Multimode and Multiphase Batch Processes Understanding and

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Multimode and multiphase batch processes understanding and monitoring based on between-mode similarity evaluation and multimode discriminative information analysis Yan Qin, Chunhui Zhao, Shumei Zhang, and Furong Gao Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b02981 • Publication Date (Web): 03 Aug 2017 Downloaded from http://pubs.acs.org on August 9, 2017

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Industrial & Engineering Chemistry Research

Multimode and multiphase batch processes understanding and monitoring based on between-mode similarity evaluation and multimode discriminative information analysis Yan Qin1, Chunhui Zhao1*, Shumei Zhang1, Furong Gao2 1. State Key Laboratory of Industrial Control Technology, College of Control Science and Engineering, Zhejiang University, Hangzhou, 310027, China 2. Department of Chemical and Biomolecular Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Special Administration Region Abstract: The high requirements of safe production and quality products have greatly spurred the development of understanding analysis and monitoring technologies of batch processes. However, for the multimode and multiphase batch processes with serious process dynamics, few process monitoring studies have been reported. To address the above issues, a novel process monitoring method based on between-mode similarity evaluation and discriminative information analysis is proposed in this article. First, within each phase, a between-mode similarity method is proposed by classifying samples into two groups, in which the group of mutual samples measures the between-mode similarity and then similar modes within each phase are merged to avoid repetitive modeling. Next, to explain the causes of between-mode differences within each phase, a multimode-Fisher discriminant analysis algorithm is developed to identify discriminative information in a sparse manner, by which discriminative variables contributing most to the discriminative information are obtained. The process understanding is thus enhanced by quantitative similarity evaluation of differences modes and accurate identification of process variables causing between-mode differences. Finally, to effectively improve process monitoring performance, multiple specific monitoring models are constructed for between-mode discriminative information and a global process monitoring model is developed for the remaining indiscriminative information, respectively. 1 ACS Paragon Plus Environment


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The proposed method provides a detailed insight into the inherent nature of multimode and multiphase batch processes and captures deep process information to enhance one’s understanding. The feasibility and performance of the proposed algorithm are illustrated through a typical multimode and multiphase batch process, i.e., injection molding. Key words: multimode and multiphase batch processes, similarity evaluation, discriminative information analysis, multimode-Fisher discriminant analysis.

1. INTRODUCTION Recently, batch processing has become an important manufacturing way with increasing focuses on high-value but rapidly changing market[1]. Without adding or removing any physical manufacturing equipment, batch processing has the advantage of producing multiple products through simply changing raw materials, operation conditions, etc. This flexible manufacturing way makes the successful applications of batch processes in both traditional fields and emerging industries, such as fine chemical industries, biomedical processes, and semiconductor manufacturing industries[2], in which production schedules have to be frequently adjusted to accommodate rapidly changing market. However, this manufacturing way may bring new problems regarding process monitoring and safe production, one of which is multimode issue because of frequent product switching, fluctuation in quality of raw materials, and equipment aging.[3] Therefore, advanced process monitoring methods are developed to solve this problem in multimode batch processes. Many process monitoring methods are usually developed based on mechanism models using energy and mass balance equations. However, as pointed out by many studies, the developments of these mechanism models are very difficult and costly. As we walk into the era of big data, data-driven process monitoring methods are preferred since they get rid of the limitations of accurate mechanism models. The well-known data driven methods are principal component analysis (PCA)[4], partial least square (PLS)[5], multi-way PCA[6], and multi-way 2 ACS Paragon Plus Environment

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PLS[7], etc., in which a low-dimensional system subspace reflecting main process variations and a residual subspace are constructed[8]. Monitoring control limit of each subspace is offline established and then used for online comparison with the calculated monitoring statistics at each sampling time. The fault signal is released if several continuous alarms are observed when monitoring statistics are over the predefined monitoring control limits. In fact, an implicit assumption of the above-mentioned idea is that the process should be operated under a steady mode, in which process characteristics, including mean, covariance, and variable correlations, are constant. Therefore, it limits the direct application of the above-mentioned methods[4-8] into multimode processes because the developed process monitoring control limits may be loose and insensitive to process faults. To solve the multimode problem, a series of methods[9-23] have been developed. According to the number of models used for describing multiple process characteristics, current methods always fall into one of two categories[14]: single model methods and multiple model methods. Single model methods concern how to integrate multimode information only using one model[15]. In the early studies of continuous processes, strict assumptions, such as similar multimode data matrices[16], common eigenvector subspace for sample variance-covariance matrices[17], are required in order to develop such single model. However, these assumptions may not always be available in practice. To relax these assumptions and directly deal with the multimode data, Choi et al.[18] introduced a Gaussian mixture model (GMM) to describe the multimode data distribution and a mixture of local Gaussian models was developed for online monitoring. Sequentially, a series of GMM based process monitoring methods[19-22] have been proposed to solve the nonlinear, time-varying and dynamic problems of the multimode process. For multiple model methods, they are developed based on an intuitive idea that the monitoring models are developed to fit each individual operation mode. For example, PCA algorithm was used to extract latent subspaces for each operation mode in Ref [23]. 3 ACS Paragon Plus Environment


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However, the above-mentioned multimode methods may not be directly used for batch processes. The multimode problem in batch processes is inherently different from that of continuous processes. Continuous processes may go through distinct regimes, in which each mode is stationary. In contrast, the multimode problem in batch processes is more complex since it presents both batch-wise dynamics related to different operation modes over batches and the time-wise dynamics related to time-varying variable trajectories within each batch[24]. Therefore, for the multimode problem of batch process, the two types of dynamics should be analyzed. Zhao et al.[25] proposed a concurrent phase partition strategy to divide the whole process into the same phases by simultaneously analyzing multimode data. Then within each phase, multiple monitoring models are developed for each mode. However, some drawbacks are observed. First, without checking similarity of each mode within a phase, it may lead to the unnecessarily repetitive modeling of similar modes. Second, understanding analysis of multimode and multiphase batch processes regarding essential causes that lead to betweenmode differences has not been studied yet. In fact, these issues will not only improve the process understanding but also benefit the quality control by eliminating between-mode differences. To solve the above-mentioned problems, in this article, a process monitoring method based on similarity evaluation and multimode discriminative information analysis is proposed. Taking the advantage of merging similar modes within each phase, the understanding analysis is further performed through extracting between-mode discriminative information, which is a sparse low-dimensional latent subspace that greatly distinguishes the multimode data. It allows for quantitatively evaluating the between-mode differences and accurately describing causes of between-mode differences using discriminative variables from datadriven view. The proposed method includes three major parts. First, a similarity evaluation method is proposed to identify mutual samples of each mode using Mahalanobis distance, 4 ACS Paragon Plus Environment

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which not only measures the distance but also reflects the variable correlations. And the ratio of mutual samples to total samples presents the between-mode similarity within each phase. Next, between-mode discriminative information is extracted using the proposed multimodeFisher discriminant analysis (FDA), in which a new objective function is defined to achieve the purposes of separating each mode as far as possible and reflecting the influences of variable variations of each mode. In this way, discriminative variables that explain the essential causes of the different modes can be sparsely compressed from discriminative directions using Lasso method. On basis of the first two steps, the original space of each mode has been divided into two subspaces, one of which is the specific subspace that presents the between-mode discriminative information and the other is a common subspace that has no differences between each mode. At last, multiple specific monitoring models are developed for each mode to monitor the mode-specific information and then a global model is developed to monitor the common information. The contribution is summarized as below: 1. A similarity evaluation index is proposed to check the specific between-mode relationship within each phase; 2. A multimode-Fisher discriminant analysis method is proposed to explore the causes of between-mode differences; 3. Multiple specific monitoring models are developed to monitor between-mode discriminative information and a global monitoring model is used to supervise the remaining indiscriminative information. The remainder of this paper is organized as follows. The details of the proposed method are described in the next section, including between-mode similarity evaluation, the specifics of between-mode discriminative information analysis, and the following online monitoring. In Section 3, the feasibility and efficacy of the proposed method are illustrated through the injection molding process. Conclusions are drawn in the last section. 5 ACS Paragon Plus Environment


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2. METHODOLOGY 2.1 Phase-wise between-mode similarity evaluation For batch processes, a steady mode is usually identified by checking the trends of process characteristic. Therefore, considering the multiphase characteristic, a steady mode will be recognized to be different from another one if different process characteristics are observed within any phase. This makes the between-mode similarity analysis necessary within each phase to merge similar modes and avoid repetitive modeling. Before introducing the specifices of similarity analysis, data preparation is presented for better understanding the following explanations. For a steady mode of batch processes, provided that J process variables are measured at each sampling time, a data matrix with dimensions J × K is yielded at the end of a batch, where K is the batch length. By collecting the number of I batches, a three-dimensional data matrix X ( I × J × K ) is obtained. Using the symbol m to indicate mode number, a data matrix

Xm ( I m × J × K m ) is formed for Mode m, where Im and Km are the batch number and batch length, respectively. It is noted that values of I m and K m may vary from mode to mode but they will not bring negative influences in the following analysis. The data structure of multimode batch processes has been illustrated in Figure 1. The phases may be different under different modes. Here, the same phase partition results over modes can be ensured through a concurrent phase partition method[25], in which multiple sequential phases are simultaneously identified for all modes. In this way, process characteristics stay similar within the same phase for different modes so that the following between-mode similarity evaluation can be readily conducted in each phase to explore the phase-specific betweenmode relationship. Thus, phase partition problem is not discussed in this article and we assume that mode and phase information of batch processes is available. Further, to make the

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three-dimensional matrix X m available, it is split along the time direction and K m time-slice data matrices with the dimensions of I m × J are obtained. Therefore, a two-dimensional matrix Xcm,k ( I m × J ) is formed at each sampling time, where c indicates the phase index,

k ∈ [1, K m,c ] is the sampling time, and Km,c is the phase length of Phase c in Mode m. In this part, the basic idea regarding between-mode similarity is to directly check the data distribution of different modes within a phase. Thus, we classify the samples in a mode into two groups: mutual samples (MSs) and the remaining independent samples. MS refers to the sample that locates in the intersecting space of two modes. In this way, it allows for quantitative calculating between-mode similarity through the ratio of MSs to total samples even if similar variable correlations or locations are observed. For each mode, a large number of MSs means a high similarity. First, the time-slice data matrix at each time is normalized to remove dominant variance of variables. Considering phase length of each mode may not be the same, the traditional normalization is revised by replacing the phase length of each mode with the minimum phase length Kc of all modes. Then time-slice matrices of each mode before time K c are picked out. Select one mode as the reference mode and normalization information at sample time k is given below, mk =

1 Im

∑

Im i =1

xic,m , k

1 δ = diag ( I m − 1) 2 k

(∑

Im i =1

(x

c i ,m ,k

− m k )( x

c i ,m,k

− mk )

T

)

(1)

where m k is the mean vector of the kth time-slice matrix Xcm ,k ( I m × J ) , δ 2k is the corresponding variance, xic,m ,k (1× J ) is the ith row of Xcm,k ( I m × J ) , Im is the number of batches in Mode m and diag gets diagonal elements of a matrix. For simplicity, we choose the first mode as reference mode here.



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For other modes, time-slice matrix at each sampling time is normalized according to the obtained normalization information. Then put the time-slice matrices from up to down according to time order and obtain an arranged two-dimensional data matrix X cm ( N m × J ) , where N m = I m J . A flowchart of the proposed method is illustrated in Figure 2 and the specifics are described as follows. Step 1. Calculation of the control limit of each mode The index DTm is defined to describe the normal region of Mode m. It is a control limit based on Mahalanobis distance, which follows the F distribution[26], DTm =

J ( N m − 1) FJ , N m − J , β Nm − J

(2)

where β is the confidence factor in F-distribution, which is given as 0.05. Thus, the percentage of covered normal data is 95. Step 2 Calculation of within-mode and between-mode distances Dk , m , m is within-mode distance which measures the distance between the sample x m , k and

the data center of Mode m. And Dk , m , m is between-mode distance that measures the distance between the sample x m , k and the data center of Mode m , where symbol m indicates a mode other than Mode m. For x m , k , distances Dk , m , m and Dk , m , m are calculated as follows, Dk ,m ,m = ( x m ,k − u m ) S m−1 ( x m , k − u m ) T

Dk ,m ,m = ( x m ,k − u m ) S m−1 ( x m , k − u m ) T

(3)

where u m and S m are the mean vector and covariance matrix of X cm ( N m × J ) , respectively. Similarly, u m and S m are the mean vector and covariance matrix of X cm ( N m × J ) . The initial values of both m and k are one. Step 3 Determination of mutual samples


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Compare the distances Dk , m , m and Dk , m , m with the control limits DTm and DTm according to the following rules: Case I: If x m , k has the relationship Dk , m，, m ≤ DTm and Dk , m , m ≤ DTm , it indicates that x m , k locates in both Modes m and m . Thus, x m , k is a MP and N MPm ,m = N MPm ,m + 1 , where N MPm ,m is the number of MPs between Modes m and m . Case II: If Dk , m , m ≤ DTm and Dk , m , m > DTm , it indicates that x m , k belongs to Mode m. In this case, this sample is an independent sample. Case III: x m , k is an outlier if the relationship Dk , m，, m > DTm holds and it should be removed in the following analysis. If all samples in Mode m have been analyzed, then move to Step 4. Otherwise, k=k+1 and repeat Steps 2 and 3. Step 4 Similarity analysis and iterative executions After identifying MPs between Modes m and m , the between-mode similarity can be evaluated using the index Rm,m = N MPm ,m N m . To analyze the remaining modes, let m=m+1 and k=1, and then iteratively execute Steps 2 through 4 until all modes are analyzed. From the above Steps 1 through 4, it outputs the between-mode similarity ratio in Phase c. Similar procedures can be performed in the remaining phases. Besides, if the value of Rm , m is larger than a threshold ζ ( ζ =0.8 in this paper), it means that there is a high similarity and these two similar modes should be merged by putting them together to avoid repetitive modeling. On the contrary, it means that these two modes are different and the analysis of causes leading to between-modes differences is necessary. Here, mode information is assumed to be known a priori which can be obtained by either some popular data analysis methods, such as clustering techniques[27], or priori process knowledge. An evaluation index is proposed to explore the between-mode similarity by a 9 ACS Paragon Plus Environment


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Mahalanobis distance using the covariance information of each mode which can evaluate the specific between-mode relationship within each phase. It is the first time that between-mode similarity analysis is introduced into multimode and multiphase batch processes which can help to understand mode characteristics for different phases. 2.2 Between-mode discriminative information analysis Based on between-mode similarity analysis, a multimode-FDA is proposed to explore the causes of between-mode differences. Novel sparse discriminant directions that distinguish the reference mode from the other ones are generated through a new defined objective function. Nonzero variables of the discriminant directions termed discriminative variables are used for explaining between-mode differences. Traditional FDA looks for discriminant directions on which data sets can be separated from each other. However, it may not be suitable for the case where data sets are similar[28]. Besides, tradition FDA is out of function when the between-mode differences are caused by different variable variations. To overcome the above-mentioned problems, a new objective function is defined by directly introducing samples of each mode. Benefiting from this, the purposes, which distinguish each mode as far as possible and reflect different variable correlations on the discrimination directions, are achieved. For simplicity, after merging similar modes within a phase, the data matrix is still noted using the same symbol X cm . Assuming that Mode m is the reference mode, matrix S mp is defined to measure the differences between Modes m and p in Phase c, c N mc N p

S mp = ∑∑ ( x cm ,i − x cp , j )( x cm ,i − x cp , j ) ( p ≠ m) T

(4)

i =1 j =1

where x cm ,i is the ith sample in matrix X cm , xcp , j is the jth sample in matrix Xcp , N mc is the number of samples in Mode m, N pc is the number of samples in Mode p, and the subscript c indicates phase number, which is omitted in the following analysis for conciseness. 10 ACS Paragon Plus Environment

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S mp is cross-class scatter matrix, which is the covariance matrix by employing all samples

of Modes m and p. To achieve the double purposes of separating points of each mode and reflecting the variable variation information, multimode-FDA is proposed by maximizing the cross-class scatter matrix considering all modes,

  J (ω) = max  ∑ ωT S mpω   p ≠m 

(5)

where the vector ω is the optimal discriminant direction. Instead of maximizing the ratio of between-class scatter matrix to within-class scatter matrix in tradition FDA, multimode-FDA maximizes cross-class scatter matrix between the reference mode and all the other modes. Thus, multimode-FDA searches for the discrimination directions, on which samples of the reference mode and the remaining modes are separated as far as possible. Besides, the mode differences caused by changing variable correlations are considered. To constrain the length of ω , the objective function can be further described as follows,   J (ω ) = max  ∑ ωT S mp ω   p ≠m  T s.t. ω ω = 1

(6)

Using a Lagrange operator, the objective function in Eq. (6) is transformed into an unconstrained extremum problem as shown in Eq. (7),

Π (ω, λm ) = ∑ ωT S mpω − λm ( ωT ω − 1)

(7)

p≠m

Perform derivation on Π (ω, λm ) with respect to ω and λm , and set both equal to zero. The following equations can be obtained, ∂Π = ∂ω

∑ 2S

mp

ω − 2λm ω = 0

p≠m

∂Π = ωT ω − 1 = 0 ∂λm


(8)


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Multiply both sides of Eq. (8) by ωT and obtain,

∑ω S T

pm

ω = λm

(9)

p≠m

Therefore, the value of the objective function in the proposed method is equal to λm according to Eq. (5). From Eqs. (5) through (9), it is clear that a vector ω that maximizes J (ω) is equal to calculate the maximum eigenvalue problem,

   ∑ S pm  ω = λmω  p≠m 

(10)

where the eigenvalue indicates the importance of the corresponding vector ω . From Eq. (10), a set of eigenvectors can be obtained one time corresponding to the R largest nonzero generalized eigenvalues to construct the matrix Wm ( J × R ) , where R denotes the number of retained discriminant components. Wm contains the novel discriminant directions that separate the reference Mode m from the other modes by focusing on betweenmode discriminative information. To more clearly explain between-mode differences, discriminative variables that mostly contribute to discriminant directions are analyzed by employing a sparse method Lasso[29-31], which includes a penalty term that constrains the values of the discriminant directions to evaluate the importance of each variable. Projecting X cm on Wm , the score matrix Tm is obtained as shown in Eq. (11), Tm = X mc Wm

(11)

Note the f th score vector of Tm is t f . The vector β f is reconstructed to explain t f in a sparse manner as shown in Eq. (12),

(

β f = min ( X cm β f − t f

) (X T

c m

β−tf )+γ βf


)

(12)

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where γ is a penalty term, which is a nonnegative parameter and determined by trial and error, and symbol | | is used to calculate the L1-norm of a vector. For the f th score vector t f , discriminative variables are determined as the variables that are nonzero in β f and contribute to distinguishing Mode m from the other ones. Repeat Eq. (12) until all scores in Tm having been analyzed, and the collections of nonzero variables are final discriminative variables. It should be noted that the value of γ has influences on the selection of discriminative variables. If the value of γ is zero, the penalty is removed and β f will converge to ω f , which is the f th column of Wm . On the contrary, if γ is close to its maximum, i.e., one, few variables will be selected. Therefore, the selection of γ is important and may need trial and error. This subsection outputs the between-mode discriminative variables, which make Mode m unique to the other ones. Removing these discriminative variables, Mode m has similar process variations in comparison with other modes. Discriminative variables are pair-wise determined based on the reference mode. Therefore, replacing Mode m with another mode, the corresponding discriminative variables are determined using the similar procedures as shown above. 2.3 Multimode modeling and online process monitoring With the between-mode discriminative information analysis, discriminative variables are separated to explain between-mode differences. Naturally, when the operation mode changes, monitoring on discriminative variables will timely release alarms by avoiding the disturbances of other indiscriminative variables. Since Mode m is the reference mode, X cm is separated into two parts Xcm,v and Xcm,v% according to discriminative variables, where the subscript v indicates discriminative variables and subscript v% indicates the remaining indiscriminative variables. In the same way, data 13 ACS Paragon Plus Environment


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matrices of the other modes are also divided into two parts and data matrices Xcm% ,v and Xcm% ,v% are formed by putting the counterparts of each mode together, where the subscript m% indicates all modes other than Mode m. Presenting similar process variations, Xcm,v% and Xcm% ,v% are merged to a new matrix X cv% , which is composed of the indiscriminative variables of all modes. Finally, two data matrices, i.e., Xcm,v and X cv% , are employed for development of monitoring models. Xcm,v refers to the specific information, i.e., the between-mode discriminative information. Besides, X cv% is the between-mode indiscriminative information of all modes. (1) Modelling of between-mode discriminative information Perform PCA on Xcm,v and calculate score matrix Tmc ,v as follows,

Tmc ,v =Xcm,v Pmc ,v

(13)

where Pmc ,v is the loading matrix in Mode m. All principal components are retained to cover process variations regrading between-mode discriminative information and thus, T2 monitoring control limit is used for monitoring. Here, the batch-wise variations at each time within the same mode can be deemed to follow Gaussian distribution[6,7], and F distribution[26] can thus be used to describe the representative normal region for all time-slices within the same phase, c 2 m,v

T

~

Vmc,v ( N mc 2 − 1) N mc ( N mc − Vmc,v )

FV c

c c m ,v , N m −Vm ,v ,δ

(14)

where N mc is the number of samples in Mode m, σ is the significance factor, which is set to be 0.05 for calculating the 95% confidence limit, and Vmc,v is the number of influential variables. Also, there are some other methods, such as the non-parametric empirical density estimate method[32], can be used to derive the confidence limit. 14 ACS Paragon Plus Environment

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(2) Modelling of between-mode indiscriminative information Perform PCA algorithm on X cv% to develop monitoring control limits for between-mode indiscriminative data,

Tv%c = Xvc% Pv%c ˆ c = Tc P cT =Xc P c P cT X v%

v%

c v%

c v%

v%

v%

v%

(15)

v%

ˆ =X ( I − P P E = X −X c v%

c v%

c v%

cT v%

)

where Tv%c and Pv%c are the score matrix and loading matrix, respectively, E cv% is the residual matrix. Similarly, Tv%c 2 control limit can be calculated with Eq. (14). Besides, a SPE control limit is computed for the residual subspace, which can be approximated by a weighted chi-squared distribution[33], SPEvc ~ g vc χ hc ,η

(16)

v

v

where g vc = vvc 2 mvc and hvc = 2( mvc ) 2 vvc , mvc is the average of the vector

∑E

c v

(i, j ) 2 , vvc is

j =1

the corresponding variance, and η is the significance factor (which is 0.05) to derive the 95% confidence limit here. Monitoring on both between-mode discriminative and indiscriminative information is achieved for Mode m. Therefore, it is easy to repeat the above procedures and develop monitoring models for other reference modes. (3) Online process monitoring For online monitoring, it involves a crucial issue: how to identify the current operation mode. Without any priori knowledge, we have to check each mode in turn and give the reliable judgments according to the following strategy. Once a new sample x new ( J ×1) is available, normalize x new using the mean and variance estimated in Eq. (1) with the help of



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time index. Since the mode of x new is unknown, process monitoring models developed under each reference mode are adopted to compute the monitoring statistics. Project x new on each subspace as follows, t cm ,v T =x newT Pmc ,v t cv% T = x newT Pv%c

(17)

e = x new ( I − P P c v%

c v%

cT v%

)

Thus, the T2-statistic and SPE-statistic are calculated as, tmc ,v 2 =(t cm ,v − tmc,v )T Σ mc ,v −1 (t mc ,v − tmc,v ) tvc% 2 = (t cv% − tvc )T Σ cv% −1 (t cv% − tv%c )

(18)

c spevc% 2 = ecT v% e v%

where tmc,v denotes the mean vector of Tmc ,v% , t cv% is the mean vector of tv%c , Σcm,v −1 is the covariance matrix of Tmc ,v% , and Σ cv% −1 is the covariance matrix of tv%c . Compare these monitoring statistics with monitoring control limits to determine the current mode. If the monitoring statistic tmc ,v% 2 only stays under the monitoring control limits of Mode m, we can say that the process operates in Mode m. However, a fault in influential variables is

detected if monitoring statistic tmc ,v% 2 is out of the normal control limit for any mode. Besides, a fault in uninfluential variables will be detected if tv%c 2 or spevc 2 above the corresponding control limits.

3. ILLUSTRATION RESULTS This section highlights the proposed method that can well evaluate the between-mode similarity and identify the discriminative variables causing the between-mode differences. Besides, process monitoring under normal case and fault case are illustrated. The performances are demonstrated using a typical multimode and multiphase batch process, i.e., injection molding. 16 ACS Paragon Plus Environment

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3.1 Process description and data preparation Injection molding is used to produce plastic parts through sequential manufacturing procedures, which is a typical multimode and multiphase batch process. It mainly includes four stages, which are filling, packing-holding, plasticization and cooling stage. Here, the description of the four stages are briefly given and more detailed information can be found in previous work[34,35]. Starting with filling stage, the plastic melt is injected into the mold cavity at a certain speed. When the mold is filled with the plastic melt, injection is stopped and the process switches into the packing-holding stage. During this stage, high packing pressure is kept to prevent the melt flows out of the mold and the additional material is compacted into the mold to make up the shrinkage associated with cooling and solidification. The process then enters into cooling stage until the part is rigid enough to be ejected. Meanwhile, plasticization stage happens in the early stage of the cooling and the polymer melts are conveyed to the front of the barrel by screw rotation in this stage[35]. Totally, eleven process variables are collected from the injection machine and the detailed descriptions of the variables are given in Table 1. In this article, the set-point values of injection velocity and packing pressure are changed to produce three different operation modes, which are described in Table 2. Injection velocity mainly takes effect in injection stage and three different speeds, including 25mm/s, 35mm/s, and 45mm/s, are given for each mode. Meanwhile, in the packing-holding stage, two different values, 25Pa and 30Pa, are given for variable packing pressure. Thirty batches are generated for each mode and it should be noted that batch length of each mode is different. The collected data sets are

X1 ( 30 ×11× 530 ) of the first mode, X2 ( 30 ×11× 510 ) of the second mode, and X3 ( 30 ×11× 498 ) of the third mode. To solve the multiphase characteristics, lots of phase partition methods are proposed based on process knowledge and data analysis[25]. For simplicity, process knowledge[35] is employed 17 ACS Paragon Plus Environment


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to partition phase and the results are shown in Table 3, where four phases are partitioned in each mode. Influenced by different values of injection velocity and packing pressure, the batch length of each mode is not the same. In Table 3, the differences of batch length are mainly caused by the first phase, i.e., filling phase.

3.2 Between-mode similarity evaluation In this part, the between-mode similarity will be analyzed. Starting with the first phase, the shortest phase length among the three modes is 41, which is chosen as the phase length. Thus, the first 41 time-slice matrices of each mode are taken out and the data matrices are

X11 ( 30 ×11× 41) , X12 ( 30 ×11× 41) , and X13 ( 30 ×11× 41) , respectively. First, X11 ( 30 ×11× 41) is selected as the reference mode and the normalization information at each time are calculated according to Eq. (1). Then, time-slice matrices of the other modes are normalized using the obtained normalization information. Next, the distribution regions of three modes are calculated using Eq. (2), in which the significant factor is 95%. Thus, the number of samples is 1230 and the number of process variables is 11. By employing the Steps 2 and 3 given in Subsection 2.1, there are 92.44% samples of Mode 1 locate in its own normal region. Meanwhile, Mode 1 does not have MPs by comparing the sample distances with the corresponding control limits of Modes 2 and 3. The remaining 7.56% samples of Mode 1 are outliers. The above results are summarized as the first row of Phase 1 in Table 4. Besides, it is further concluded that Mode 1 has no similar samples with the other two modes. Taking Modes 2 and 3 for analysis, the results are presented in the second and the third rows of Phase 1 in Table 4, respectively. No samples in Mode 2 locates in the normal region of the other two modes indicating that Mode 2 also do not have any similar samples with the other two modes. There are 93.66% samples of Mode 2 are IPs and the remaining ones are outliers. The similar conclusions are drawn for Mode 3. Thus, in the first phase, all the three modes are different with other. In fact, it agrees well with the real case since we introduce totally 18 ACS Paragon Plus Environment

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different injection speeds for each mode and this causes the data distributions vary greatly according to process knowledge[35]. Some different phenomena are observed when we analyze the remaining Phases. In Phase 2, 23.02% samples of Mode 1 are MPs by analyzing the between-mode similarity with Mode 2. Similarly, MPs are observed in Mode 2 since 29.94% samples of Mode 2 locate in the normal region of Mode 1. However, Mode 3 is still independent of Modes 1 and 2. Since process conditions of Modes 1 and 2 are the same in Phase 2, it is natural that similar samples appear in Phase 2. Moreover, from Phases 3 to 4, the similarity between Mode 1 and 2 becomes larger. Especially, in Phase 4, the similarity between Modes 1 and 2 is large than 0.8, which means these two modes are almost the same and should be merged during the following between-mode discriminative information analysis.

3.3 Between-mode discriminative information analysis To analyze the discriminative variables, Modes 1 through 3 is selected as the reference mode in turn and then multimode-FDA is performed to extract the discriminant directions in each phase. By solving the eigenvalue decomposition function in Eq. (10), the discriminant direction is determined as the one whose eigenvalue is large than zero. In Phase 1, only one discriminant direction is available to distinguish Mode 1 from the other modes when Mode 1 is the reference mode. Then project the data of all three modes onto this discriminant direction and the obtained discriminant component is plotted using box plot, which is a useful tool to describe the variability of data set by defining quartiles. The whisker indicates the normal variations of data and box covers the distribution from the first quartile to the third quartile. The second quartile is just the band inside the box, which indicates the median. Outliers are plotted as individual points, which are described by the symbol plus. In Figure 3(a), the main variation region of the discriminant component of Mode 1, which is covered by a blue box, can be well separated from the other two modes. Similarly, if we 19 ACS Paragon Plus Environment


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change the reference mode as Modes 2 and 3, both of them can be separated by the discriminant directions, as shown in Figure 3(b) and Figure 3(c), respectively. From above analysis, we know all the three modes can be separated in Phase 1. Along these directions, we try to analyze which variable is crucial to make between-mode differences by employing a spare method, Lasso. If the tunable parameter λ of Lasso is large, the coefficients in discriminant direction will be greatly compressed and a few variables are retained. On the contrary, more variables will be retained if the value of λ is small. Figure 4 shows results based on several different values of λ. In Figure 4(a), when the value of λ is 0.5153, only Variable 3 is retained as a discriminative variable. If we decrease the value of λ to 0.3898, Variables 3 and 4 are discriminative variables. However, if the value of λ is further decreased to 0.3552, the discriminative variables keep the same. Thus, in Phase 1, the discriminative variables are Variables 3 and 4 (Screw stroke and Screw velocity) that make the Mode 1 different from the other two modes. When Mode 2 is the reference mode, discriminative variables are still Variables 3 and 4 according to the same analysis. The similar conclusions can be drawn for Mode 3. According to the process knowledge, injection velocity causes the difference between each mode. Injection velocity is difficult to be directly measured from the machine. However, screw stroke and screw velocity have a direct relationship with injection velocity by controlling the movement speed of screw. In Phase 2, perform multimode-FDA from Modes 1 through 3 in turn. Discriminant components of Modes 1 and 2 have overlap with each other as shown from Figure 5(a) through 5(c), where the values of the box are close to each other. Thus, Modes 1 and 2 in Phase 2 can not be separated. However, the box of Mode 3 locates outside the range of the other two modes, which indicates that Mode 3 is independent. The discriminant directions are extracted and compressed based on Lasso method. In Figure 6(c), Variable 8 (Injection pressure) and Variable 5 (Ejector stroke) are obvious the discriminative variable in Mode 3 20 ACS Paragon Plus Environment

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that makes the difference. In practice, to increase the pressure of mold cavity, the value of packing-pressure will increase and this leads to large injection pressure and position of ejector stroke. The similar analyses are performed in Phases 3 and 4, which are not directly influenced by the changes of set-points as shown in Table 2. In Figure 7, it is observed that Modes 1 and 2 have overlap region with each other, which indicates they can not be separated from each other. However, Mode 3 is still independent with the other two modes because the box of Mode 3 locates outside the range of the other two modes. After performing Lasso method, a discriminative variable that makes the difference between Mode 3 and Modes 1 and 2 in Phase 3 is Variable 5 (Ejector stroke), as shown in Figure 8. In Phase 4, according to the before between-mode similarity analysis, Modes 1 and 2 are merged together. As shown in Figure 9, Modes 1 and 2 are well separated from Mode 3, and the discriminative variable is Variable 5 (Ejector stroke), as identified in Figure 10. 3.2 Online process monitoring In each phase, monitoring models of each reference mode are developed based on discriminative variables and indiscriminative variables, respectively. When measurements are available, these corresponding models are adopted with the help of time index. Here, two cases are introduced to illustrate the monitoring performance. In the first case, a new normal batch of Mode 1 is tested, which is used to evaluate whether correct mode can be determined using between-mode discriminative information. Since no priori knowledge is available, each mode is checked in turn and the correct mode will be chosen if all monitoring statists are in the normal region of the control limits. As shown in Figure 11, monitoring statistics based on the two subspaces are all under their normal regions. Further, Modes 2 and 3 are tested and the results are shown in Figures 12 and 13, respectively. In Figure 12, it is obvious that the monitoring statistics in the subspace of influential variables 21 ACS Paragon Plus Environment


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are above its control limits in Phase 1. However, the subspace of indiscriminative variables is normal. Thus, the normal batch of Mode 1 is recognized as abnormal when the monitoring models of Mode 2 are adopted. Excepting the early stage of Phase 2, the monitoring performances become normal in Phase 2 for Mode 1 and Mode 2, which meet well with the real case. Then, Mode 3 is adopted to check the testing batch. In Figure 13, the monitoring statistics of all phases in the subspace of influential variables are above its control limits. From the above analysis, the testing batch is determined to be in Mode 1, which meets well with the real case. Besides, a fault with 1.5% step increase is introduced in Variable 8 of Mode 1 at the 51st sampling time and removed at the 150th sampling time. It is observed that fault occurs in the discriminative variable of Mode 1 in Phase 2. Therefore, the only subspace of discriminative variables is employed for process monitoring. In Figure 14(a), the abnormality is immediately detected by the T2 statistics at the 51st sampling time. Therefore, the proposed monitoring models can provide timely detection of a fault to avoid aggregation and taking corrective measures. After removing the fault at the 150th sampling time, the process goes back to the normal state, and monitoring statistics stay well within their control limits. In Figures 14(b) and (c), monitoring results are totally wrong. From the above illustrations, the between-mode discriminative information is correctly identified and corresponding monitoring models are developed. By using two different cases, including a normal case and a fault cause, the correct mode is selected and abnormality can be timely indicated.

4. CONCLUSIONS Considering the process monitoring of multimode and multiphase batch process is lacking, this article proposes a method based on between-mode similarity evaluation and multimode 22 ACS Paragon Plus Environment

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discriminative information analysis. It quantitatively evaluates the between-mode similarity within each phase and accurately describes the causes of between-mode differences from data-driven view. Through distinguishing mutual samples from each mode, the betweenmode similarity analysis is easily derived using the ratio of mutual samples to the total samples. In this way, the similar modes within each phase are merged to avoid repetitive modeling. Moreover, the understanding analysis is achieved by identifying discriminative information in a sparse manner, by which discriminative variables contributing most to discriminative information are extracted. Finally, to provide more effective process monitoring performance, multiple specific monitoring models are developed for betweenmode discriminative information and a global process monitoring model is developed for the indiscriminative information, respectively. The above analysis clearly explains the causes of between-mode differences through a typical multimode and multiphase batch process, injection molding and the results meet well with the priori process knowledge.

AUTHOR INFORMATION Corresponding Author *Email: [email protected]; Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS The authors gratefully acknowledge the support from the following foundations: This work is supported by the National Natural Science Foundation of China (No. 61422306 and No. 61433005).



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[2]. Qin, Y.; Zhao, C.H.; Wang, X.Z.; Gao, F.R. Subspace decomposition and critical phase selection based cumulative quality analysis for multiphase batch processes. Chem. Eng. Sci. 2017, 166, 130-143. [3]. Yao, Y. Multivariate statistical monitoring and fault diagnosis of dynamic batch processes with two-time-dimensional strategy. Ph.D. thesis, The Hong Kong University of Science & Technology, 2009. [4]. Jolliffe, I.T. Principal Component Analysis; Springer-Verlag: New York, 1986. [5]. Dayal, B.S.; MacGregor, J.F. Improved PLS algorithms. J. Chemom. 1997, 11, 73-85. [6]. Nomikos, P.; MacGregor, J.F. Monitoring batch processes using multi-way principal component analysis. AIChE J. 1994, 40, 1361-1375. [7]. Nomikos, P.; MacGregor, J.F. Multi-way partial least squares in monitoring batch processes. Chemom. Intell. Lab. Syst. 1995, 30, 97-108. [8]. Tan, S.; Wang, F.L.; Peng J.; Chang, Y.Q.; Wang S. Multimode process monitoring based on mode identification. Ind. Eng. Chem. Res. 2012, 51, 374-388. [9]. Zhao, C.H.; Yao, Y.; Gao, F.R.; Wang F.L. Statistical analysis and online monitoring for multimode processes with between-mode transitions. Chem. Eng. Sci. 2010, 65, 5961-5975. [10]. Ma, Y.X.; Shi, H.B. Multimode process monitoring based on aligned mixture factor analysis. Ind. Eng. Chem. Res. 2014, 53, 786-799. [11]. Ma, H.H.; Hu, Y.; Shi, H.B. A novel local neighborhood standardization strategy and its application in fault detection of multimode processes. Chemom. Intell. Lab. Syst. 2012, 118, 287-300. 24 ACS Paragon Plus Environment

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[14]. Song, B.; Tan, S.; Shi, H.B. Key principal components with recursive local outlier factor for multimode chemical process monitoring. J. of Process Control 2016, 47, 136-149. [15]. Zhang, S.M.; Wang, F.L.; Tan, S.; Wang, S.; Chang, Y.Q. Novel monitoring strategy combing the advantages of the multiple modelling strategy and Gaussian mixture model for multimode processes. Ind. Eng. Chem. Res. 2015, 54, 11866-11880. [16]. Hwang, D.H.; Han, C.H. Real-time monitoring for a process with multiple operating modes. Control Eng. Pract. 1999, 7, 891-902. [17]. Lane, S.; Martin, E.B.; Kooijmans, R.; Morris, A.J. Performance monitoring of a multiproduct semi-batch process. J. of Process Control, 2001, 11, 1-11. [18]. Choi, S.W.; Park, J.H.; Lee, I.B. Process monitoring using a Gaussian mixture model via principal component analysis and discriminant analysis. Comput. Chem. Eng. 2004, 28, 1377-1387. [19]. Yoo, C.K.; Villez, K.; Lee, I.B.; Rose´n, C.; Vanrolleghem, P.A. Multimodel statistical process monitoring and diagnosis of a sequencing batch reactor. Biotechnol Bioeng, 2007, 96, 687-701. [20]. Yu, J.; Qin, S.J. Multimode process monitoring with Bayesian inference-based finite Gaussian mixture modes. AIChE J. 2008, 54, 1811-1829. [21]. Xie, X.; Shi, H.B. Dynamic multimode process modeling and monitoring using adaptive Gaussian mixture models. Ind. Eng. Chem. Res. 2012, 51, 5497-5505.



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[22]. Yu, J. A new fault diagnosis method of multimode processes using Bayesian inference based Gaussian mixture contribution decomposition. Eng. Appl. of Artif. Intel. 2013, 26, 456466. [23]. Zhao, S.J.; Zhang, J.Y.; Xu, M. Monitoring of processes with multiple operating modes through multiple principal component analysis models. Ind. Eng. Chem. Res. 2004, 43, 70257035. [24]. Qin, Y.; Zhao, C.H.; Gao, F.R. An iterative two-step sequential phase partition (ITSPP) method for batch process modeling and online monitoring. AIChE J. 2016, 62, 2358-2373. [25]. Zhao, C.H. Concurrent phase partition and between-mode statistical analysis for multimode and multiphase batch process monitoring. AIChE J. 2014, 60, 559-573. [26]. Qin, S.J. Statistical process monitoring: basic and beyond. J. Chemom. 2003, 17, 480502. [27]. Peng K.X.; Zhang K., You B.; Dong J.; Wang Z.D. A quality-based nonlinear fault diagnosis framework focusing on industrial multimode batch processes. IEEE Trans. Ind. Electron. 2016, 63, 2615-2624.

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List of Figure Captions Figure 1 Descriptions of the data structure of multimode and multiphase batch processes. Figure 2 Procedures of between-mode similarity evaluation (Dk,m,m is the distance between x m , k and Mode m, Dk , m , m is the distance between x m , k and Mode m ).

Figure 3 The discriminant component in Phase 1 when reference mode is (a) Mode 1, (b) Mode 2, and (c) Mode 3. Figure 4 Discriminative variables in Phase 1 when reference mode is (a) Mode 1, (b) Mode 2, and (c) Mode 3. Figure 5 The discriminant component in Phase 2 when reference mode is (a) Mode 1, (b) Mode 2, and (c) Mode 3. Figure 6 Discriminative variables in Phase 2 when reference mode is (a) Mode 1, (b) Mode 2, and (c) Mode3. Figure 7 The discriminant component in Phase 3 when reference mode is (a) Mode 1, (b) Mode 2, and (c) Mode 3. Figure 8 Discriminative variables in Phase 3 when reference mode is (a) Mode 1, (b) Mode 2, and (c) Mode 3. Figure 9 The discriminant component in Phase 4 when reference mode is (a) Modes 1 and (b) Mode 2. Figure 10 Discriminative variables in Phase 4 when reference mode is (a) Modes 1 and (b) Mode 2. Figure 11 Monitoring charts of the normal case when Mode 1 is adopted in (a) subspace of discriminative variables and (b) subspace of indiscriminative variables (red dashed line: 95% control limits, black dot line: the online monitoring statistics).


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Figure 12 Monitoring charts of the normal case when Mode 2 is adopted in (a) subspace of discriminative variables and (b) subspace of indiscriminative variables (red dashed line: 95% control limits, black dot line: the online monitoring statistics). Figure 13 Monitoring charts of the normal case when Mode 3 is adopted in (a) subspace of discriminative variables and (b) subspace of indiscriminative variables (red dashed line: 95% control limits, black dot line: the online monitoring statistics). Figure 14 Monitoring charts of the fault case in (a) Mode 1, (b) Mode 2, and (c) Mode 3 (red dashed line: 95% control limits, black dot line: the online monitoring statistics).



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Figure 1 Descriptions of the data structure of multimode and multiphase batch processes.


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Dk , m, m

N mpm ,m = N mpm ,m + 1

m

Figure 2. Procedures of between-mode similarity evaluation (Dk,m,m is the distance between x m , k and Mode m, Dk , m , m is the distance between x m , k and Mode m ).



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Figure 3. The discriminant component in Phase 1 when reference mode is (a) Mode 1, (b) Mode 2, and (c) Mode 3.

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(a) Coefficient

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λ = 0.6080 λ = 0.3479 λ = 0.0862

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Figure 4. Discriminative variables in Phase 1 when reference mode is (a) Mode 1, (b) Mode 2, and (c) Mode 3.



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(a) Coefficient

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λ = 0.1266 λ = 0.1153 λ = 0.0455

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Coefficient

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λ = 0.3376 λ = 0.2327

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5

0

0 -1 -2 -3 -4 -5 -6

Modes 1 and 2

Mode3

Mode3

Mode 1 and 2

Figure 9. The discriminant component in Phase 4 when reference mode is (a) Mode 1 and (b) Mode 2.


Page 39 of 48

Coefficient

(a)

0.5

λ = 0.4925 0

λ = 0.1613

-0.5 -1 1

(b)

2

3

4

5

6

7

8

9

10

11

0.5

Coefficient

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


λ = 0.4110 0

λ = 0.1952

-0.5 -1 1

2

3

4

5

6

7

8

9

10

11

Variable No.

Figure 10. Discriminative variables in Phase 4 when reference mode is (a) Mode 1 and (b) Mode 2.



T 2m,v

(a)

15 10 5 0

(b)

0

50

100

150

200

250

300

350

400

450

500

0

50

100

150

200

250

300

350

400

450

500

0

50

100

150

200

250

300

350

400

450

500

15

T 2v

10 5 0 20

SPEv

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 40 of 48

10

0

Samples

Figure 11. Monitoring charts of the normal case when Mode 1 is adopted in (a) subspace of discriminative variables and (b) subspace of indiscriminative variables (red dashed line: 95% control limits, black dot line: the online monitoring statistics).


(a)

10

0

-10

(b)

0

50

100

150

200

250

300

350

400

450

500

0

50

100

150

200

250

300

350

400

450

500

0

50

100

150

200

250

300

350

400

450

500

5

T 2v

0 -5 -10 4

SPEv

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


T 2m,v

Page 41 of 48

2 0 -2

Samples




(a) T 2m,v

10

5

0

(b)

0

50

100

150

200

250

300

350

400

450

500

0

50

100

150

200

250

300

350

400

450

500

0

50

100

150

200

250

300

350

400

450

500

5

T 2v

0 -5 -10 4

SPEv

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 42 of 48

3 2 1

Samples



Page 43 of 48

(a) Mode 1

T 2v

10 0 -10 0

50

100

150

200

250

300

350

400

450

500

300

350

400

450

500

300

350

400

450

500

(b) Mode 2

T 2v

10 0 -10 0

50

100

150

200

250

(c) Mode 3 10

T 2v

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


5 0 0

50

100

150

200

250

Samples

Figure 14. Monitoring charts of the fault case in (a) Mode 1, (b) Mode 2, and (c) Mode 3 (red dashed line: 95% control limits, black dot line: the online monitoring statistics).



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 44 of 48

Table 1. Description of process variables used in injection molding No. Variable’s descriptions Unit 1

Pressure valve

%

2

Flow valve

%

3

Screw stroke

mm

4

Screw velocity

mm/sec

5

Ejector stroke

mm

6

Mold stroke

mm

7

Mold velocity

mm/sec

8

Injection pressure

Bar

9

Temperature 3

°C

10

Temperature 2

°C

11

Temperature 1

°C

ACS Paragon Plus Environment

44

Page 45 of 48

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 2. Three operation modes with different operation conditions Mode No. Injection velocity (mm/s) Packing pressure (Pa) 1

25

25

2

35

25

3

45

30


45


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 46 of 48

Table 3. Phase partition results of the concerned modes using phase duration Phase Name Filling Packing pressure Plasticization Cooling Mode No. phase phase phase phase 1 1-73 74-233 234-325 326-530 2

1-53

54-213

214-304

305-510

3

1-41

42-201

202-292

293-498


46

Page 47 of 48

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48


Mode No. 1

Table 4. Between-mode similarity (%) of concerned modes in each phase Phase1 Phase2 Phase3 Phase4 2

3

1

1

92.44 0

0

71.63 23.02 0

44.29 53.55 0

18.05 80.46 0

2

0

93.66 0

29.94 64.33 0

54.07 42.67 0

84.36 11.56 0

3

0

0

92.85 0

2

0

3

1

92.77 0

2

3

1

11.03 84.29 0

2

8.18

3

83.43


47


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 48 of 48

TOC graphic page


48

Multimode and Multiphase Batch Processes Understanding and

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