Mathematical Problems in Engineering

Volume 2019, Article ID 2474909, 9 pages

https://doi.org/10.1155/2019/2474909

## Robust Optimization of Industrial Process Operation Parameters Based on Data-Driven Model and Parameter Fluctuation Analysis

Correspondence should be addressed to Zhiqiang Liao; moc.621@oailgnaiqihz

Received 6 June 2019; Revised 26 August 2019; Accepted 5 September 2019; Published 8 October 2019

Academic Editor: Waldemar T. Wójcik

Copyright © 2019 Taifu Li and Zhiqiang Liao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The fluctuation of industrial process operation parameters will severely influence the production process. How to find the robust optimal process operation parameters is an effective method to address this problem. In this paper, a scheme based on data-driven model and variable fluctuation analysis is proposed to obtain the robust optimal operation parameters of industrial process. The data-driven modelling method: multivariate Gaussian process regression (MGPR) based on Bayesian statistical learning theory can map the process operation parameters to objective performance with the flexibility in nonparameter inferring and the self-adaptiveness to determinate hyperparameters. According to the minimum variance criterion, the parameter fluctuation analysis can be performed through multiobjective evolutionary algorithm based on the MGPR model. To analyze the robustness influence of a single parameter, cross validation is applied to evaluate the model output with 2% fluctuation. After that, the robust optimal process operation parameters can be obtained and applied to guide the production. The effectiveness and reliability of the proposed method have been verified with the hydrogen cyanide production process and compared with other model methods and single objective optimization method.

#### 1. Introduction

With increasing attention paid to controlling production quality and costs in industrial production, various studies on reducing the costs and increasing production benefits have been widely explored in recent years [1, 2]. The operation parameter optimization is an effective method to promote the profit of industrial production processes. It works on a process parameter optimization algorithm under the built process model to choose the optimal operation parameter [3, 4]. The critical step of the operation parameter optimization method is to build a precise production process model. But most of the industrial processes are nonlinear, and it is difficult to describe them with a mathematical model. In recent years, with the rapid development of automation and digitalization, abundant data have been kept in most of industrial production processes. The data carries production information which can reflect the operation condition. The data-driven method based on machine learning is an effective analysis method and has been widely used in many papers [5–7]. It can describe the system model without too much prior knowledge. Based on the data-driven model, the optimization method can be performed to optimize the operation parameter. Because the industrial production process generally has nonlinear characteristics, the traditional optimization methods are not suitable for obtaining the optimal operation parameters. The evolutionary optimization algorithms can find the optimal solutions without knowing too many complicated mathematical models [8–11].

After the optimal operation parameters are obtained and set in the industrial production equipment, the deviation may happen due to the harsh environment and equipment performance degradation. Because most of the optimal objectives are very sensitive, the real objective performance tends to have some fluctuations. Therefore, fluctuation analysis is needed. The robustness of parameters can reflect the fluctuation characteristic, and the robust optimal operation parameters can decrease the fluctuation of objective performance. There are many studies focusing on robust optimization: A sigma point method for robust multiobjective dynamic optimization of chemical processes was presented in reference [12]. A parametric uncertainty Bayesian description method was used for optimizing the chemical processes by solving a robust optimization problem in reference [13]. By applying the Taguchi method, a robust design method was used for optimizing the bakers’ yeast process in reference [14]. In the above methods, robust optimization needs to consider both optimal operation parameters performance and robustness metric at the same time. Hence, the fluctuation analysis of operation parameter is a multiobjective problem. Multiobjective evolutionary optimization algorithm, through imitating biological evolution process and searching for Pareto frontier, can efficiently and quickly solve multiobjective optimization problems.

In this paper, a scheme based on data-driven model and fluctuation analysis is proposed to obtain the robust optimal operation parameters of industrial processes. The Gaussian process regression based on the Bayesian statistical learning theory can build the model between the operation parameter and objective performance with the flexibility in nonparameter inferring and the self-adaptiveness to determinate hyperparameters. This data-driven method does not need to include too much information about the operating mechanism. To analyze the fluctuation of parameters, 2 objective functions are designed to perform the multiobjective method NSGAII: (1) maximization of the expected objective performance and (2) minimization of robust metric based on the minimum variance criterion. To analyze the single variable robustness influence and obtain the more accuracy decision information for production, cross validation is applied to evaluate the mean of model output with 2% fluctuation. The effectiveness and reliability of the proposed method have been verified through the hydrogen cyanide (HCN) production process.

This paper is organized as follows: the data-driven model Gaussian process regression method, NSGAII, multiobjective robust optimization design criterion, and fluctuation analysis are introduced in Section 2. The verification experiments and comparison results are given in Section 3. Section 4 presents the conclusion of the proposed method.

#### 2. Robust Optimization of Operation Parameter Design

##### 2.1. Data-Driven Multivariate Gaussian Process Regression Theory

Multivariate Gaussian process regression is a GPR model with multiple inputs and multiple outputs. Optimization modelling means that a model is built to calculate the optimal solution. and are the input and output vector of the training data, where .

The Gaussian process regression mode is to build a model between input vectors and output vectors. If a new sample is available, the predicted can be obtained based on the model built.

The Gaussian process regression model [15] assumes that there is a zero-mean Gaussian prior distribution regression function; the function is shown in the following equation:wherein is the covariance matrix with its element ; here, is the commonly used squared exponential covariance function. The function is given in the following equation:where only when *i* = *j* otherwise, , , and *l* is length scale, and and are signal and noise variance, respectively.

An appropriate hyperparameter set is crucial for a GPR model to make the prediction of a dominant variable more accurate. Hence, the hyperparameters need to be optimized in the training process through maximizing the likelihood function; it is shown in the following equation:

Once the optimal hyperparameter set is obtained, the GPR model is available to make a prediction of the distribution of for the corresponding . If comes, according to the property of the multivariate Gaussian distribution, the posterior distribution of the output can be obtained, where and are the mean and variance of the distribution which can be calculated in the following equations:

Finally, the expectation of the posterior distribution is taken as the predicted result of the GPR model.

The accuracy of the optimization model is assessed by using the mean square error (MSE) and the mean average relative error (MARE) of the test samples.where is the expected output, is the predicted output of the built GPR, and *N* is the length of data.

##### 2.2. Robust Optimization and Criterion Design

The robustness of the parameters can be illustrated in Figure 1. Both *A* and *B* are operation parameters in the parameter solution set. When the parameters in *A* have a fluctuation Δ*X*, the objective performance will decrease Δ*Y*_{1}. When the parameters in *B* have a fluctuation Δ*X*, the objective performance will decrease Δ*Y*_{2}. It can be seen that the Δ*Y*_{2} is less than Δ*Y*_{1} which means that *B* is much robust than *A*, and *B* is the robust parameter and *A* is the optimal parameter in solution set.