Soft-Transition Sub-PCA Fault Monitoring of Batch Processes

Article pubs.acs.org/IECR

Soft-Transition Sub-PCA Fault Monitoring of Batch Processes Jing Wang,* Huatong Wei, Liulin Cao, and Qibing Jin College of Information Science and Technology, Beijing University of Chemical Technology, Beijing 100029, China ABSTRACT: Inaccurate substage division problems often emerge when multiway principal component analysis is applied in fault monitoring of multistage batch processes. A new two-step stage division method based on support vector data description (SVDD) is proposed in order to avoid the hard-division and misclassification problems. The loading matrices of the MPCA model are modified using the idea of combining the mechanism knowledge with field data in the rough division step. The model differences are increased by introducing the sampling time to loading matrices, which can avoid division mistakes caused by the fault data. Detailed stage separation is realized here based on the SVDD hypersphere distance to divide the process strictly into steady or transition stages. Then a soft-transition sub-PCA model is given based on the hypersphere distance. The method is applied to monitoring a penicillin fermentation process online. Simulation results show that the proposed method can describe transition stage information in more detail. It can detect the fault earlier and avoid the false alarm compared with traditional subPCA monitoring.

1. INTRODUCTION Batch or semibatch processes have been utilized to produce high-value-added products in the biological, food, and semiconductor industries. Batch processes, such as fermentation, polymerization, and pharmacy, are highly sensitive to abnormal changes in operating conditions. Monitoring of such processes is extremely important in order to get higher productivity. However, it is more difficult to develop an exact monitoring model of batch processes than that of continuous processes due to the common natures of batch processes: nonsteady, timevarying, finite duration, and nonlinear behaviors. The lack of exact monitoring model in most batch processes makes it so that operators cannot identify the faults when they occur. Therefore, effective techniques for monitoring batch processes exactly are necessary in order to remind an operator to take some corrective action before the situation becomes more dangerous. Benefitting from the rapid development of process instrumentation and data acquisition technology, multivariate statistical methods such as principal component analysis (PCA) and partial least-squares (PLS) have been successfully used in the modeling of multivariate continuous processes. The conventional PCA method cannot be used in batch processes directly, though it is performed well in continuous processes. The main reason is that conventional PCA can only deal with two-dimension data, while the data of batch processes is threedimensional. Thus, several extensions of the conventional PCA/PLS to batch processes have been reported, such as multiway principle component analysis (MPCA) and multiway partial least squares (MPLS). The three-dimension data is unfolded into two-dimension space, and then the data through a whole batch is monitored1−4 in these modified methods. However, the traditional MPCA method still has many limitations when it is used in online monitoring of batch processes. For example, the future values until the end of each batch must be estimated. Furthermore, it is difficult to reveal the changes of process correlations from one substage to another, because the entire batch data is taken as a single © 2013 American Chemical Society

object. Thus, the monitoring performance of the traditional MPCA is poor when it is used in a time-varying process or the initial stage of a batch when many measurements are not available. Therefore, the unique process correlation information of different phases is unreasonable, which will not only be difficult to demonstrate the process nature but also reduce the monitoring efficiency. Generally, many batch processes are carried out in a sequence of steps, which are called multistage or multiphase batch processes. Different phases may have different inherent natures. Thus, it is desirable to develop stage-based models that each model represents a specific stage and focuses on a local behavior of the batch process. This kind of monitoring method based on multiphase models can effectively increase process understanding and enhance monitoring reliability. There are many methods to divide batch processes into phases,5 which are generally called phase or stage-based sub-PCA methods. Some of the literature has divided batch processes into multistages based on mechanism knowledge. A process can be divided into several segments according to the different processing units or the distinguishable operation phases inside each unit.6,7 It is suggested that the batch process data could be naturally divided into groups before modeling and analysis. This kind of phase division can directly reflect the process operation status, while prior knowledge may not always be sufficient to divide processes into phases reasonably. Muthuswamy and Srinivasan8 identified several division points according to the process variable features described in the form of multivariate rules. Undey and Cinar9 used an indicator variable that contained significant landmarks to detect completion of each phase. Doan and Srinivasan10 did phase division based on singular points in some known key variables, which were also some landmarks. Kosanovich, Dahl, and Received: Revised: Accepted: Published: 9879

November 22, 2012 June 6, 2013 June 12, 2013 June 13, 2013 dx.doi.org/10.1021/ie3031983 | Ind. Eng. Chem. Res. 2013, 52, 9879−9888

Industrial & Engineering Chemistry Research

Article

Piovoso11 pointed out that the changes in the process variance information explained by principal components could indicate the division points between the process stages. Much work has been done on this method.12−15 However, these methods did not give a clear strategy to distinguish the steady phase and transition phase. An improved online sub-PCA method for multiphase batch processes is proposed here. A new two-step stage dividing algorithm based on the support vector data description (SVDD) technique is used in stage dividing and soft-transition partitioning. Then multiphase batch processes may be divided into several “operation” stages reflecting their inherent process correlation nature. Mechanism knowledge is considered in the first step of sub-PCA modeling by introducing the sampling time into the loading matrices of PCA model, which can avoid segmentation mistakes caused by the fault data. Then the support vector data description method is used to strictly refine the initial division for further improving the division accuracy, which is called soft-transition substage distinction between stability and transition periods. A representative model can be built for each substage, and an online fault monitoring algorithm is given based on the division techniques above. This method can detect fault earlier and avoid false alarms because of more precise stage division, compared with the conventional sub-PCA method. The remainder of this paper is organized as follows: in section 2, the conventional stage-based sub-PCA monitoring method is briefly introduced. Then, stage division and the novel soft-transition sub-PCA monitoring method are described in section 3. In section 4, the superiority of the proposed monitoring method over the conventional sub-PCA is illustrated by being applied to the fed-batch penicillin process. Finally, conclusions are given in section 5.

principal component number of each phase could be calculated based on the relative cumulative variance. Lastly, T2, SPE statistics, and their corresponding control limits were calculated according to the sub-PCA models. Which phase the new data was located in was determined by checking the Euclidean distance to the clustering center during online monitoring. Then, the corresponding sub-PCA model was used to monitor the online process data. Fault warning would be given out if there was a fault detected by T2 or SPE.

3. FAULT MONITORING BASED ON SVDD SOFT-TRANSITION SUB-PCA Generally speaking, industry batch processes operate in a variety of states. Some of them are steady states, while others include grade changes, startup, shutdown, and maintenance operations. A transitional region between neighboring stages is very common in multistage process, which shows the gradual changeover between two different patterns. Regularly, transitional phase shows the underlying characteristic more similar to the previous steady stage at first and then more similar to the next steady stage at the transition end. Different transitions go through different trajectories from one steady pattern to another, whose process characteristics changing along sampling time are more obvious and complex than those within one stage. Therefore, valid process monitoring during transitions is very important. To date, few investigations about transition modeling and monitoring have been reported16 to improve transition process monitoring performance using multivariate statistical methods. Here a new transition identification and monitoring method based on the SVDD division method is proposed. 3.1. Data Preprocessing and Time-Slice PCA Modeling. Historical data of batch process are composed of a threedimension array X(I × J × K), where I is the number of batches, J is the number of variables, and K is the number of sampling times. The three-dimension process data X(I × J × K) has to be unfolded into two-dimension forms Xk(I × J) (k = 1, 2, ..., K) before performing PCA. Then a time-slice matrix is placed beneath one another but not beside,17,18 as shown in Figure 1. Sometimes different batches have different lengths, i.e., sampling times K are different. Process data need to be aligned before unfolding. There are many data alignment methods raised by former researchers, such as directly filling zeros to missing sampling time19 and dynamic time warping.20 This unfolding approach does not require estimation of unknown future data for online monitoring. By subtracting the

2. WHAT IS STAGE-BASED SUB-PCA MPCA still has many problems to be solved, such as data uneven length and online monitoring. Many data alignment methods, i.e., directly sampling time cutting and dynamic time warping, have been proposed recently. Furthermore, many methods like the multiphase PCA and phase-based sub-PCA methods were employed in a multiphase batch process or semibatch process. These methods used a series of statistical models to extract different process characteristics in each process stage. There were several major steps in sub-PCA modeling procedures, namely, batch process data matrix splitting, phase division, and sub-PCA model building. Original data X should be conveniently rearranged into two-dimensional data matrices prior to developing statistical models. Different unfolding methods would highlight the different variances or covariance structures of the original data. Two traditional methods were widely applied: the batch-wise unfolding and the variable-wise unfolding, in which the most used method is batch-wise unfolding. The three-dimensional matrix X should be cut into K time-slice matrices after the batch-wise unfolding was completed. Then PCA treatment on these split time-slice matrices was implemented, and the variable correlation information was contained in K time-slice loading matrices. Phase division could be accomplished by the k-means clustering algorithm to identify all phases. Then sub-PCA models for all phases were built by taking the average of the time-slice PCA models in the corresponding phase. The

Figure 1. Batch-wise unfolding. 9880

dx.doi.org/10.1021/ie3031983 | Ind. Eng. Chem. Res. 2013, 52, 9879−9888


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vector with a repeated value of current sampling time. The sampling time would have an obvious effect on phase separation. The influence of sampling time will not change too much with the ongoing batch process. We define the Euclidean distance of extended loading matrix P̂k as

grand mean of each variable over all time and all batches, unfolding matrix Xk is centered and scaled Xk =

[Xk − mean(Xk )] σ (X k )

(1)

where mean(Xk) represents the mean value of matrix Xk and σ(Xk) is the standard variance value of matrix Xk. The main nonlinear and dynamic components of every variable are still left in the scaled matrix. The PCA method can be applied to analyze each time-slice matrix after the data is unfolded. The data matrix Xk can be projected into principal component subspace by loading matrix Pk to obtain the scores matrix Tk Xk = TkPk + Ek

2 || Pi ̂ − Pĵ || = [ Pi − Pj ti − tj ][ Pi − Pj ti − tj ]T

= || Pi − Pj ||2 + || ti − tj ||2

(3)

Then the batch process can be first divided into S1 stages using the k-means clustering method to cluster the extended loading matrices P̂k. It is obvious that the Euclidean distance of extended loading matrix contains both data differences and sampling time differences. Data in different stages are of great difference in sampling time. Thus, the great difference from sampling times would make the final Euclidean distance stay in a large value when a noise disturbance makes the data in different stages the same or similar. Because the wrong data has great difference in sampling time from other phases, transition phase data has little change in sampling time. We could easily distinguish the wrong division made by the transition phase from those led by noise. 3.3. Detailed Stage Division Based on SVDD. The timeslice loading matrices P̂ k represent the local covariance information and underlying process behavior as mentioned before, so they are used in determining the operation stages by proper analyzing and clustering procedures. The process is divided into different stages, and each separated process stage contains a series of successive samples. Moreover, it is unsuitable to be forcibly incorporated into one steady stage because of the variation complexity of process characteristics in transition regions. The transiting alteration of process characteristics imposes disadvantageous effects on the accuracy of stage-based sub-PCA monitoring models. Furthermore, it deteriorates fault detecting performance if just a steady transition sub-PCA model is employed to monitor the transition stage. Consequently, a new method based on a support vector data description (SVDD) is proposed to separate the transition regions after the original stage division which is determined by the k-means clustering. The support vector data description21 is a relatively new data description method, which was originally proposed by Tax and Duin22 for the one-class classification problem. SVDD has been employed for damage detection, image classification, one-class pattern recognition, etc. Recently, it has also been applied in monitoring continuous processes. However, SVDD has not been used for batch process phase separating and recognition to date. The loading matrix P̂ k of each stage is used to train the SVDD model of the transition process. The SVDD model first maps the data from original space to feature space by a nonlinear transformation function, which is called the kernel function. Then a hypersphere with minimum volume can be found in the feature space. To construct such a minimum volume hypersphere, the following optimization problem is obtained

(2)

where Ek is the residual. The first few components in principal component subspace which represent major variation of original data set Xk are chosen. The original data set Xk is then divided into the score matrix X̂ k = Tk Pk and the residual matrix Ek. Here X̂ k is PCA model prediction. Many other techniques, such as the cross-validation, have been used to determine the most appropriate retained numbers of principal components. Then the loading matrix Pk and singular value matrix Sk of each time-slice matrix Xk can be obtained. 3.2. Rough Stage-Division Based on Extended Loading Matrix. It is customary to analyze process data in order to identify the adequate model structure when a continuous process is modeled. Generally, all variables of continuous processes are supposed to stay around certain preset values and the correlation between these variables remains relatively stable. Non-steady-state operating conditions, time varying, and multiphase behavior are typical characteristics of a batch process. The process correlation structure might change due to process dynamics and time-varying factors. The MPCA model is ill suited and may cause lots of false alarms when monitoring a multiphase batch process. This is because it takes the entire batch data as a single object, and the process correlation among different stages cannot be captured effectively. Multiphase PCA (MPPCA) is aimed at employing a separate model for a forthcoming specific process period, instead of using a single modeling structure for the entire process. The whole batch is divided into several appropriate phases, which are characterized by their unique correlation structures. As the loading matrix Pk reflects the process variables’ correlations, it usually is used to identify the process stage. Sometimes disturbances brought by measurement noise or other reasons will lead to wrong division, because the loading matrix just obtained from process data is hard to distinguish between wrong data and transition phase data. Generally, different phases in the batch process could be first distinguished according to mechanism knowledge. Here sampling time is the simplest. The sampling time is added to the loading matrix in order to divide the process exactly. Sampling time is a continuously increasing data set, so it must also be centered and scaled before adding to the loading matrix. Generally, the sampling time is centered and scaled not along the batch dimension like process data X but along the time dimension in one batch. Then the scaling time tk is changed into a vector tk by multiplying the unit column vector. Thus, the new time-slice matrix is written as P̂ k = [Pk tk], in which tk is a 1 × J column

min ε(R , A, ξ) = R2 + C ∑ ξi i

(4)

2

s. t. || Pi ̂ − A || ≤ R2 + ξi , ξi ≥ 0, ∀ i 9881



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batch i (i = 1, 2, ..., I), stage s, I is the number of batches, and α is the significant level. 3.4. PCA Modeling of Transition Stage. Now a softtransition multiphase PCA modeling method based on SVDD is presented according to the above discussion. It uses the SVDD hypersphere radius to determine the range of transition region between two different stages. Meanwhile, it introduces a concept of membership grades to evaluate quantitatively the similarity between current sampling time data and transition (or steady) stage models. Then sub-PCA models for steady phases and transition phases are established which greatly improve the accuracy of models. Moreover, they reflect the characteristic changing during the different neighboring stages. Time-varying monitoring models in transition regions are established relying on the concept of membership grades, which are the weighted sum of nearby steady phase and transition phase submodels. Membership degree values are used to describe the partition problem with ambiguous boundary, which can objectively reflect the process correlations changing from one stage to another. Here the hyperspace distance Dk,s is defined from the sampling data at time k to the center of the sth SVDD submodel. This distance is used as a dissimilarity index to evaluate quantitatively the changing trend of process characteristics. Correlation coefficients λl,k are given as the weight of the soft-transition submodel, which are defined as

where R and A are the radius and center of the hypersphere respectively, C gives the trade-off between the volume of the hypersphere and the number of error divides, ξi is a slack variable which allows the probability that some of the training samples can be wrongly classified. Another form of the optimization problem (eq 4) can be rewritten as min ∑ αiK (Pi ̂ , Pi ̂) −

∑ αiαjK (Pi ̂ , Pĵ)

i

(5)

i,j

s. t. 0 ≤ α ≤ Ci where K(P̂ i,P̂j) is the kernel function and αi is the Lagrange multiplier. Here a Gaussian kernel function is selected as the kernel function. The general quadratic programming method is used to solve the optimization question (eq 5). The hypersphere radius R can be calculated according to the optimal solution αi n

n

R2 = 1 − 2 ∑ αiK (Pk̂ , Pi ̂) + i=1

αiαjK (Pĵ , Pi ̂)

∑

(6)

i = 1, j = 1

Here the loading matrices P̂ k are those corresponding to nonzero parameter αk. It means that they have an effect on the SVDD model. Then the transition phase can be distinguished from the steady phase by inputting all time-slice matrices P̂k into the SVDD model. When new data P̂new is available, the hyperspace distance from the new data to the hypersphere center should be calculated first

⎧λ ⎪ s − 1, k = (Dk , s + Dk , s + 1) ⎪ /(2*(Dk , s − 1 + Dk , s + Dk , s + 1)) ⎪ ⎪ λ = (D k , s − 1 + Dk , s + 1) ⎨ s,k /(2*(Dk , s − 1 + Dk , s + Dk , s + 1)) ⎪ ⎪ ⎪ λs + 1, k = (Dk , s − 1 + Dk , s) ⎪ /(2*(Dk , s − 1 + Dk , s + Dk , s + 1)) ⎩

2

2

̂ − a || D = || Pnew n

n

̂ , Pi ̂) + = 1 − 2 ∑ αiK (Pnew i=1

∑ i = 1, j = 1

αiαjK (Pi ̂ , Pĵ ) (7)

If the hyperspace distance is less than the hypersphere radius, i.e., D2 ≤ R2, the process data P̂ new belongs to steady stages; otherwise, that is, D2 > R2, the data will be assigned to transition stages. The whole batch is divided into S2 stages at the detailed division, which includes S1 steady stages and S2−S1 transition stages. The mean loading matrix P̅s can be adopted to get the subPCA model of this stage because the time-slice loading matrices in one stage are similar. P̂s is the mean matrix of the loading matrices Pk in one stage. The principal components number a can be obtained by calculating the relative cumulative variance of each principal component until it reaches 85%. Then the mean loading matrix is modified according to the obtained principal components. The sub-PCA model can be described as

where l = s − 1, s, and s + 1 is the stage number, which represents the last steady stage, current transition stage and next steady stage, respectively. The correlation coefficient is inverse proportional to hyperspace distance. The larger the distance is, the less the influence of hyperspace distance has. The monitoring model of transition stage at every time interval can be obtained by the weighted sum of sub-PCA models, that is s+1

P′k =

⎧ Tk = Xk Ps̅ , ⎪ ⎪ ⎨ X̅k = Tk(Ps̅ )T , ⎪ ⎪ E = X − X̅ ⎩ k k k

(I − as , i)

(8)

Fas ,i , I − as ,i , α

SPEk , α = gk χh2 , α , gk = k

λl , k Pl̅

(11)

The soft-transition PCA model in eq 11 properly reflects the time-varying transiting development. Also, the score matrix T′k and the covariance matrix S′k can be obtained at each time instance. The SPE statistic control limit is still calculated by eq 9. Different batches have some differences in transition stages. The mean T2 limit of all batches is used to monitor this process in order to enhance the robustness of the proposed method. The T2 statistic control limit could be calculated by data and correlation coefficients of historical batches

In addition, T and the SPE statistic control limit are calculated as , i(I − 1)

∑ l=s−1

2

Tα2s , i ≈

(10)

(9)

vk 2mk2 , hk = 2mk vk

s+1

Tα2 ′ =

where mk and vk are the average and variance of all batches data at time k, as,i is the number of retained principal components in

I

∑ ∑ λl ,i,kTα2 /I s,i

l=s−1 i=1

9882

(12)



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Figure 2. Illustration of soft-transition sub-PCA modeling.

(4) Inputting again the extended loading matrices P̂k into the original SVDD model to divide explicitly the process into S2 stages: the steady stage and transition stage; then retraining the SVDD classifier for these new S2 stages. The mean loading matrix P̅ s of each new steady stage should be calculated, and the sub-PCA model is built in eq 8. The correlation coefficients λl,k are calculated to get the soft-transition stage model P′k in eq 11 for transition stage, t. (5) Calculating the control limit of SPE and T2 to monitor new process data. The whole flow chart of improved sub-PCA modeling based on SVDD soft transition is shown in Figure 2. This modeling process is off-line, which is dependent on the historical data of I batches. 3.6. Online Monitoring Based on the Soft-Transition Sub-PCA Model. The following steps should be adopted during online process monitoring. (1) Whenever getting a new sampling time-slice data xnew, centering and scaling it based on the mean and standard deviation of prior nomal I batches data. (2) Calculating the covariance matrix xTnew xnew, the loading matrix Pnew can be obtained based on singular value decomposition. Then adding sampling time tnew into it to obtain the extended matrix P̂ new. Inputting the new

where i (i = 1, 2, .., I) is the batch number and T2αs,i is the substage T2 statistic control limit of each batch which is calculated by eq 9 for substage s. Now the soft-transition model of each time interval in transition stages is obtained. The batch process can be monitored efficiently by combining with the steady stage model given in section 3.3. 3.5. Flow Chart of Soft-Transition Sub-PCA Modeling. The whole batch process has been divided into several steady stages and transition stage after the two-step stage dividing, shown in section 3.2 and 3.3. The new soft-transition sub-PCA method is applied to get a detailed submodel shown in section 3.4. The detailed modeling step is given as follows. (1) Getting normal process data of I batches, unfolding them into two-dimension time-slice matrix, and then centering and scaling each time-slice data as in eq 1. (2) Performing PCA on the normalized matrix of each timeslice and getting the loading matrices Pk, which represent the process correlation at each time interval. Adding sampling time t into the loading matrix to get the extended matrices P̂k. (3) Dividing roughly the process into S1 stages using k-means clustering on extended loading matrices P̂ k; then training the SVDD classifier for the original S1 steady process stages. 9883



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matrix P̂ new into the SVDD model to identify which stages this new data belongs to. (3) If current time slice data belongs to a transition stage, the weighted sum loading matrix P′new is employed to calculate the score vector tnew and error vector enew ⎧ tnew = xnewP′new ⎪ ⎨e = x − x new new ̅ ⎪ new = xnew(I − P′new P′Tnew ) ⎩

fermentor under open-loop operation during the fed-batch mode. There are 14 variables considered in this model as shown in Table 1: five input variables (1−4 and 14) and nine process Table 1. Variables Used in the Fed-Batch Penicillin Fermentation Process

(13)

or if it belongs to a steady one, the mean loading matrix P̅s would be used to calculate the score vector tnew and error vector enew ⎧ tnew = xnewPs̅ ⎪ ⎨ ⎪ T ⎩ enew = xnew − xnew s̅ s̅ ) ̅ = xnew(I − PP

(14)

(4) Calculating the SPE and T2 statistics of current data as follows 2 T ⎧ ⎪ Tnew = tnewSs̅ tnew ⎨ ⎪ T ⎩ SPEnew = enewenew

(15)

no.

variable

1 2 3 4 5 6 7 8 9 10 11 12 13 14

aeration rate (L/h) agitator power input (W) substrate feed rate (L/h) substrate feed temperature (K) dissolved oxygen concentration (%) culture volume (L) carbon dioxide concentration (mmol/L) pH temperature in the bioreactor (K) generated heat (kcal/h) cooling water flow rate (L/h) penicillin concentration hydrogen ion concentration substrate concentration

variables (5−13). In addition, there are five quality variables that cannot be measured online, which are not illustrated in Table 1. Generally, there are four physiological phases (lag, exponential cell growth, stationary, and cell death) and two operational modes in penicillin fermentation. The first operational mode is batch operation, where cultivation is carried out in a batch mode to promote biomass growth resulting in high cell densities. The second is fed-batch operation. Glucose will be fed until the operation end when glucose is consumed by growing cells. The penicillin process with multiple phases and nonlinear dynamics characteristic is a typical multistage process. Cells begin to self-dissolve, and the production rate will drop in the cell death phase. This phase is not conducive to product synthesis, so only the first three physiological phases are considered to ensure higher final penicillin concentration. A reference data set of 10 batches is simulated under nominal conditions with small perturbations. The completion time is 400 h. All variables are sampled every 1 h, so that one batch will offer 400 sampling time data in it. 4.2. Stage Identification and Modeling. The first step division result based on the k-mean method is shown in Figure 4. Originally, the batch process is classified into 3 steady stage, i.e., S1 = 3. Then SVDD classifier with Gaussian kernel function was used here for detailed division. The hypersphere radius of original 3 stages is calculated, and the distances from each sampling time data to the hypersphere center are shown in Figure 5. It is shown in Figure 5 that the sampling data between two stages, such as the data during 28−42 and 109−200 h, are obviously out of the hypersphere. This means the data at this two time regions have significant difference from that of other steady stage. Therefore, these two stages are considered as a transition stage. The process was further divided into 5 stages according to the detailed SVDD division, shown in Figure 6. It is obviously that the stages during 1−27, 43−109, and 202−400 h are steady stages. The hyperspace distance of stage 28−42 and 109−200 h exceeded the radius of hypersphere

(5) Judging whether the SPE and the T2 statistics of current data exceed the control limit. If one of them exceeds the control limit, alarm abnormal; if none of them exceeds, current data is normal.

4. CASE STUDY 4.1. Fed-Batch Penicillin Fermentation Process. The fed-batch penicillin fermentation process data is generated using a simulator named PenSim developed by the monitoring and control group at Illinois Institute of Technology.23 The flow sheet of the fermentation process is shown in Figure 3.

Figure 3. Flow diagram of the penicillin fermentation process.

The entire duration of each batch is 400 h with a sampling interval of 1 h. two PID controllers are adopted to control the pH and temperature by adjusting the flow rates of acid/base and cold/hot water in this fermentor, respectively. On the other hand, the substrate of glucose is continuously fed into the 9884



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4.3. Monitoring of Normal Batch. Monitoring results of the improved sub-PCA methods for the normal batch are presented in Figure 7. The blue line is the statistic

Figure 4. First-step division result.

Figure 7. Monitoring plots for a normal batch: (a) squared prediction error chart, (b) Hotelling T2 chart.

corresponding to online data, and the red line is the control limit with 99% confidence, which is calculated based on the normal historical data. It can be seen that as a result of a large change of the hyperspace distance at about 30 h in Figure 5, the T2 control limit drops sharply. The T2 statistic of this batch still stays below the confidence limits. Both of the monitoring systems (T2 and SPE) do not yield any false alarms. It means that this batch behaves normally during running. 4.4. Monitoring of Fault Batch. Monitoring results of the proposed method are compared with that of the traditional subPCA method8 in order to illustrate the effectiveness. Here two faults are used to test the monitoring system. Fault 1 is the agitator power variable with a decreasing 10% step at 20−100 h. It is shown in Figures 8 and 9 that SPE statistic increases sharply beyond the control limit in both methods, while T2 statistic which in fact reflects that changing of the sub-PCA model was not beyond the control limit in the traditional sub-PCA method. This means the proposed soft-

Figure 5. SVDD stage classification result.

Figure 6. Detailed process division result based on SVDD.

obviously. Thus, the two stages are separated as the transition stage. Then the new SVDD classifier model is rebuilt. The whole batch process data set is divided into five stages using the phase identification method proposed in this paper, that is, S2 = 5.

Figure 8. Proposed soft-transition monitoring for fault 1: (a) squared prediction error chart, (b) Hotelling T2 chart. 9885



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Figure 9. Traditional sub-PCA monitoring for fault 1: (a) squared prediction error chart, (b) Hotelling T2 chart.

Figure 11. Projection in principal components space of the traditional sub-PCA.

transition method made a more exact model than the traditional sub-PCA method. The difference between these two methods can be seen directly at the projection map, i.e., Figures 10 and 11. The blue

Figure 12. Proposed soft-transition monitoring for fault 2: (a) squared prediction error chart, (b) Hotelling T2 chart.

Figure 10. Projection in principal components space of the proposed method.

dot is the projection of data in 50−100 h to the first two principal components space, and the red line is the control limit. Figure 11 shows that none of the data are out of the control limit using the traditional sub-PCA method. The reason is that the traditional sub-PCA does not divide the transition stage. The proposed soft-transition sub-PCA can effectively diagnose the abnormal or fault data, shown in Figure 10. Fault 2 is a ramp decreasing with 0.1 slopes which is added to the substrate feed rate at 20−100 h. Online monitoring results of the traditional sub-PCA and proposed method are shown in Figures 12 and 13. It can be seen that this fault is detected by both two methods. The SPE statistic of the proposed method is out of the limit about at 50 h and the T2 values alarms at 45 h. Then both of them increase slightly and continuously until the end of the fault. It is clearly shown in Figure 13 that the SPE statistic of traditional sub-PCA did not alarm until about 75 h, which lags far behind that of the proposed method. Meanwhile, the T2

Figure 13. Traditional sub-PCA monitoring for fault 2: (a) squared prediction error chart, (b) Hotelling T2 chart.

statistic has a fault alarm at the beginning of the process. This is a false alarm caused by the process initial state changing. In comparison, the proposed method has fewer false alarms, and 9886



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Table 2. Monitoring Results for Other 12 Faults soft-transition sub-PCA

traditional sub-PCA

fault ID

variable no.

fault type

magnitude/ slope

fault time (h)

time detected(SPE)

time detected (T2)

1 2 3 4 5 6 7 8 9 10 11 12

2 2 3 3 1 1 3 2 1 3 2 1

step step step step step step ramp ramp ramp ramp ramp ramp

−15% −15% −10% −10% −10% −10% −5% −20% −10% −0.2% −20% −10%

20 100 190 30 20 150 20 20 20 170 170 180

20 100 190 48 20 150 28 31 24 171 181 184

28 100 199 45 20 151 40 45 30 171 195 188

FA

time detected(SPE)

time detected (T2)

FA

0 0 0 0 0 0 0 0 0 0 0 0

20 100 190 81 20 150 28 44 21 170 177 185

none 101 213 45 48 151 41 34 28 173 236 185

9 1 11 5 1 2 1 6 10 3 1 2

identifying process dynamic transitions for an unknown batch or semibatch process.

the fault alarm time of the proposed method is obviously ahead of the traditional sub-PCA. The monitoring results for the other 12 different faults are presented in Table 2. The fault variable nos. 1, 2, and 3 represent the aeration rate, agitator power, and substrate feed rate, respectively, as shown in Table 1. Here FA is the number of false alarms during the operation life. It can be seen that the false alarms of the conventional subPCA method are obviously higher than that of the proposed method. In comparison, the proposed method shows good robustness. The false alarms here are caused by the small change of the process initial state. The initial states are usually different in a real situation, which will lead to changes in the monitoring model. Many false alarms are caused by these little changes. The conventional sub-PCA method shows poor monitor performance in some transition stages and even cannot detect these faults because of the inaccurate stage division.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (61174128), Beijing Natural Science Foundation (4132044), and Fundamental Research Funds for the Central Universities, China (ZZ1223).

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REFERENCES

(1) Geladi, P.; Kowalski, B. R. Partial least-squares regression: a tutorial. Anal. Chim. Acta 1986, 185, 1. (2) Nomikos, P.; MacCregor, J. F. Monitoring batch processes using multiway principal component analysis. AIChE J. 1994, 40, 1361. (3) Nomikos, P.; MacCregor, J. F. Multivariate SPC charts for monitoring batch processes. Technometrics. 1995, 37, 41. (4) Nomikos, P.; MacCregor, J. F. Multi-way partial least squares in monitoring batch processes. Chemom. Intell. Lab. Syst. 1995, 30, 97. (5) Yao, Y.; Gao, F. R. A survey on multistage/multiphase statistical modeling methods for batch processes. Annu. Rev. Control. 2009, 33, 172. (6) Dong, D.; McAvoy, T. J. Batch tracking via nonlinear principal component analysis. AIChE J. 1996, 42, 2199. (7) Reinikainen, S. P.; Hoskuldsson, A. Multivariate statistical analysis of a multistep industrial processes. Anal. Chim. Acta 2007, 595, 248. (8) Muthuswamy, K.; Srinivasan, R. Phase-based supervisory control for fermentation process development. J. Process Control 2003, 13, 367. (9) Undey, C.; Cinar, A. Statistical monitoring of multistage, multiphase batch processes. IEEE Control Syst. Mag. 2002, 22, 40. (10) Doan, X. T.; Srinivasan, R. Online monitoring of multi-phase batch processes using phase-based multivariate statistical process control. Comput. Chem. Eng. 2008, 32, 230. (11) Kosanovich, K. A.; Dahl, K. S.; Piovoso, M. J. Improved process understanding using multiway principal component analysis. Ind.l Eng. Chem. Res. 1996, 35, 138. (12) Camacho, J.; Pico, J. Multi-phase principal component analysis for batch processes modeling. Chemom. Intell. Lab. Syst. 2006, 81, 127. (13) Camacho, J.; Pico, J.; Ferrer, A. Multi-phase analysis framework for handling batch process data. J. Chemom. 2008, 22, 632. (14) Lu, N.; Gao, F. R.; Wang, F. L. Sub-PCA modeling and on-line monitoring strategy for batch processes. AIChE J. 2004, 50, 255.

5. CONCLUSION The correlation among process variables changes with stage shifting in multiple phase batch process. This makes the MPCA and traditional sub-PCA method insufficient for process monitoring and fault diagnosis. Here a new stage identification method is proposed to identify the steady and transition stages clearly. Each stage usually has its own characteristic dynamics and deserves individual treatment. The transition stage between two steady stages especially has its own dynamic transition characteristic, while it is hard to be identified. Here two techniques are adopted to overcome these problems. First, inaccurate stage division caused by fault data will be avoided in the rough division by introducing the sampling time into the loading matrix. Then, the transition stage can be identified from the nearby steady stage according to the distance from process data to the SVDD hypersphere center. Different sub-PCA models for these steady and transition stages are given. In particular, the soft-transition sub-PCA model is the weighted sum of the last steady stage, current transition stage, and next steady stage, which can reflect the dynamic characteristic changing during the transition stage. Lastly, this method is applied to the penicillin fermentation process. Simulation results show the effectiveness of the proposed method. Also, this method can be applied to the monitoring problems for any batch or semibatch process without detailed process information. It is helpful when 9887



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(15) Yao, Y.; Gao, F. R. Phase and transition based batch process modeling and online monitoring. J. Process Control 2009, 19, 816. (16) Zhao, C. H.; Wang, F. L.; Lu, N. Y.; Jia, M. X. Stage-based softtransition multiple PCA modeling and on-line monitoring strategy for batch processes. J. Process Control 2007, 17, 728. (17) Westerhuis, J. A.; Kourti, T.; Macgregor, J. F. Comparing alternative approaches for multivariate statistical analysis of batch process data. J. Chemom. 1999, 13, 397. (18) Wold, S.; Kettaneh, N.; Friden, H.; Holmberg, A. Modelling and diagnosis of batch processes and analogous kinetic experiments. Chemom. Intell. Lab. Syst. 1998, 44, 331. (19) Arteaga, F.; Ferrer, A. Dealing with missing data in MSPC: several methods, different interpretations, some examples. J. Chemom. 2002, 16, 408. (20) Kassida, A.; MacGregor, J. F.; Taylor, P. A. Synchronization of batch trajectories using dynamic time warping. Am. Inst. Chem. Eng. J. 1998, 44, 864. (21) Tax, D. M. J.; Duin, R. P. W. Support vector domain description. Pattern Recogn. Lett. 1999, 20, 1191. (22) Tax, D. M. J.; Duin, R. P. W. Support vector data description. Mach. Learn. 2004, 54, 45. (23) Birol, G.; Undey, C.; Cinar, A. A modular simulation package for fed-batch fermentation: penicillin production. Comput. Chem. Eng. 2002, 26, 1553.

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Soft-Transition Sub-PCA Fault Monitoring of Batch Processes

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