Inferential Model for Industrial Polypropylene Melt Index Prediction

Article pubs.acs.org/IECR

Inferential Model for Industrial Polypropylene Melt Index Prediction with Embedded Priori Knowledge and Delay Estimation Haichuan Lou, Hongye Su, Lei Xie,* Yong Gu, and Gang Rong State Key Laboratory of Industrial Control Technology, Zhejiang University, 310027 Hangzhou, P.R. China ABSTRACT: Melt index inferential model plays an important role in the control and optimization of polypropylene production. This study proposed a novel multiple-priori-knowledge based neural network (MPKNN) inferential model for melt index prediction. The prior knowledge from the industrial propylene polymerization process is fully exploited and embedded into the construction of multilayer perceptron neural network in the form of nonlinear constraints. Meanwhile, an adaptive PSO-SQP (particle swarm optimization-sequential quadratics programming) is proposed to optimize the network weights. The proposed MPKNN model has good fitting and prediction ability. Meanwhile, it can avoid unwanted zero value and wrong signal of the model gains. By embedding priori knowledge, the model ensures the safety in the quality control of melt index. In addition, a hybrid model combining the MPKNN model with a simplified mechanism model is proposed to enhance the extrapolation capability. A normalized mutual information method is employed to estimate the delay between independent variables and dependent variables. The proposed hybrid inferential model is validated on recorded data from an industrial double-loop propylene-polymerization reaction process.

1. INTRODUCTION As a polymer of propylene monomer polymerization, polypropylene (PP) is an excellent general-purpose plastic and widely used in packaging, manufacturing, textile, and other areas of civilian consumption. Melt index (MI) is a key quality indicator of PP product that reflects the performance of resin flow and determines the resin grades. Since operation conditions of propylene polymerization reactor may change frequently, the modeling and control of MI is crucial to produce high quality, multigrade polypropylene. The main difficulties of polypropylene quality modeling include the following: (i) Lack of online analyzers. MI can only be measured by off-line analysis, which usually costs 2−4 h. Such large time delay cannot meet the requirements for realtime control. (ii) The propylene polymerization process involves multiscale and nonlinear physical and chemical reactions. Different product grades require significant shift in the operating conditions, and it is difficult to utilize one simple soft-sensor or inferential model to describe such changes. The development of first principle model (FPM) of PP MI has been widely discussed in the literature.1−4 The main advantage of the first principle model is its good extrapolation ability; however, it is always time-consuming to obtain a good model that can explain the extremely complex behaviors of an industrial polyolefin polymerization process. To overcome the disadvantages of first principle model, datadriven MI modeling has attracted much attention in the past decade. Ohshima and Tanigaki5 used a wavelet-based neural network for MI modeling of high-density polyethylene, but the model accuracy significantly reduces when it works outside the training operating conditions. Taking the multiscale process characteristics into account, Shi6,7 combined multiscale analysis, independent principal component with RBF neural network for MI modeling. Li8 proposed a hybrid evolutionary algorithm based on PSO and simulated annealing for the RBF network © 2012 American Chemical Society

optimization and obtained a better prediction performance than Shi.6 Although the data-driven inferential models have been widely used in the prediction of PP MI, most of them suffer from the lack of extrapolation capabilities. In addition, the datadriven models are easily subjected to unwanted zero value and wrong signals of gains which may cause safety issues of closed loop systems.9 Applications of both the first principle and data-driven models reveal that both of them have pros and cons. Kadleca10 suggested combining these two models together to ensure the trend tracking as well as the prediction accuracy. The advantages of such a hybrid model were also validated in other chemical areas of inferential model development.11−13 There are three typical structures of hybrid inferential model: (1) Series form,14 in which the data-driven approach is employed to identify the parameters of nonlinear mechanism model, (2) parallel form,15 in which the data-driven approach compensates the residuals of mechanism model, and (3) integrated form, in which the priori process knowledge is embedded into the data-driven model construction.16 Because of the intrinsic nonlinearity and complexity of the PP process, the development of the hybrid model for PP MI has not been fully addressed. In this paper, the main contributions on inferential modeling for polypropylene melt index are as follows. (1) A novel multiple-priori-knowledge based neural network (MPKNN) inferential model is presented, in which priori knowledge is fully exploited from process of propylene polymerization reactions and embedded into the neural network in the form of nonlinear inequality constraints. Meanwhile, an effective adaptive PSO-SQP Received: Revised: Accepted: Published: 8510

December 11, 2011 May 30, 2012 June 7, 2012 June 7, 2012 dx.doi.org/10.1021/ie202901v | Ind. Eng. Chem. Res. 2012, 51, 8510−8525

Industrial & Engineering Chemistry Research

Article

2.2. Simplified Mechanism Model of PP Melt Index. The quality of the PP products is mainly characterized by the melt index (MI). The MI is usually measured every four hours and analyzed in the laboratory. An online soft sensor model plays an important role in monitoring the MI change and determining the transition time from one grade to another. Owing to the complex process mechanism of propylene polymerization, a detailed mechanism model is difficult to obtain. In this study, simplified mechanism models are considered which allows MI to be estimated by the concentration of monomer, comonomer, hydrogen, and reaction temperature. Such models have been widely used in the literature1−3,17 and are described as follows. Instantaneous MI model of the first loop reactor (R201):

algorithm with constraint handling mechanism based on augmented Lagrange multiplier for MPKNN weight optimization is proposed. (2) A hybrid approach combining MPKNN and the simplified first principle model is proposed for polypropylene MI prediction. (3) A normalized mutual information method is utilized for the delay estimation between input and output variables. The paper is organized as follows. Section 2 briefly introduces the double-loop propylene polymerization reactions process and the simplified MI properties mechanism model. Section 3 presents the MPKNN inferential model, how to introduce the prior knowledge into MPKNN is discussed. Section 4 introduces the hybrid inferential model combined with the MPKNN and the simplified MI mechanism model. On the basis of the normalized mutual information method, the delay estimation between independent variables and dependent variables is discussed in section 5. Section 6 presents a comprehensive model performance study on recorded data from an industrial double-loop propylene-polymerization reaction process. Section 7 provides the conclusions of this article.

⎛C ⎞ ⎛ CH ⎞ ln(MIi,R201) = k 0 + k1ln⎜ 2 ⎟ + k 2 ln⎜ Cat ⎟ + k 3 ln(T ) ⎝ CM ⎠ ⎝ CM ⎠ (1)

Instantaneous MI model of the second loop reactor (R202): ⎛ CH ⎞ ⎛ CH ⎞ ln(MIi,R202) = k 0 + k1ln⎜ 2 ⎟ + k 2 ln⎜ 2 ⎟ ⎝ CM ⎠R202 ⎝ CM ⎠R201

2. DOUBLE-LOOP PP PROCESS AND MI MECHANISM MODEL 2.1. Double-Loop Propylene Polymerization Reactions Process. The liquid PP process of SINOPEC (China Petroleum & Chemical Corporation Limited) located in Zhejiang, China, is studied. The process includes two main polymerization loop reactors, R201 and R202, as shown in Figure 1 and Figure 2. The fresh catalyst, including cocatalyst

⎛C ⎞ ⎛C ⎞ + k 3 ln⎜ Cat ⎟ + k4 ln⎜ Cat ⎟ ⎝ CM ⎠R202 ⎝ CM ⎠R201 + k5 ln(T )

(2)

where k0, k1, k2, k3, k4, and k5 are weight coefficients to be determined, CH2, CM, and CCat represent the concentrations of hydrogen, monomer, and catalyst, respectively, and T is the reaction temperature. Cumulative MI model of the first loop reactor (R201): d ln(MIc ,R201) dt

=

1 1 ln(MIi ,R201) − ln(MIc ,R201) τ1 τ1

(3)

Cumulative MI model of the second loop reactor (R202) d ln(MIc , R 202) dt

=

1 1 ln(MIi ,R202) + ln(MIc ,R201) τ2 τ1 ⎛1 1⎞ − ⎜ + ⎟ln(MIc ,R202) τ2 ⎠ ⎝ τ1

(4)

where τ1,τ2 are the residence time of polypropylene. The simplified instantaneous MI property mechanism models given in eqs 1 and 2 are adopted during the hybrid inferential modeling in section 5 through the mechanism and priori knowledge analysis of the industrial process.

Figure 1. The actual industrial plant of double-loop propylene polymerization reactions.

3. MULTIPLE-PRIORI-KNOWLEDGE BASED NEURAL NETWORK INFERENTIAL MODEL Compared with the first principle model, the data-driven inferential model is relatively simple, convenient, and easy to satisfy the modeling accuracy, but the reliability is questionable in the actual application. For instance, it is difficult for datadriven models to follow the frequently changing operating modes of the PP process. However, if the prior knowledge is embedded into data-driven models,18−20 which enforce the models to match the process mechanism to a certain extent, the model generalization and extrapolation performance can be greatly improved.

triethylaluminum (TEAL) and electron donor, is injected into the prepolymerization reactor R200 after passing the preexposure tank D201. At the same time, the fresh propylene monomer and hydrogen are also supplied to R200. After the prepolymerization, the reactants of R200 are fed into the first loop reactor R201 to react with fresh propylene monomers and hydrogen. In R201, the conversion rate of polymers is about 60%−70%. The mixture of remaining liquid propylene and polymer slurry from R201 are continually poured into the second loop reactor R202, where the remainder of the polymerization reactions take place. 8511

dx.doi.org/10.1021/ie202901v | Ind. Eng. Chem. Res. 2012, 51, 8510−8525


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Figure 2. Block diagram of double-loop propylene polymerization reactions plant.

3.1. Multiple-Priori-Knowledge Based Neural Network Structure Description. In the context of this work, the prior knowledge refers to any known relevant information of the target system, including knowledge of process mechanism, engineer experience, etc. In the industrial practice, how to introduce the prior knowledge to the data-driven model varies according to specific applications. The typical process priori knowledge includes the maximum, minimum, and monotonicity of process gain and the concavity or convexity of the process characteristic curve. This study will focus on the neural network and discuss how to embed the prior knowledge into the soft inferential model in the form of constraint terms imposed on the objective function. As depicted in Figure 3, the multiple-priori-knowledge based neural network (MPKNN) consists of input, hidden, and

Φj1(x) =

1 , 1 + e −x

Φj2(x) = x ,

j1 = 1, 2, ···, m

j 2 = m + 1, ···, j

(5)

where j denotes the total number of hidden nodes and m is number of nonlinear hidden nodes. The hidden layer outputs are given by hj1 = Φj1(∑ xiwj1i + bj1) i

hj2 =

∑ xiwj2i + bj2 (6)

i

The MPKNN outputs are yk =

∑ Φj1(∑ xiwj1i + bj1)wkj1 j1

+

i

∑ (∑ xiwj2i + bj2)wkj2+ ∑ wkixi + bk j2

i

(7)

i

where wki is the weight between input and output layers, wj1i,wj2i are the weights between input and hidden layers, wkj1,wkj2 denote the weights between hidden and output layers, bj1,bj2, bk denote the thresholds of hidden and output layers, and the subscript j1,j2 denote nonlinear and linear nodes, respectively. Taking into account of the actual situation of propylene polymerization process, the following priori knowledge is considered to improve the performance of neural networks. (i). Process Gain and Its Characteristic. In the continuous process industries, such as refining and chemical manufacturing, maximum and minimum process gains can be obtained from the process operators and engineers according to their physical understanding or experience of the process. To approve approximate bounds on the process gains, such priori knowledge is introduced by adding constraints in the neural network model. Gains expression of MPKNN can be derived by solving partial derivatives of outputs with respect to inputs from eq 7,

Figure 3. Multiple-priori-knowledge based neural network structure.

output layers, which is similar to the multilayer perceptron (MLP) neural network. The main difference between MPKNN and MLPN lies in the direct connection of MPKNN between the input and output layers, which can improve model capability in describing the relationship between the input and output variables. Moreover, the hidden layer of MPKNN consists of both nonlinear and linear nodes for avoiding zero gain and gain reverse problems. In MPKNN, the hidden nodes use both the nonlinear sigmoid and linear activation functions as follows,

∂yk ∂xi 8512

=

∑ wj1ihj1(1 − hj1)wkj1 + ∑ wj2iwkj2+wki j1

j2

(8)



Article

As 0 ≤ hj1(1 − hj1) ≤ 1/4, the maximum gain bound and minimum gain bound can be derived as follows.

N

min f (w) =

∑ (yk

− yk̅ )2

k=1

⎧ ⎛ ∂y ⎞ 1 ⎪⎜ k ⎟ = ∑ wj1iwkj1 + ∂ x 4 j1 ⎪ ⎝ ⎠ i max ⎪ ⎨ ⎪ ⎛ ∂yk ⎞ ⎪ ⎜ ⎟ = ∑ wj2iwkj2+wki ⎪ ⎝ ∂xi ⎠ j2 ⎩ min ⎧ ⎛ ∂y ⎞ ⎪⎜ k ⎟ = ∑ wj2iwkj2+wki ⎪ j2 ⎪ ⎝ ∂xi ⎠max ⎨ ⎪ ⎛ ∂yk ⎞ 1 ⎪ ⎜ ⎟ = ∑ wj1iwkj1 + ⎪ ⎝ ∂xi ⎠ 4 j1 ⎩ min

s.t.

∑ wj2iwkj2+wki j2

yk =

, wj1iwkj1 < 0

j1

+

∂yk

j2

∂xi ∂ 2yk

⎛ ∂yk ⎞ ⎜ ⎟ > mi ⎝ ∂xi ⎠min

∂xi 2

≥ 0,

> 0,

and or

or

or

⎛ ∂yk ⎞ ⎜ ⎟ < Mi ⎝ ∂xi ⎠max

wj1iwkj1 < 0 ∂yk ∂xi

≤0

∂ 2yk ∂xi 2

0 and wj1iwkj1 < 0 can guarantee the gain monotonic decay and growth respectively. (ii). Process Monotonicity. Monotonicity is a common phenomenon in the process. It belongs to the prior knowledge of physical feature and can be determined from the experience or process characteristic curve. For example, in the propylene polymerization process, the relationship between MI value and the hydrogen concentration is monotonically increasing. The MPKNN monotonicity can be ensured by introducing firstorder derivative constraints. More precisely, ∂yk/∂xi > 0 and ∂yk/∂xi 0,

∑ wj2iwkj2+wki

Therefore, the MPKNN gain constraints are given by

∂xi 2

i

⎛ ∂yk ⎞ ⎜ ⎟ > mi , ⎝ ∂xi ⎠min

, wj1iwkj1 > 0

⎛ ∂yk ⎞ ⎜ ⎟ < Mi ⎝ ∂xi ⎠max

i

∑ (∑ xiwj2i + bj2)wkj2+ ∑ wkixi + bk j2

(9)

∂ 2yk

∑ Φj1(∑ xiwj1i + bj1)wkj1

(11)

if ∂2yk/∂xi2 > 0, the process characteristic curve is concave, and convex if ∂2yk/∂xi2 < 0. 3.2. The Objective Function Design of MPKNN and APSO-SQP Optimization Algorithm. In MPKNN, the priori knowledge is embedded into the neural network by imposing nonlinear constraints on the objective function. 8513



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Figure 4. APSO-SQP optimization framework. (a) Main flow of APSO-SQP algorithm, (b) flowchart of APSO algorithm.

Figure 5. Hybrid inferential model based on MPKNN and simplified first principles model.

disti =

1 N−1

N

∑ k = 1, k ≠ i

number of particles, D is the dimensional search space, t is the evolution generation. (ii) Obtain the adaptive factors, including the inertia weight w, cognitive factors c1 and social factors c2,

D

∑ (xij − xkj) j=1

distmin = arg min{disti} Ef t =

w(t ) =

distmin ,0 distmin , t

1

(

1 + 6 × exp ‐0.15 × (13)

G t

)

c1(t ) = (c1(t − 1) − c1, s) × Ef + c1, s

where, the subscript ″i″ denotes Ith particle, and ″j″ denotes jdimensional particles, xi is the ith particle position, k denotes the kth particle difference from the ith particle, N is the total

c 2(t ) = 0.6 × (c 2, s − c 2(t − 1)) × Ef + c 2(t − 1)

(14)

where G is the total evolution generation, c1(t), c1(t − 1), c2(t), c2(t − 1) are cognitive factors and social factors of the current 8514



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model is another particular priori knowledge that can be integrated into the MPKNN in the hybrid form. The parameters of MPKNN and SFPM are optimized, respectively, in the MPKNN-SFPM hybrid model, where the coefficients, k0−k5 in the SFPM are optimized by AN unconstrained optimization algorithm (e.g., the genetic algorithm or steepest descent algorithm). Weight optimization of MPKNN is carried out by solving the objective function with nonlinear constraints by THE APSO-SQP algorithm introduced in section 3.2.

and previous generation, respectively, which are used to adjust the particle’s optimal flight path. c1,s and c2,s are steady-state cognitive factors and social factors, which are set according to the actual situation. (iii) Evolution of velocity and position. For the evolution of the ith particle in the next generation, velocity and position of the particle are updated, vij(t + 1) = ω(t )vij(t ) + c1(t )r1j(t )(pij (t ) − xij(t )) + c 2(t )r2j(t )(pgj (t ) − xij(t )) xij(t + 1) = xij(t ) + vij(t + 1)

5. DELAY ESTIMATION BASED ON NORMALIZED MUTUAL INFORMATION During the inferential modeling process of propylene MI, the true value of MI usually comes from off-line analysis, which results in a large delay corresponding to each independent variable. Hence, in order to maintain consistency between independent variables and dependent variables, the time delay should be estimated in advance. In the engineering application, the time delay is commonly obtained through detail analysis of the process mechanism (e.g., residence time, length of pipe, etc). However, in many cases there is no clear process knowledge; only operational data is available. To determine the time delay from the observation data, several methods have been proposed, for instance, the maximum correlation coefficient method for delay estimation of linear process,26 genetic algorithms to optimize linear delay model for delay selection of variables,27 and delay optimization for nonlinear process by cross-entropy method combined with the GA high-dimensional.28 As a useful information measure, mutual information, which measures the correlation between two nonlinear events, can be used for feature as well as nonlinear process delay estimation.29 Therefore, we introduce the normalized mutual information method (see Appendix B in detail) to estimate the time delay between independent variables and dependent variables of MI inferential model. The key point using the NMI method to estimate the time delay between independent variables and dependent variables is to transfer the direct estimation of time delay into the selection of important features. The detail steps are described as follows. Step 1. Set the sampling period of dependent variable MI, Ts = T, and independent variables ui, I = 1, 2, ··· n, ts, respectively. Then the maximum step size of delay, Dmax is determined as Dmax = Ts/ts − 1. The samples ratio of dependent variables and independent variables is NMI/Nu = 1/(Dmax+1). Step 2. Considering the different step sizes for each delay, samples of independent variables are rearranged as uil(t), uil(t − 1), ···, uil(t − d), ···, uil(t − max), where l = 1, 2, ···, NMI, and d is the delay of each variable to be determined. Hence, Dmax + 1 subvariables are obtained for each variable; Step 3. On the basis of the normalized mutual information method, values of NMI between Dmax + 1 subvariables and dependent variables are derived, and then the corresponding delay between the independent variable and dependent variable is obtained by maximizing NMI,

(15)

where xij(t) is the ith particle position, vij(t) is the ith particle velocity,, pij(t) is the best individuals position, pgj(t) is the global best position. r1 ≈ U(0,1) and r2 ≈ U(0,1) are two independent random functions. To reduce the possibility of particles leaving the search space during the evolutionary process, vij is usually limited to a certain extent; that is, if the search space is limited in [−xmax, xmax], it can set vmax = kxmax, 0.1 ≤ k ≤ 1.0. In summary, the flowchart of APSO-SQP algorithm is shown in Figure 4.

4. HYBRID INFERENTIAL MODEL COMBINED WITH MPKNN AND SFPM The multiple-priori-knowledge based neural network inferential model can overcome the deficiency of general data-driven models. Moreover, in order to ensure the model accuracy, extrapolation ability as well as the trend tracking ability, we further put forward a new hybrid inferential model. Different from the existing methods,11−16 the proposed model is a combination of the MPKNN and the simplified first principle model (SFPM) for polypropylene MI prediction by using the harmonic mean approach (see Figure 5). MIout = w1 × MINN + w2 × MIFPM

(16)

where, MINN, MIFPM, MIout, denote the prediction outputs of MPKNN model, simplified MI mechanism model and hybrid model, respectively. w1,w2 are weight factors, which can be optimized by optimization methods. For simplicity, the harmonic mean method is chosen here to obtain the two weights. w1 =

1 σ12 1 σ12

+

1 σ2 2

,

w2 =

1 σ2 2 1 σ12

+

1 σ2 2

(17)

where σ12 and σ22 represent the variances of submodel, σ12 = σ12 =

n

1 n

∑ (MINN −

1 n

∑ (MIFPM −

MINN)2

i=1 n

MIFPM)2

i=1

The prediction output of hybrid model MIout can guarantee unbiased estimation, and its variance σ2 satisfies

σ 2 < min(σ12 , σ22)

(18)

arg max{NMIj , i} → di ,

In this simple and intuitive hybrid method, the simplified first principle model of MI can ensure the overall tracking ability for the process trend, and the MPKNN model is used to compensate its prediction accuracy. In fact, the first principle

i = 1, 2, ···n;

j = 1, 2, ···, Dmax + 1

Eventually, the corresponding delay is obtained as Td,i = di × ts. 8515



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Figure 6. The raw data of independent variables of propylene polymerization reactions plant.

6. RESULTS AND DISCUSSION 6.1. Data Acquisition and Preprocessing. 6.1.1. Data Collection. In the continuous PP polymerization process, MI is mainly affected by factors like the catalytic flow rate, hydrogen concentration, hydrogen flow rate, propylene monomer flow rate, reactor temperature, jacket water temperature, reaction pressure, density etc. After analyzing the reaction mechanism of the double-loop liquid propylene polymerization process, the following independent variables of the inferential model are selected to predict the melt index: the hydrogen concentration (CH2,R201, CH2,R202), hydrogen flow rate (FH2,R201, FH2,R202), propylene monomer flow rate (FM,R201, FM,R202), jacket water temperature (TCW,R201, TCW,R202), and the catalyst flow rate (FC1, FC2). The modeling data is collected from a double-loop liquid propylene polymerization plant of SINOPEC, which is located in Zhejiang, China. Data of independent variables are collected and stored in the real-time database of a DCS server station, the sampling time is 6 min. A total of 4600 samples are collected, as is shown in Figure 6. On the other hand, the MI values of the second reactor R202 are obtained from the analyzing laboratory at a period of every 4 h. A total of 115 samples within two grades are collected, as is shown in Figure 7. Note that, to reduce the uncertainty of sample data source to a certain extent, it should communicate with the operators before soft sensor projects are carried out. 6.1.2. Data Preprocessing. Measurement data in the industrial process are sampled, transferred and collected by sensors, transmitters and other instruments, which can be easily deteriorated by instrument accuracy, high-frequency noise, measurement methods, etc. Hence, the measurement data inevitably contain errors. The errors may seriously mislead the operator, and cause fluctuations in production. Therefore, it is necessary to preprocess the raw data to improve model accuracy and reliability. Here, the preprocessing involves three steps, that is; wavelet threshold denoising, robust scale outlier detection, and the max−min normalization method. The results are shown in Figure 8. (i) Wavelet Threshold Denoising. The linear moving average method of signal denoising is commonly used in the

Figure 7. MI laboratory analyzing value of the second loop reactor R202.

industrial field, but this method is not suitable for nonlinear process data since it will cause the operating point drift. In contrast, the wavelet denoising method30 is very suitable for denoising nonlinear process data. The focal point of one-dimensional wavelet signal denoising is how to choose the most critical threshold, and how to quantify the threshold. Here the Haar function is selected as the wavelet base function, and Birge−Massart strategy is used as the threshold selection rules of the wavelet coefficients. (ii) Robust Scale Outlier Detection. In the actual measurement, abnormal outliers occur due to serious mistakes of measuring and recording, or sudden fluctuations of instrumentation. Commonly used outlier detection methods include manual observation, statistical discrimination method, like 3σ edit rules, etc.31 In the robust scale outlier detection method,32 the normalized distance di of each sample is computed as u − um di = i sm 8516



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Figure 8. The preprocessing results of raw data of independent variables.

where, um is the median of variables {ui}, and sm is the median absolute deviation from the median sm = 1.4826 median{|ui − um|}

Before constructing the inferential model, the time delay of independent variables should be determined to maintain the time synchronization between independent variables and dependent variable. According to the analysis in section 5, the corresponding delays between variables based on the NMI method were estimated. The sampling period of MI is Ts = 4 h = 240 min, the sampling time of each input variable is ts = 6 min, the maximum fixed step size of delay is Dmax = 39. So the step size of delay di of independent variables are obtained as

(i = 1, 2, ···, n)

According to the 3σ edit rules, the sample with |di| > 3 is regarded as an outlier and should be removed. (iii) Max−Min Data Normalization. The robust scale outlier detection method can also be used for the data normalization, but the max−min normalization method is chosen to ensure the convergence of MPKNN training as quickly as possible,

[C H2,R201(t − 34), C H 2,R202(t − 27), FM,R201(t − 29)

u − min u (max u ′ −min u ′) + minu ′ u′ = max u −min u

, FM,R201(t − 19), FC1,R201(t − 33), FC2(t − 30)]

Hence, the corresponding time delay isTd,i = di × ts. In the practical process, it has been realized that data preprocessing is a vital and inevitable step for the success development of soft sensors,10 otherwise the process and measurement uncertainty including noises, sample delays, outliers, and missing values may deteriorate the soft sensor performance significantly. More recent research on handling the common process and measurement in Bayesian inference framework refers to Yu.33 6.2. Model Performance Evaluation. 6.2.1. Model Evaluation Indices and Simulation Parameters Initialization. Comprehensive performance evaluation indices of inferential models, including the root-mean-square error function (RMSE), the relative mean square error function (RE), the mean absolute error function (MAPE), and normalized mean square error function (NMSE),34 are selected here.

where, u, minu, and maxu, are the uninominal data, maximum and minimum sample values, respectively, and u′,minu′ and maxu, are nominal sample data, nominal minimum, and maximum sample values, respectively. 6.1.3. Input Variable Selection and Time Delay Estimation. Ten independent variables related to MI are initially chosen in section 6.1.1 as the input variables of the inferential model. However, due to process constraints, the variables are often highly colinear, so dimensionality reduction should be performed to choose reasonable independent variables. Production of industrial polypropylene is commonly under stable conditions. As the chain transfer agent, hydrogen can cause the transferring of the active center of polymer molecular chains, thus terminating the chain growth to control the molecular weight of polypropylene. For this reason, the hydrogen concentration is the main influence factor of grades production. In the closed-loop quality control of polypropylene, MI usually acts as the controlled variable and the hydrogen concentration as the manipulated variable; other factors such as liquid propylene flow rate and catalyst flow rate are considered as disturbance variables. Thus, based on analysis of process mechanism of polypropylene, hydrogen concentration in the loop reactors R201 and R202 are selected as the manipulated variables, while liquid propylene flow rate of R201 and R202, the catalyst flow rate as disturbance variable, hence a total of six independent variables are determined .

Root mean squared error: N

RMSE =

∑i = 1 (yi − yi ̂ )2 N

Relative root mean squared error: N

RE = 8517

∑i = 1 [(yi − yi ̂ )/yi ]2 N dx.doi.org/10.1021/ie202901v | Ind. Eng. Chem. Res. 2012, 51, 8510−8525


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Figure 9. MPKNN modeling and prediction curves with delay and without delay estimation: (a) modeling curves, (b) prediction curves.

Table 1. Results of Modeling and Prediction Performance of MPKNN Model with Delay and without Delay Estimation MPKNN with delay estimation MPKNN without delay estimation

RMSE

RE

MAPE

NMSE

RMSE_t

RE_t

MAPE_t

NMSE_t

0.0265 0.0283

0.0094 0.0100

0.0065 0.0072

0.3049 0.3253

0.0422 0.0704

0.0147 0.0241

0.0104 0.0160

0.4586 0.7644

Mean absolute percentage error:

1 MAPE = N

N

∑ i=1

Normalized mean square error:

(yi − yi ̂ )

NMSE =

yi 8518

N 1 ∑i = 1 (yi − yi ̂ )2 N 1 N ∑i = 1 (yi − y ̅ )2 N−1 dx.doi.org/10.1021/ie202901v | Ind. Eng. Chem. Res. 2012, 51, 8510−8525


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Figure 10. Approximation curve of relationship between hydrogen concentration and MI.

where the first three indicators reflect the accuracy of model, and the RMSE is usually as the main evaluation indicator in industrial process. While the NMSE reflects both the tracking trend of model and the model accuracy, is also an important indicator in the actual production process. First, in order to illustrate the importance of delay estimation between dependent variables and independent variables, performance comparison of inferential modeling before and after the delay estimation based on the NMI method is carried out (see Figure 9 and Table 1, the MPKNN inferential model is applied here). Figure 9 and Table 1 indicate that delay estimation before modeling can significantly improve the model prediction accuracy and overall performance after timing synchronization between independent variables and dependent variables. Next, comprehensive performance comparison of inferential models involving MPKNN, simplified first principles model (SFPM), MPKNN-SFPM hybrid model, and least squares support vector machine (LS-SVM) is executed to illustrate the particular performance of the MPKNN and MPKNN-SFPM model. The initial simulation parameters of models are set as follows: a. MPKNN model. The MPKNN has a structure of 1 hidden layer, 13 nonlinear hidden nodes, and 1 linear hidden node. In the MPKNN model, the hydrogen concentrations of loop reactors R201 and R202 are the main manipulated variables introducing relevant prior knowledge. The detail priori knowledge is described as follows: (i) The approximate curve (Figure 10)shows that the relationship between hydrogen concentration and MI is monotonically increasing, that is ∂(MI)/∂CH2 > 0, which is in accordance with process mechanism. (ii) Process gain about MI with respect to hydrogen concentration, known from the experience value of field engineers, is in the range of (0, 1.0]. Meanwhile, in order to guarantee the right decaying trend of the process gain, make sure wj1iwkj1 > 0.

Nevertheless, priori knowledge about the process concave is not considered here to avoid over strictly constraints affecting prediction accuracy, due to various unpredictable disturbances in the industrial process. The parameters of APSO-SQP algorithm for weight optimization are set as: population size, N = 35, and population dimension, D = 97, which are determined by the weight of MPKNN; position and velocity range, [−1,1]; total evolution generation, G = 200; initial weight w = 1.0; the initial acceleration parameters c1 = 2.5, c2 = 1.8, c1,s = c2,s = 2.0, in addition to the Lagrange multipliers λ0 = 0.1, penalty factor r0 = 1000, and the number of iterations k = 5. b. Simplified First Principles Model (SFPM). The MI value of the loop reactor R202 is collected from laboratory analysis. Because only the data of hydrogen concentration has been acquired by the field chromatographic analysis, those concentrations of hydrogen and propylene are taken into account for the simplified MI mechanism model of R202. ⎛ C H ,R201 ⎞ ⎛ C H ,R202 ⎞ ⎟⎟ ⎟⎟ + k 2 ln⎜⎜ 2 ln(MIi)2 = k 0 + k1ln⎜⎜ 2 ⎝ CM,R201 ⎠ ⎝ CM,R202 ⎠

where CH2,R201,CH2,R202 denote the hydrogen concentration of reactors R201 and R202. Under the steady state condition, the polymerization temperature is maintained at a certain value (about 70 degrees), and the concentration of liquid propylene is almost unchanged. According to the chemical formula: molar concentration = density/molecular weight, liquid phase propylene concentration of loop reactors R201 and R202, CM,R201 and CM,R202, are obtained. Therefore, the unknown factors are identified by the unconstrained optimization methods (steepest descent and genetic algorithms), [k 0 , k1 , k 2] = [1.2390, 0.0361, 0.0102]

c. MPKNN-SFPM Hybrid Model. Parameters setting of the hybrid model are referred to the initialization of MPKNN and SFPM model. Besides, weights of two models are obtained by utilizing the harmonic mean method. 8519



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Figure 11. Model training and prediction curves: (a) model training curves; (b) model prediction curves.

Table 2. Comprehensive Performance Evaluation of Models MPKNN SFPM MPKNN- SFPM LS-SVM

RMSE

RE

MAPE

NMSE

RMSE_t

RE_t

MAPE_t

NMSE_t

0.0281 0.0365 0.0309 0.0295

0.0100 0.0130 0.0100 0.0105

0.0070 0.0093 0.0070 0.0075

0.3230 0.4203 0.3557 0.3396

0.0300 0.0393 0.0334 0.0421

0.0108 0.0141 0.0120 0.0147

0.0074 0.0097 0.0074 0.0119

0.2887 0.3786 0.3212 0.4057

⎛ ||x − xi|| ⎞ ⎟ K (x , xi) = exp⎜ − ⎝ 2σ 2 ⎠

d. LS-SVM Model. The standard LS-SVM, which is an extension of support vector machines,35 is a method used for solving linear equations with fast convergence ability. It has some potential applications in the inferential modeling.36 The essential of LS-SVM is to find the best regularization parameters and kernel parameters for the best model selection. Radial basis function is selected as kernel function,

where ||x − xi|| = (∑k n= 1(xk − xki )2)1/2, the initial regularization parameter c = 5, and the kernel parameters σ = 1. 6.2.2. Comparison of Basic Model Performance. The process data sets collected from the plant are divided into three 8520



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Figure 12. Behaviors of adaptive factors.

Figure 13. Objective function optimization of MPKNN model.

sets, that is, training set, test set, and validation set, for comprehensive evaluation of basic model performance. The training set contains 72 samples, and the test set contains 18 samples. The training and test set are used to construct the model, where the optimal model parameters are determined by 5-fold cross validation. The remaining 25 samples are used as the validation set to verify the model extrapolation ability. (1). Comparison of Fitting and Prediction Accuracy. A good inferential model should have good generalization ability in addition to good fitting accuracy. Since the operating point and operation domain change frequently in actual applications, engineers and operators are more concerned with the generalization ability of the inferential model. Here, the fitting and prediction performance of the four models is compared, and the training and prediction curves are shown in Figure 11. From Table 2 we can see that all the four models meet the accuracy requirements of an inferential model; that is,

performance evaluation indices of an inferential model, for example, mean square error (RMSE/RMSE_t), should be less than 10% . The SFPM model is not as good as LS-SVM model with respect to fitting accuracy; however, it has better generalization ability. In contrast, the MPKNN model is the best; it outperforms other models for all the criteria, that is, fitting accuracy, prediction accuracy, and trend-tracking performance. Similar to the semimechanism model, the MPKNN model embedded with some priori knowledge can substantially increase the model generalization ability within the training region. Yet, to avoid blind extrapolation of the MPKNN model outside the region of training sample data, the MPKNN-SFPM hybrid model is proposed and implemented. The detail analysis will be demonstrated in section 6.2.3. (2). APSO-SQP Optimization Method. As there are more than one hundred of nonlinear inequality constraints 8521



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Figure 14. Model extrapolation curves.

introduced in the MPKNN and MPKNN-SFPM model, the optimization of network weights is a high-dimensional nonlinear problem. Whether the optimization algorithm is good or bad, directly determines the model accuracy and generalization performance. Here the APSO-SQP algorithm is proposed to optimize network weights, where APSO is used as the global optimal searcher, and SQP is used to accelerate the local optimal convergence. The key point of the optimization algorithm is to maintain effective coordination of adaptive factors in all stages (including exploration, development, and convergence phase) of APSO, as is shown in Figure 12. Figure 13 shows the optimizing process for objective function of MPKNN model, which has fast convergence to the global optimum f * = 0.0862. Meanwhile, the SQP algorithm (f mincon function in the toolbox of Matlab 7.2 is adopted here) is employed as the remedy for further convergence to the final local optimum, and ultimately get the optimal solution f ′ = 0.0472. In the case of nonlinear inequality constraints, this solution is perfectly within the engineering precision. Note that the SQP is employed to improve the APSO results only when its global exploration ability has been exhausted. 6.2.3. Characteristic Analysis of MPKNN and MPKNNSFPM Models. (1). Multimode Operation and Extrapolation Ability. It is a common practice to operate chemical processes in multimodes to satisfy the different quality requirements of different products.37 A model generalization ability of a blackbox data-driven inferential model may deteriorate in grade transition due to the lack of training samples.38 There has been some research tackling this issue by integrating the Gaussian mixture model with Bayesian inference strategy.37,38 The proposed MPKNN and MPKNN-SFPM hybrid model, embedded with priori knowledge and process mechanism, can adapt well to multimode operation with good extrapolation performance. A validation set is employed to test the extrapolated ability of difference models and the results are shown in Figure 14 and Table 3. As shown in Figure 14 and Table 3, the LS-SVM has good prediction ability within the training region, but is subjected to

Table 3. Extrapolation Performance Evaluation of Models validation region LS-SVM MPKNN SFPM MPKNN- SFPM

extrapolation region

RMSE_v

NMSE_v

RMSE_e

NMSE_e

0.1125 0.0446 0.0394 0.0217

0.8519 0.3379 0.2974 0.1644

0.1732 0.0630 0.0495 0.0233

3.3936 1.2344 0.9703 0.4562

a sharp decline in the prediction performance outside the sample region when grade transition happened. The MPKNN model, as the semimechanism model has a similar extrapolation ability as the mechanism model. However, the process complexity, especially the propylene polymerization reaction process with unpredictable interference or abrupt change in the large operating region (described in the Figure 14), limits its extrapolation performance. In sharp contrast, the MPKNNSFPM model has excellent accuracy and good trend tracking ability in the extrapolation region, and hence adapts to multimode operation very well. (2). Model Gain and Safety Performance. Among various data driven inferential models in the development and application of actual process, neural networks are the mostly commonly selected.26,39,40 However, in actual applications, the safety issue of neural networks cannot be ignored.9 The process gain is related to the controller gain. Traditional neural networks (e.g., Bp neural network based on sigmoidal activation function) may be subjected to zero gain or gain reversals if gain constraints or process mechanism is not considered during model training (as shown in Figure 15, fitting and prediction gains of model without priori knowledge constrained). For example, when an MI inferential model based on neural network is directly used for real-time closed-loop control, an error control signal is inclined to mislead the manipulated variables (e.g., hydrogen flow control valve) of a concentration controller. If the impact of gain direction is not considered, the operating mechanism may be violated by a neural networks based inferential model, resulting in process instability and increasing product quality fluctuations. Whereas, 8522



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Figure 15. Model fitting and prediction gains without priori knowledge constrained.

Figure 16. Model fitting and prediction gains with priori knowledge constrained.

compared with the general data-driven inferential model (LSSVM) and simplified MI mechanism model, the MPKNN inferential model not only has better fitting and prediction ability, but also can avoid zero gain and gain reversal. In addition, a simplified first principles model (SFPM) of MI, which can be regarded as particular priori knowledge, is integrated with MPKNN model into a hybrid model utilizing the harmonic mean method. In the integrated model, the SFPM follows the overall process-tracking trend, and the MPKNN model improves the overall prediction performance by local accuracy compensating. This method is simple and intuitive, it ensures the effective prediction accuracy and adapt to the multimode operation, and enhances the extrapolation ability, hence achieving a good unity between model extrapolation and prediction accuracy. Furthermore, normalized mutual information method for delay estimation between the independent variables and dependent variables of inferential model is also introduced. This method can ensure the timing synchronization of

once gain constraints or other priori knowledge are introduced, either the process mechanism can be reflected or zero gain or gain reversal occurring are avoided, hence ensuring reliability and safety of the industrial process (as shown in Figure 16, model fitting and prediction gains with priori knowledge constrained are positive).

7. CONCLUSIONS Melt index modeling and prediction of PP is a challenging and meaningful problem. Developing a good MI inferential model is of great benefit for PP quality closed-loop control and resin production. In this study, a novel multiple-priori-knowledge based neural network inferential model is presented, and priori knowledge of polypropylene industrial process is fully exploited and embedded into the MLP neural network in the form of nonlinear constraints. APSO-SQP algorithm with higher optimization accuracy and efficiency based on the constraint handling mechanism of an augmented Lagrange multiplier method is proposed for network weights optimization. Experiment with actual industrial process data shows that, 8523



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If P(X = xi, Y = yj) = p(xi,yj), then the joint entropy H(X,Y) is defined as

independent variables and dependent variables, hence enhancing the model prediction accuracy. The MPKNN and MPKNN-SFPM inferential model proposed here possess a very good potential value in the actual application. During actual operation, the MPKNN model can be applied within the sample-training region, and automatically switched to the MPKNN-SFPM hybrid model once the process data outside the region is detected. However, the updating of the soft sensor after the implementation of closed-loop control will definitely require the closed-loop delay estimation deserving further study. Closed-loop process identification theory may help here.

■

H (X , Y ) =

∑ p(xi , yj )log i,j

1 p(xi , yj )

The normalized mutual information is expressed as NMI = =

I (X ; Y ) I (X ; Y ) · H (X ) H (Y ) I (X ; Y ) H (X ) · H (Y )

APPENDIX

where NMI ∈ [0,1]; it notes that NMI is larger, the correlation between two variables is stronger.

A. Constraint-Handling Mechanism Based on Augmented Lagrange Multiplier Method

■

Using the augmented Lagrange multiplier method21 to deal with the inequality constraint in eq 10, L (x i , λ , r ) = f (x i ) +

*E-mail: [email protected].. Tel.: +86-571-87952233-8237.

∑ (λjθj(x i) + rjθj2(x i)) (A.1)

j=1

Notes

The authors declare no competing financial interest.

where θj(x ) = max[gj(x) ,(−λj)/(2rj)], j = 1, 2j, ···mi, f(x ) is the objective function, xi denote the weight variables, gj(xi) denote the violation of inequality constraints, λj denote the augmented Lagrange multiplier, and rj denote the penalty factor. Therefore, a series of unconstrained optimization subproblems is solved. In each iteration of sub-problem sequence, the multiplier λ and penalty parameter rj are fixed, and the augmented Lagrange multipliers and penalty factor should be updated, i

i

i

λjv + 1 = λjv + 2r jvθj(x i)

■

ACKNOWLEDGMENTS This work is supported by the State Key Program of National Natural Science of China (Grant No. 60904039, 61134007), the MOE Program of Introducing Talents of Discipline to Zhejiang University on Information and Control Science (111 Project) (B07031) and the Fundamental Research Funds for the Central Universities.

■

(A.2)

⎧ 2r v , if g ̂ (x v) > g ̂ (x v − 1) ∧ g ̂ (x v) > ε g j j j ⎪ j ⎪ r jv + 1 = ⎨ 1 r jv , if g ̂ (x v) ≤ εg j ⎪2 ⎪ v ⎩ r j , otherwise

(A.3)

where, ĝj(x ), j = 1, 2···, mi denote the violation of inequality constraints. Besides lower thresholds of the penalty factor need to be set to ensure its effective amplitude range. The thresholds are given as follows: 1 2

|λ j | εg

B. Normalized Mutual Information Method

Assume two events X = {x1,x2,···xn}, Y = {y1, y2,···, ym} are two discrete random variables. The mutual information is defined as I (X ; Y ) = H (X ) + H (Y ) − H ( X , Y )

where, H(X),H(Y) are the entropy of random variables, defined as n

H (X ) =

∑ p(xi) log i=1 m

H (Y ) =

∑ p(yj ) log j=1

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Inferential Model for Industrial Polypropylene Melt Index Prediction

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