An Iterative Modeling and Trust-Region Optimization Method for Batch

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An Iterative Modeling and Trust-region Optimization Method for Batch Processes Jinjin Zhao, Yi Yang, Xi Chen, and Furong Gao Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie505064g • Publication Date (Web): 11 Mar 2015 Downloaded from http://pubs.acs.org on March 19, 2015

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Industrial & Engineering Chemistry Research

An Iterative Modeling and Trust-region Optimization Method for Batch Processes Jinjin Zhao1, Yi Yang1, Xi Chen1∗, Furong Gao2 1 State Key Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang University, Hangzhou, China 2 Department of Chemical and Biomolecular Engineering, The Hong Kong Univ. of Science and Technology, Clear Water Bay, Hong Kong

Abstract: Batch process optimization is of great significance in industrial applications. Considering the difficulty of obtaining industrial process models in traditional methods, this paper proposes an iterative modeling and trust-region optimization (IMTO) method for batch processes. The key factors of the method and IMTO algorithm are proposed and illustrated in detail. The characteristics of the method are demonstrated through several numerical simulations, where surrogate models are used as objective functions or constraints. The proposed method is successfully implemented in the quality control and operation optimization of an injection molding process with satisfactory performance and high efficiency.

1. INTRODUCTION Batch processing is widely used in chemical processes, such as those performed to produce specialty chemicals, bio-pharmaceuticals, and food. Contrary to traditional continuous processes in the industry, batch process presents two distinct characteristics, namely, dynamic and iterative. In a single batch, the work profile is dynamic from beginning to end. Between batches, the process repeats iteratively during production. Several studies on iterative control and monitoring of batch process based on certain pre-defined optimal dynamic profile have been reported. 1-3

∗ To whom correspondence should be addressed. Tel: +86-571-87953966; Email: [email protected]

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However, obtaining the optimal profile requires dynamic optimization. Therefore, batch process optimization is an important endeavor. To utilize numerical methods with computers, dynamic optimization of batch process can be converted into static optimization through parameterization. Iterative characteristic is another important feature that may be utilized during optimization. Model-based optimization (MBO) is widely used as a traditional method for batch process optimization. This approach initially establishes a process model, and then uses an optimization algorithm to determine the optimal solution. Model accuracy is thus of great importance for MBO implementation. A number of studies on batch process modeling techniques have been reported, and the models employed in these studies can be categorized into two types. The first modeling method is the first-principle modeling technique, which acquires an accurate process model through process knowledge. Bonvin4-5 studied the characteristics of batch process reactors and key factors to build knowledge-driven models. Rafizadeh et al.6 and Kamal et al.7 developed first-principle models for injection molding process as a typical batch process to achieve optimal operation. Such models show good extrapolation. The second modeling method is the data-driven modeling technique, which aims to achieve an accurate data-driven model correlating operating conditions and product quality based on historical data. Kulkarni et al.8 and Yu and Li9 proposed modeling techniques with neural networks for batch processes. Kourti10 discussed multivariate dynamic data modeling for batch processes. Good data-driven models are simple, easy to apply in process optimization, and present characteristics nearly similar to those of the original first-principle model. However, considering that a batch process requires a wide operation range, building an accurate model is difficult. High model accuracy is usually obtained at high cost with a large number of experiments.

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Moreover, for a flexible batch process that requires frequent changes in either feedstock or products, model extension to new conditions is difficult to realize. Model-free optimization (MFO) has been proposed to overcome the challenges of modeling difficulties in MBO. Instead of building an accurate process model, online measurements are directly used to search for the optimal solution via a certain model-free algorithm. Georgakis11 and Martinez et al.12 proposed model-free methodologies using dynamic experiment designs to solve batch process optimization problems. Kong et al.13 recently proposed a specific MFO method for a batch process to solve its quality control problem. Such batch processes, which feature short cycle time and low operational cost in a single batch, allow online experimentation during optimization. Zhao et al.14 proposed a natural gradient based on MFO for quality control of batch processes from the aspect of Riemannian manifold and stochastic gradient optimization. Zhu et al.15 later extended this method to process optimization problems by addressing convergence and noise. The advantages of MFO include no modeling cost and easy implementation, especially for typical batch processes with short cycle time and low operational cost. However, because of the lack of available process models, no accurate derivative information with which to generate a good search direction is available and extra experimental trials are necessary to verify a good step length, which also increases the process optimization cost. As described above, the model in MBO offers process information and model derivatives, thereby allowing easy optimization. Online experiments in MFO allow iterative adjustment of the optimal solution. However, obtaining an accurate model in MBO is difficult, and lack of global information without a model is a major drawback of MFO. Thus, combination of these two methods may offer improved performance. Surrogate-based optimization (SBO), which was reviewed by Queipo et al.16, is a

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method used in simulation environments to address engineering optimization problems where an accurate process model is unavailable but the simulation platform is available to replace the exact process and generate data. The basic SBO17, 18 is as follows. A surrogate model is initially built based on process (simulator) samples, after which optimization is conducted to obtain the next iterate. After simulation under the estimated iterate, the surrogate model is updated with new samples generated by resampling through simulation until the convergence is achieved. Some recent advances in surrogate-based optimization were also reviewed by Forrester et al.19 Most simulation works on engineering optimization have focused on determining how to update the surrogate model for better optimization. The main work in this process is determination of how to use the resampling strategy to improve the accuracy of the surrogate model. Using the trust region approach, Caballero and Grossmann20 proposed an algorithm using learning surrogate models to solve modular flow sheet optimization problems; here, model refinement was accomplished by resampling in a shrunk or expanded sampling region. Alexandrov et al.21 proposed a trust region framework to manage approximation models in engineering optimization; in this case, the model is updated by resampling in an updated trust region based on the current model accuracy. There also exist some basic research work about trust region method and the application.22, 23 Although the reported applications are not for batch processes, the concept may be applied in the present case. However, experimental costs must be seriously considered when dealing with industrial engineering optimization problems for batch processes where the industrial process takes the place of the simulator (in simulation optimization) as a black box to generate input and output data online. Considering such practical problems, this paper proposes a practical and low-cost

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method for industrial engineering optimization of batch processes. The proposed method is called iterative modeling and trust-region optimization (IMTO). The detailed steps of IMTO are addressed in this paper. The characteristics of the method are demonstrated in detail through numerical simulations and its industrial application in injection molding is implemented successfully with good performance and high efficiency.

2. IMTO METHOD The general optimization problem can be described as follows: min

()

s.t. () ≤ 0 ∈ A ⊂

(1)

where () is a cost function to be minimized, ( ) ≤ 0 is a set of constraints to be satisfied, and A is a box constraint including lower and upper boundaries. In many cases, the process model is not given, resulting in the fact that either () or ( ) is

unknown. Some system knowledge is also required to find the box constraints A where safe operation is assured. In this project, we focus on the process optimization using surrogate models as min

()

s.t. ( ) ≤ 0 ∈ A ⊂

(2)

where () and () are surrogate models for ( ) and ( ), respectively; and A is the box constraints considering the safe operation. Note that sometimes only one of these functions is required to be replaced with a surrogate model. To avoid difficulties in building an accurate process model for the whole range in traditional MBO methods, IMTO uses local models for optimization. The flowchart of the proposed method is shown in Fig. 1. Given limited experimental data, an initial 5



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model is built by data-driven modeling, and this model is accurate only within a limited range. Optimization is then conducted with a bounded trust region to ensure a reasonable step. After experimental testing of the derived solution, the model is updated by iterative modeling with newly obtained data. Thus, trust-region optimization can be conducted again within an updated region. The method stops when the local model is no longer updated. Techniques for iterative modeling and trust-region optimization are keys to the proposed method and elaborated as follows.

2.1 Iterative modeling technique A key to the proposed method is using local models for optimization. An iterative modeling technique is used to ensure increases in local model accuracy and movement of the local model toward the optimal solution through iterative optimization. When working with local surrogate models, maintaining balance between simplicity of optimization and complexity of representation of the nonlinear characteristics of the true process model is essential. Linear models are too simple to capture the nonlinear information of the complex process despite the obvious advantages of the former during analysis and optimization. Higher-order models normally require more experimental data to achieve good fitting performance. Thus, a quadratic polynomial model is selected in this paper. When data are available, model structures and parameters can be determined by simultaneously applying a model basis selection strategy and linear least-squares regression. Model basis functions are composed of linear terms, quadratic terms, and cross terms, as shown in Eq. (3):

= () 6


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() ⊆ {1, , , … , , , , … , , , … , , , … , } = ( , , … , ( ( )⁄ ) )

(3)

where is the n-dimension decision variable, k is the iteration number, ()

represents the model structure, is the coefficient vector, and is the surrogate

model used for objective function or constraint as shown in Eq. (2). As shown in Eq. (3), a model structure with some of the basis functions is determined by a model basis selection strategy. After least-squares regression, the surrogate model is expressed as a linear combination of basis functions with corresponding coefficients. The model may not be completely accurate in the current modeling region, but it can be improved by iterative modeling to achieve acceptable model accuracy. Iterative modeling is realized through model structure reselection, data set updating, and model residual verification. The detailed strategies are described below. Model structure selection Model structure selection helps build the best model with a suitable structure. This strategy is applied at the beginning of modeling when data are insufficient to obtain a complete quadratic polynomial model. Model complexity (i.e., model basis term number) depends on the available data number. Therefore, the number of model terms cannot be larger than the data number; otherwise, the regressed model is not unique. Suppose the available data number is N, then the term number in the model structure should be no larger than N. The most suitable model structure is determined through the model basis selection strategy, which includes two layers. The outer layer sets the term number and the inner layer chooses the best combination of terms under the preset term number by mean square error criteria. First, in the outer layer, the term number of the model is set. Then, in the inner layer, the best combination of model terms under the preset term number is selected based on the mean square error criteria. 7



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The above steps will repeat until all the combinations have been compared traversed and the best one is selected out as the ultimate model structure. A notable rule in IMTO is that as many terms as possible must be included to present more information correlating the variables and results. Therefore, the term number is set as N with N available data in layer one. By comparing the most suitable combinations under such term number, the best model structure is obtained in layer two. Data set updating During traditional model updating, the data set is updated by resampling in the current region. This model does not make full use of the available data, which could result in high experimental costs. The proposed data set updating strategy utilizes as much available data as possible to update the surrogate model. At each iteration, the new experimental data are added. Old data that are farthest from the current region are discarded. This strategy includes two main steps. 1.

Data set updating. Denoting ! ={ , ( )} as the new data, the data set

is updated as follows: " where

" ∪ {! } ; & ' < ) =# " ∪ {! } \{!+,-./0- } ; & ' = )

123ℎ563 = arg max:;:< ∥ > −

(4)

∥

is the iterate newly obtained by optimization, (n+1)(n+2)/2 is the term

number for a complete quadratic polynomial model with n variables, U is the upper number limit of data used for modeling that satisfies U > (n+1)(n+2)/2, N is the current data size, " is the data set, and !+,-./0- is the data farthest from the current iterate. According to Eq. (4), new data are initially added to the data set. Whether or not old data are discarded depends on the current data size. If the current data size is 8


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smaller than the upper boundary, no data are discarded. Otherwise, the farthest data are replaced by the new data to maintain the size of the data set. 2.

Criticality step. As the model accuracy increases with iteration, trust region

optimization will result in similar optimal solutions in several iterations. Thus, the data used for efficient modeling will become insufficient for obtaining a unique model. The criticality step is used to generate extra data by interpolation to ensure a unique model. It works by interpolating on the input and obtaining the output data via experiment response. Model residual verification (MRV) Model residual verification is used to obtain a well-regressed model based on limited data and achieve a model with small residual error by data set updating. The model residual, which is the sum of errors between the surrogate model and unknown function values on the sampling data, is defined in Eq. (5). If the residual is larger than a preset tolerance @, the farthest data from the current iterate [the same definition in Eq. (4)] are discarded. This procedure is repeated until the model residual decreases into the tolerance as shown in Eq. (5).

MR=∑BC∈DE( (> ) − (> ))

" = " \G!+,-./0- H ; & I > @ (5) " ; & I ≤ @ If the sample set size falls below (n+1)(n+2)/2 due to residual verification, critical " = F

step will work to ensure a unique model and later model structure selection will work to ensure the best model with current data.

2.2 Trust-region optimization and updating In each iteration, the surrogate model is built with limited data, making the model potentially unreliable for a large region. Thus, rather than search for the optimal solution on the whole range, optimization within a trust region is proposed to restrict 9



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the optimal solution from changing too far away from the current solution.

In the traditional trust-region method, the objective function, , is given. The

approximate model is commonly chosen as the second Taylor-series expansion of . Optimization is conducted on the approximate model with a trust region.

The region size and region center are updated based on the agreement between the

approximate model and the original model within two iterates. The trust

region not only restricts the step length, but also guarantees good reduction in . The

irregular contours in Fig. 2 illustrate the curvature of the original model . The

elliptical contours illustrate the approximate model, , which is built around the

current iterate, . If no trust region exists, the line search method based on the approximate model searches along the step direction to the minimizer of as a

new iterate K . However, this new iterate results in an increase in the objective

model . When bounded by a trust region, as shown by the dotted circle, the line

search steps to the minimizer of with a significant reduction in as a better iterate .

Contrary to traditional trust-region optimization, the original model in this case is unknown. However, we can still borrow this idea for the iterative optimization. After the local surrogate model is built based on experimental data, optimization can be conducted on the local surrogate model, which takes the role of the approximate model in traditional trust-region optimization, to obtain the new iterate. The local model can be updated with newly obtained experimental data and optimization continues. This process is repeated until the model can no longer be updated. The key aspects relating to the trust region definition, optimization, and updating are described as follows. 1.

Trust region is defined by a region center and radius. The region center is the 10


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current iterate. The region radius is where the model can be trusted to conduct optimization. The region center is updated along batches based on Eq. (6),

= /L

(6)

New solutions obtained by optimization are always accepted as the next iterate. The region radius is updated according to Eq. (7), O △- ; | ( ) − ( )| > Q △- = NR △- ; | ( ) − ( )| < S △- ; others

(7)

where Q, S, O, R are constants, Q>1, R>1,0

An Iterative Modeling and Trust-Region Optimization Method for Batch

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