Prediction interval-based modelling of polymerization reactor: A new modelling strategy for chemical reactors

Highlights

• A totally new modelling technique (prediction interval (PI)-based modelling) for chemical reactors is introduced.

• A PI-based neural network (PI-NN) model is developed for chemical processes to quantify the process disturbances.

• This technique bears extra information such as prediction accuracy rather than traditional point forecast models.

• To further improve the quality of the model, a new method is proposed for aggregation of PIs from individual NN models.

• Results reveal that ensemble PI-NNs performance is superior to those individual PI-NN models in terms of PI quality.

Abstract

Precise and reliable modelling of polymerization reactor is challenging due to its complex reaction mechanism and non-linear nature. Researchers often make several assumptions when deriving theories and developing models for polymerization reactor. Therefore, traditional available models suffer from high prediction error. In contrast, data-driven modelling techniques provide a powerful framework to describe the dynamic behaviour of polymerization reactor. However, the traditional NN prediction performance is significantly dropped in the presence of polymerization process disturbances. Besides, uncertainty effects caused by disturbances present in reactor operation can be properly quantified through construction of prediction intervals (PIs) for model outputs. In this study, we propose and apply a PI-based neural network (PI-NN) model for the free radical polymerization system. This strategy avoids assumptions made in traditional modelling techniques for polymerization reactor system. Lower upper bound estimation (LUBE) method is used to develop PI-NN model for uncertainty quantification. To further improve the quality of model, a new method is proposed for aggregation of upper and lower bounds of PIs obtained from individual PI-NN models. Simulation results reveal that combined PI-NN performance is superior to those individual PI-NN models in terms of PI quality. Besides, constructed PIs are able to properly quantify effects of uncertainties in reactor operation, where these can be later used as part of the control process.

Introduction

Traditionally, free radical polymerization (FRP) processes are modelled by using mass and energy balance equations as well as differential algebraic equations [1]. Traditional modelling techniques used for industrial polymerization processes are reviewed by Muller et al. [2]. They divide the polymerization model into four sub-models. The four sub-models are reaction kinetics, thermodynamics, mass transport and particle kinetics model. Among these four sub-models, the mathematical models of reaction kinetics and mass transport are still limited. The reactions of polymerization are complex and diffusion limitation (glass and gel effect in high temperature) in mass transport affects both chemical rate constant and interphase mass transport.

In 1972, Ray presented a complete set of mathematical equations to describe the batch FRP systems under some assumptions [3]. In the following years, many research studies appeared in the literature to improve the mathematical model of polymerization reactors. Over the last century or so, researchers developed extensive mathematical models to present the dynamic behaviour of different polymerization systems [4], [5], [6], [7], [8]. Kiparissides [9] summarized the recent development of mathematical modelling for FRP in batch and continuous reactor. Many researchers also reviewed the modelling of FRP processes [10], [11].

Polystyrene (PS) polymerization in batch reactor is taken into account as a case study in this work. As mentioned earlier, Ray [3] described a complete set of mathematical equations for PS polymerization reactions. Many researchers followed this model for control study purpose due to the lack of available mechanistic models and to avoid the complexity of batch FRP reactors [12], [13], [14].

In the mathematical model, the major uncertainties lie in obtaining reasonable estimates of the termination rate parameters and the rate constants for transfer reactions, especially the ones representing transfer to large polymer molecules. There are lots of difficulties involved in kinetic modelling, such as systematic kinetic investigations are required, and often the type of data required is difficult to obtain due to poor precision, lack of well-developed procedures, and frequently practical problems such as the insolubility of branched polymers in common solvents. Therefore, researchers usually consider a lot of assumptions to develop a kinetic model for polymerization reactors due to above reasons [15], [16]. Moreover, there are no universal kinetic parameters for FRP and many researchers estimated and used different kinetic parameters for the same systems. Another disadvantage of mathematical model for polymerization system is that most available models fail to describe the reactor dynamics in the presence of disturbances [17]. In addition, the complete mathematical model cannot be utilized in control system due to equation complexity and huge computational load. This limitation leads to searching alternative modelling techniques such as data-based or hybrid modelling strategy to describe nonlinear batch reactors [15], [18].

Many studies have shown the importance of data-based modelling techniques to model industrial polymerization processes in the last two decades [19], [20], [21], [22], [23]. The traditional mathematical model cannot describe the kinetic model of FRP properly due to the complex reaction mechanism [17]. In the termination stage of FRP reaction, most of the reactions are neglected in traditional modelling techniques to avoid mathematical complications as well as lack of detailed knowledge of termination reaction behaviour. This limitation can easily be eliminated with the help of data-based modelling. Usually data-based modelling techniques such as neural networks (NNs) are used to model unknown parameters as well as reaction kinetics of FRP.

In a recent study, Noor et al. [17] reviewed the NN application in various polymerization processes. They started from the initial ideas of NNs and described the recent development of NNs for better performance. They also summarized different types of NNs that are used in polymerization systems. These include feed forward NN, staked NN, bootstrap aggregated NN, recurrent NN, internal recurrent NNs, and neural fuzzy networks. NNs can be used to model and describe the nonlinear, complex and unknown parts of the polymerization processes as a standalone model or part of a bigger model. A hybrid model for batch polymerization reactor utilizing artificial NN in mechanistic model is developed by Hosen et al. [15]. They used a NN modelling strategy to describe the complex kinetic parameters. Many similar hybrid applications and NN individual models can be found in the literature for chemical reactors [24], [25], [26], [27], [28], [29].

It is well known that the performance of NNs significantly drops in the presence of large process disturbances [30]. In addition, there is no indication of performance accuracy in traditional NN forecasts. To eliminate these limitations, a new modelling technique, named prediction interval-based NN (PI-NN) model is proposed here for polymerization reactor. In contrast to traditional modelling techniques that just produce point forecasts, this strategy predicts an interval (consists of an upper and lower bound) with a prescribed confidence level (CL). Studies on construction, optimization, and applications of PIs in different fields of science and engineering can be found in the literature [31], [32], [33].

PI-based NNs allow us to quantify uncertainties due to the disturbing presence in non-linear system. Literature has proposed four methods for construction of NN-based PIs. These include bootstrap [34], Bayesian [35], mean-variance estimation [36], and delta methods [37]. Regardless of their implementation differences, the main principle is the same for these four methods. All of these methods use a traditional error-based cost function to train NNs. After developing NN models, PIs are constructed using these trained NNs without changing their structure. It has been argued that the quality of PIs constructed in this way is questionable [38]. In addition, the main strategy of all these methods is to minimize the prediction error, instead of trying to improve the PIs quality (such as PI coverage probability and width).

To effectively cope with these problems Khosravi et al. [38] recently developed another method named lower upper bound estimation (LUBE) to construct PIs. They used a PI-based cost function rather than error-based cost function to train a NN with two outputs. These outputs are the upper and lower bounds of a PI. Moreover, this technique optimizes the NN structure for a particular CL to construct PIs. The PI-based cost function not only increases the PIs coverage probability, but also decreases the PI width to ensure quality PIs.

In the present work, several traditional NNs are developed with and without including disturbances present in the polymerization process system. As NN prediction is poor, mean absolute percentage error (MAPE) > 20% in the presence of disturbances, an extended version of LUBE method is used to generate quality NN-based PIs. Metaheuristic optimization methods, such as simulated annealing (SA) is applied to optimize the PI-based cost function in LUBE method. It is established that NN performance intimately depends on initial parameters as well as perturbation of NN parameters. Due to this, NN prediction performance fluctuates from one replicate of training to the other one. Hence, aggregation of forecasts obtained from a couple of NNs is the best solution for improving forecast accuracies compared to the case of using individual NNs [39]. Many researchers claim that an ensemble of NNs can greatly improve the overall representation accuracy, generalization, and robustness of NN predictions [40], [41]. The effects of a poor prediction from one NN in combined networks is simply minimized by effects of good predictions obtained from the other NNs [24].

As the key contribution of this paper, we extend the concept of forecast aggregation to the field of PI construction. This is done with the purpose of enhancing the quality of PIs constructed using individual NNs in the LUBE method. The arithmetic mean and median are applied as the aggregation tool for developing combined PIs. Firstly, several NNs for the polymerization reactor are developed using the LUBE method to construct PIs for 90% CL. Then, NNs are ranked and shortlisted based on the quality of PIs constructed for the validation set. The performance of the proposed method for generating quality aggregated PIs is compared with the performance of individual NNs for construction of PIs.

The rest of this paper is organized in the following manner. Section 2 briefly describes the PI-NN techniques to model nonlinear system, followed by a discussion about PI-based cost function. The procedure for construction of combined PIs is discussed in Section 3. Section 4 describes the polymerization reactor system and its parameters used in this work. This section also provides the data and parameters used in proposed method. Section 5 demonstrates and discusses the simulation results of the traditional point-based and PI-based NN model. Section 6 concludes the finding in the present study with some future directions.