Cm), the concentration of solvent in the solvent feed (Cs) and the initiator concentration in the initiator feed streams (Ci), the volumetric flow rates of the inlet solvent (Fi), the inlet monomer (Fm), the inlet initiator (Fs), and the inlet feed temperature (Ti). The measured variables change from one operational condition to another. The measured variables consist of four data sets sampled in four different operational conditions. The steady state concentrations of polymers are unmeasured variables. The heat of reaction coming from the propagation reaction is considered as the model parameter to be estimated. The DPRE problem with four operational conditions (C = 4) and one parameter (P = 1) for this free radical polymerization of the styrene system is tested here. In this test, the solutions to the simulation problem P0 are given as the initial values. P0 is obtained from the nominal operational conditions. The initial values of polymer concentration for the free radical polymerization of styrene are shown in Figure 7. The optimal values of the DRPE problem with four operational conditions are also included in Figure 7. After checking all the distribution of polymer concentration in Figure 7, it is found that the initial values of the optimization variables and other distribution with four operational conditions are different. If the large-scale NLP algorithm (IPOPT) is used to directly handle the DRPE problem with those initial values, it cannot successfully solve the DRPE problem and yield convergent results for the large-scale DRPE problem. From the solution, it is known that the solutions to a special DRPE problem P0 = P(c = 0,K) in the normal operational condition are not good initial guesses for the current solution to

or the most sensitive parameter are iteratively added using the following steps: Step 1: Set the solutions to the problem P0 as the initial guesses and k ← 0, c ← C, A = {P0}. Step 2: Calculate sensitivity σθi, i = 1,...,K − k using eq 15 and select the most sensitive parameter (ik). Step 3: Set k ← k + 1. Add the ikth parameter to construct the subproblem P(C, k). Set x0c = xc*, u0c = uc*, c = 1,...,C, θ0i = θi*,i = 1,...k − 1 and θ0k = θ*0,k. Step 4: Use the large-scale NLP algorithms to solve the subproblem P(C, k) with the initial guesses (x0c , u0c , c = 1,...,C and θ0i , i = 1,...,k). If the solutions to the subproblem are converged, set A = A∪{P(c, k)}; otherwise, go to Step 5. The convergent solutions (x*c , u*c , c = 1,...,C and θ*i , i = 1,...,k) to this subproblem are used as the initial guesses of the next subproblem P(C, k + 1) and go to Step 6. Step 5: For c = 1:C (i) Calculate similarity SIMi,j, i = 0,1,...,c; j = 1,...,C − c using eq 8 and select the most similar condition (jc). Add the measured data in the jcth operational condition to construct subproblem P(c, k). (ii) Use the large-scale NLP algorithm to solve the subproblem P(c,k). The procedure of selecting initial guesses is the same as that of the IO-DRPE programming. (iii) When the solutions to the subproblem are converged, set A = A ∪ {P(c,k)}. The convergent solutions to this subproblem are used as the initial guesses of the next subproblem P(c + 1,k). If the subproblem cannot be solved, go to Step 7. End Step 6: If k ≤ K, go to Step 2; otherwise, the solutions of the large scale problem (P(C, K)) are obtained. Step 7: The HIPO-DRPE programming with the parameterfirst search algorithm is not converged; try other sequential incremental-scale subproblems strategies.

4. NUMERICAL RESULTS In this section, two industrial applications are presented, respectively. In section 4.1, a typical polymerization process, free radical polymerization of styrene, is used. Because of the dynamic change of the market, the grade of the polymerization product needs to be changed frequently. The grade switch-over in the polystyrene process would cause fluctuation of product quality. Moreover, unlike the quality output in conventional processes, the product output in the polymerization process has the nature of a distribution output which significantly influences the product quality. In section 4.1, a real industrial process, the purified terephthalic acid (PTA) oxidation process system, is investigated. PTA is made by a reaction between the secondary petroleum product paraxylene (PX) and acetic acid. It is a raw material used in making high-performance multipurpose plastics, such as polybutyl terephthalate, polyethylene terephthalate, polytrimethylene terephthalate, and the new Bioplastic. It has been garnering attention in recent years. All the process models in the above two cases are large scale and nonlinear. Through the above cases, the programming strategies of sequential incremental-scale subproblems are compared with the traditional DRPE method in terms of performance. 4.1. Free Radical Polymerization of Styrene. The free radical polymerization of styrene is simulated. The output of this process has the nature of distribution, such as molecular weights. This type of output is called the distribution output. Control of the distribution output is crucial as it significantly influences 5704


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the DRPE problem P(4,1). The initial objective is 1.2955 × 103 with the initial guesses provided by problem P0. It is larger than the final objective (9.9680) of the DRPE problem P(4,1). The value of primal infeasibility (2.09 × 103) and the value of dual infeasibility (9.94 × 107) are also very large. This suggests that full solution with IPOPT does not work because of the bad initial guesses. As there is only one parameter in this case, the proposed IODRPE algorithm is used to achieve a warm start for IPOPT, which partitions the original DPRE problem (P(4,1)) into a series of incremental-scale subproblems (P(1,1),P(2,1), ..., P(4,1)) based on the operational conditions. The subproblem P(1,1) is first solved by the measured data with one operational condition only and it is the smallest scale optimization problem among all the subproblems. It is easiest to be solved. Then the solutions to the current subproblem P(1,1) are used as initial guesses of the next subproblem P(2,1) and then solve the next subproblem P(2,1). The solutions to the subproblem P(2,1) are used as initial guesses of the next subproblem P(3,1) and so on. By partially deriving good initial guesses step by step, the warm

Figure 7. Comparisons of the initial values of polymer concentration with optimal values of polymer concentration after DRPE optimization.

Table 1. Main Information of the Initial Values, Measured Values, and the Reconciled Values for Free Radical Polymerization of Styrene Cm operational condition 1 2 3 4 operational condition 1 2 3 4 operational condition 1 2 3 4 operational condition

Cs

measured value

initial value

reconciled value

measured value

initial value

reconciled value

9.304893 9.694608 8.397110 9.921820

7.352360 7.352360 7.352360 7.352360 Cm

9.304863 9.694603 8.397126 9.921819

7.725985 7.475132 6.860171 6.600002

8.490063 8.490063 8.490063 8.490063 Cs

7.725985 7.475132 6.860171 6.600002

measured value

initial value

reconciled value

measured value

initial value

reconciled value

9.304893 9.694608 8.397110 9.921820

7.352360 7.352360 7.352360 7.352360 Cm

9.304863 9.694603 8.397126 9.921819

7.725985 7.475132 6.860171 6.600002

8.490063 8.490063 8.490063 8.490063 Cs

7.725985 7.475132 6.860171 6.600002

measured value

initial value

reconciled value

measured value

initial value

reconciled value

9.304893 9.694608 8.397110 9.921820

7.352360 7.352360 7.352360 7.352360 Ci

9.304863 9.694603 8.397126 9.921819

7.725985 7.475132 6.860171 6.600002

8.490063 8.490063 8.490063 8.490063 Fi

7.725985 7.475132 6.860171 6.600002

measured value

initial value

reconciled value

measured value

initial value

reconciled value

1 2 3 4

6.285465 7.852452 7.398524 5.848812

8.694404 8.694404 8.694404 8.694404 Ci

6.285465 7.852452 7.398524 5.848812

1.39850 × 10−05 1.76087 × 10−05 1.65886 × 10−05 1.70688 × 10−05

1.50234 × 10−05 1.50234 × 10−05 1.50234 × 10−05 1.50234 × 10−05 Fi

1.87071 × 10−05 1.66995 × 10−05 1.85679 × 10−05 1.69522 × 10−05

operational condition

measured value

initial value

reconciled value

measured value

initial value

reconciled value

1 2 3 4

6.285465 7.852452 7.398524 5.848812

8.694404 8.694404 8.694404 8.694404

6.285465 7.852452 7.398524 5.848812

1.39850 × 10−05 1.76087 × 10−05 1.65886 × 10−05 1.70688 × 10−05 Ti

1.50234 × 10−05 1.50234 × 10−05 1.50234 × 10−05 1.50234 × 10−05

1.87071 × 10−05 1.66995 × 10−05 1.85679 × 10−05 1.69522 × 10−05


measured value

initial value

reconciled value

1 2 3 4

313.3145 308.4689 335.4087 347.3934

318.5628 318.5628 318.5628 318.5628

313.3145 308.4689 335.4087 347.3934

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Figure 8. Flowchart of PTA oxidation process system.

concentration in the inlet monomer feed stream, and the initiator concentration in the outlet initiator feed stream are included. Therefore, the number of measured variables is increased. The overall measured variables also have four data sets sampled in four different operating conditions. The number of model parameters is also increased. Three constants of the reaction rate in the decomposition reaction, including the propagation reaction and the termination reaction are also considered as model parameters to be estimated. Therefore, the number of model parameters is increased to be 4. The DPRE problem with four operational conditions (C = 4) and four parameters (P = 4) would be solved. This DRPE problem is directly solved by the large-scale NLP algorithm (IPOPT), and it still cannot yield convergent results. Also, the proposed (objective based and parameter based) sequential subproblem programming strategies also fail to solve this large scale DRPE problem. Therefore, the hybrid sequential incremental-scale programming strategy for the subproblems is required and used in this case study to achieve a warm start for IPOPT, which partially derive good initial guesses step by step by solving a series of incremental-scale subproblems. With the use of hybrid sequential incremental-scale subproblem programming strategy, there is an accessible path from P0 to P(4,4) by the objective-first search algorithm. ACC(P0,P(4,4)) is shown as follows:

start for IPOPT can be achieved. Finally, the optimum values of the original DRPE problem P(4,1) is obtained because the good initial guesses are provided by solving the series of incrementalscale subproblems. This means that an accessible path (ACC(P0,P(4,1))) from P0 to P(4,1) with a sequentially incremental size of the objective is represented by 95

27

29

21

41.55s

19.63s

39.14s

39.94s

P0 ⎯⎯⎯⎯⎯→ P(1, 1) ⎯⎯⎯⎯⎯→ P(2, 1) ⎯⎯⎯⎯⎯→ P(3, 1) ⎯⎯⎯⎯⎯→ P(4, 1)

where the figure above the arrow is the number of iterations for solving each sequential incremental-scale subproblem, and the figure below the arrow is the corresponding solving time. The total number of iterations is 172, including 95 times in P(1,1), 27 times in P(2,1), 29 times in P(3,1), and 21 times in P(4,1). The total solving time is 140.26 s, which is no more than 3 min. The proposed method is efficient to the large scale optimization problem with 8 014*4 nonlinear constraints. The proposed IODRPE algorithm can achieve a warm start for IPOPT, and the initial objective is 9.96802 with the initial guesses provided by problem P(3,1). It is very close to the final objective (9.96798). The values of the primal infeasibility (3.49 × 10−5) and the values of the dual infeasibility (3.88 × 103) are also much smaller. This suggests that the full solution with IPOPT can work because of the good initial guesses. The reconciled values of the measured variables after DRPE optimization is shown in Table 1. The reconciled values are quite different from the initial values. The comparisons also indicate that the NLP algorithm failed to directly solve the large-scale DRPE problem with those initial values, but the proposed IODRPE algorithm can get the converged solutions in these four operational conditions even if the initial values of the variables are far away from the optimal values. Case 2: From the solution in Table 1, it is also found that some measured variables are nonredundant measured variables, because the reconciled values are almost the same as the measured values. For industrial applications, more redundantly measured information should be included. In four different operating conditions in this case study, the weight-average molecular weight, the reactor temperature, the monomer

As we can see, the total solving time (648.63 s) is much longer than the time used in the first case study. Thus, the sequential subproblem programming strategies are not suitable for online DRPE problem (i.e., model updating for real time optimization) but useful to the off-line DRPE problem (i.e., model design). With the use of hybrid sequential incremental-scale subproblem programming strategy, the parameter-first search 5706


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Industrial & Engineering Chemistry Research Table 2. Main Information of the Initial Values and the Measured Values in PTA Oxidation Process System flow rate of product TA/kg h−1 operational condition

measured value

1 2 3 4 5

measured value

1 2 3 4 5

2094.31 2273.14 2412.93 2512.93 2722.86

measured value

58086.6 64043.6 68703.0 72041.8 79022.4

initial value 21.31 21.31 21.31 21.31 21.31

mass fraction of oxygen in stream 26/% 1 2 3 4 5

offset

69815.2 11728.6 75772.2 11728.6 80431.6 11728.6 83770.4 11728.6 90751.0 11728.6 flow rate of consumption acetic acid/kg h−1



mass fraction of product 4-CBA/PPM

initial value

initial value

offset

2464 445117 2498 445117 2643 445117 2541 445117 2600 445117 mass fraction of oxygen in stream 29/%

offset

measured value

2073.00 6.01 2251.83 5.70 2391.62 5.69 2491.62 5.90 2701.55 6.00 mass fraction of carbonic oxide in stream 26/%

initial value

442653 442619 442474 442576 442517 offset

10.90 4.89 10.90 5.20 10.90 5.21 10.90 5.00 10.90 4.90 mass fraction of carbon dioxide in stream 26/%

measured value

initial value

offset

measured value

initial value

offset

measured value

initial value

offset

4.04 4.03 4.06 4.11 4.00

15.84 15.84 15.84 15.84 15.84

11.80 11.81 11.78 11.73 11.84

0.472 0.479 0.473 0.474 0.470

0.0004 0.0004 0.0004 0.0004 0.0004

0.472 0.479 0.473 0.474 0.470

1.44 1.43 1.43 1.41 1.45

0.058 0.058 0.058 0.058 0.058

1.382 1.372 1.372 1.352 1.392

algorithm can find an accessible path from P0 to P(4,4), and ACC(P0,P(4,4)) is shown as follows:

reaction rate constants in the above 10 reactions. The measured variables include the flow rate of the product TA, mass fraction of the product 4-CBA, the flow rate of consumption acetic acid, mass fraction of oxygen in stream 26, mass fraction of carbonic oxide in stream 26, mass fraction of carbon dioxide in stream 26, and mass fraction of oxygen in stream 29. All the measured data are collected from the industrial plant. There are 5 different operational conditions in the operating plant, so there are 5 data sets for the sampled measured variables. The flow rates, pressures, temperatures, and fraction vary in different operational conditions. The DPRE problem uses 7 measured variables with 5 data set conditions (C = 5) and 10 model parameters (P = 10). There are 18 488 equations in this DRPE problem and some of the equations are nonlinear. Therefore, this DRPE problem is a large scale and nonlinear problem. First, the problem is solved based on Aspen Plus with the EO mode.26 The DMO solver is selected to implement a variant of the SQP algorithm to solve small or large-scale optimization problems. The sensitivity script command, supplied by Aspen Plus OOMF Script Language, is used to analyze the sensitivity information. In this test, the solutions to simulation problem are used as the initial values. The main information on the initial values and measured values is shown in Table 2. The initial values of the optimization variables may be far away from the optimum, because the offsets between measured values and initial values are so large. The large-scale NLP algorithm (SQP) is used to solve the DRPE problem directly with those initial values. However, it cannot yield convergent results for this large scale DRPE problem. Therefore, it fails to directly solve the DRPE problem. The proposed sequential incremental-scale programming strategies for the subproblems are used in this DRPE problem. There is no accessible path from P0 to P(5,10) using the objective based sequential incremental-scale subproblem strategy. However, an accessible path from P0 to P(5,10) using the parameter based sequential incremental-scale subproblem strategy can be derived. ACC(P0,P(5,10)) is shown as follows:

Using either the objective-first search algorithm or the parameter-first search algorithm, the final three solving procedures (P(4,1) → P(4,2) → P(4,3) → P(4,4)) are still the same, but the total solving time in this case study by the parameter-first search algorithm (595.37 s) is less than that by the objective-first search algorithm (648.63 s). 4.1. PTA Oxidation Process System. In the second example, the proposed strategies for sequential subproblem programming are used in the industrial PTA oxidation process system. The flowchart of the PTA oxidation process system is shown in Figure 8. The PTA oxidation process system includes the following ten elementary reactions: C8H8O + 5.5O2 → 4H2O + 8CO C8H6O3 + 4O2 → 3H2O + 8CO C2H4O2 + O2 → 2H2O + 2CO

C8H8O + 9.5O2 → 4H2O + 8CO2

C8H6O3 + 8O2 → 3H2O + 8CO2

C2H4O2 + 2O2 → 2H2O + 2CO2

C8H10 + O2 → C8H8O + H2O C8H80 + 0.5O2 → C8H8O2

C8H8O2 + O2 → C8H6O3 + H2O C8H6O3 + 0.5O2 → C8H6O4

The steady-state models of this system consist of material balances, equilibrium relations, fraction summation, enthalpy balances, and chemical balances, etc. The model parameters are 5707


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Industrial & Engineering Chemistry Research Table 3. Solutions of the Measured Variables Using Sequential Incremental-Scale Sub-Problem Programming Strategies flow rate of product TA/kg h−1 operational condition

measured value

1 2 3 4 5

mass fraction of product 4-CBA/PPM

reconciled value

offset

69815.2 69642.7 75772.2 75623.5 80431.6 80160.3 83770.4 83490.1 90751.0 90457.1 flow rate of consumption acetic acid/kgh−1

operational condition 1 2 3 4 5

measured value

172.5 148.7 271.3 280.3 293.9

2464 2522 2498 2484 2643 2580 2541 2615 2600 2589 mass fraction of oxygen in stream 29/%

measured value

reconciled value

offset

measured value

2094.31 2273.14 2412.93 2512.93 2722.86

2081.31 2294.14 2418.93 2501.93 2711.86

13 21 6 11 11

6.01 5.70 5.69 5.90 6.00

mass fraction of oxygen in stream 26/%

reconciled value

mass fraction of carbonic oxide in stream 26/%

reconciled value

offset 58 14 63 74 11 offset

6.10 0.09 5.60 0.1 5.75 0.06 5.97 0.07 5.89 0.11 mass fraction of carbon dioxide in stream 26/%


measured value

reconciled value

offset

measured value

reconciled value

offset

measured value

reconciled value

offset

1 2 3 4 5

4.04 4.03 4.06 4.11 4.00

4.00 4.03 4.06 4.07 4.07

0.04 0 0 0.04 0.07

0.472 0.479 0.473 0.474 0.470

0.471 0.477 0.473 0.471 0.471

0.001 0.002 0 0.003 0.001

1.44 1.43 1.43 1.41 1.45

1.44 1.44 1.43 1.42 1.42

0 0.01 0 0.01 0.03

subproblem programming strategies, the large-scale NLP algorithms can solve the large scale DRPE problems successfully. Also, the offsets of the measured variables decrease considerably after DRPE. A more consistent process model can be derived.

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From the solving procedures, it can be found that the number of iterations in the last four solving procedures are very small, which suggests that these solving procedures tune the model parameters only a little. In fact, the solutions to subproblem P(5,6) are almost the same as the solutions to the last subproblem P(5,10). The solutions of measured variables and parameters can be seen in Table 3. In Table 3, the offsets of the measured variables decrease considerably after DRPE. The offset of each measured variable is small. Therefore, the offsets between the outputs of the tuned process model and the measured values from the real plant are small. The tuned process model is consistent with the real plant.

AUTHOR INFORMATION

Corresponding Author

*Phone: +886-3-2654107. Fax: +886-3-2654199. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge National Science Council, R.O.C. (NSC 102-2811-E-033-001), the National Natural Science Foundation of China (No. 61374167; 51207112), and the Natural Science Foundation of Zhejiang Province (No. LQ14F030006) for financial support.

5. CONCLUSIONS In order to solve the large scale DRPE problem with multioperational conditions efficiently, three sequential incrementalscale programming strategies for subproblems are proposed in this paper. On the basis of the characteristics of the DRPE optimization problem, a series of incremental-scale subproblems based on the objective and model parameters are constructed. They decompose the original DRPE optimization problem in two directions, including the direction of incremental objectives of the DRPE problem and the direction incremental parameters of the DRPE problem. The solutions to each subproblem are good initial values for the optimum of the next subproblem. By using three sequential incremental-scale subproblem programming strategies to derive an accessible path for the original large scale DRPE optimization problem, the optimum solution can be found successfully. The proposed strategies are used in the two industrial processes: free radical polymerization of styrene and purified terephthalic acid (PTA) oxidation process systems, both of which cannot be efficiently handled by conventional methods. On the basis of the proposed sequential incremental-scale

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REFERENCES

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