(78a) An Extensive Comparison Study of Batch Process Monitoring Approaches
AIChE Spring Meeting and Global Congress on Process Safety
Tuesday, April 28, 2015 - 8:00am to 8:30am
Batch processes play a central role in modern industrial plants, namely in the production of high valued products and fine chemicals, due to their added flexibility and scalability. The operation of batch processes is also intrinsically dynamic and non-stationary, bringing more degrees of freedom to their operation but increasing the complexity of the optimization and monitoring tasks. Regarding process monitoring, batch processes generate data with a 3-way structure, X (I x J x K), where I is the number of batches, J is the number of variables and K is the number of measurements made during a batch. One of the first methods proposed for addressing this problem is Multiway Principal Component Analysis (MPCA), which is based on the batch-wise unfolding of the 3-way data matrix followed by the application of a 2-way latent variable methodology, such as PCA or PLS. Further developments led to the variable-wise unfolding and 3-way approaches. More recently, dynamic methods were also introduced with the intent to explicitly model the process dynamics.
MPCA for batch process monitoring was first proposed by Nomikos and MacGregor . The methodology starts by unfolding the 3-way data matrix into a 2-way matrix X (I x JK): vertical slices (I x J) are placed side by side such that the first slice corresponds to all variables at the first time point and subsequent slices, regarding subsequent times, are arranged to the right of the existing ones. The variables of the extended X matrix are then mean centered and their variance scaled to unity. This corresponds to a batch-wise normalization, which essentially removes the batch mean non-stationary trend. The remaining variation is then modeled by PCA. As a result of the unfolding procedure, during online monitoring it is necessary to predict future observations in order to compute the scores. To do so, several in-filling approaches are available such as: zero deviation, current deviation and projection data.
The class of 3-way approaches is based on the use of different trilinear decompositions of X (I x J x K) performed by several chemometric methods, such as PARAFAC and Tucker3. Tucker3 provides a decomposition of the 3-way array into three orthogonal loadings (A, B and C) and a core matrix (G) plus a residual matrix (E). The PARAFAC model is written in terms of the product of the three loading matrices (A, B and C) plus a matrix of residuals (E). PARAFAC differs from Tucker3, mainly because each of its three dimensions cannot be flexibly combined as occurs in the Tucker3 model, due to the nature of its core matrix. Furthermore, PARAFAC also requires that all loading matrices share the same number of components.
Regarding the class of dynamic methodologies, Chen and Liu  proposed a Batch Dynamic PCA (BDPCA) approach that extends the Dynamic PCA method of Ku, Storer  to batch processes. In this method an extended data matrix of time-shifted variables is constructed for each batch and used to compute an extended sample covariance matrix. A standard PCA model is then build based on the average sample covariance matrix.
Alternatively, instead of monitoring the measured observations, Choi, Morris  proposed the monitoring of their residuals with resort to an autoregressive PCA (ARPCA) method. Basically, this method fits a multivariate autoregressive model to the data and then uses it to compute one-step-ahead predictions of the current measurements. Afterwards, the residuals are determined and organized variable-wise prior PCA analysis.
Based on the BDPCA scheme, the application scope of Dynamic PCA with Decorrelated Residuals (DPCA-DR) proposed by Rato and Reis  can be easily adapted to the case of batch monitoring. To apply this approach a DPCA model is constructed as in the BDPCA methodology. However, in DPCA-DR the monitored quantities are the one-step-ahead predictions of the scores and observations, which are obtained through the application of missing data imputation techniques , assuming that the current observations are missing. As a result, the monitoring statistics present low autocorrelation levels and have a similar behavior to those of ARPCA but without the need of fitting a multivariate time series model.
As can be observed from the discussion above, a wide range of methods for online monitoring are currently available. However, only very few studies have been conducted to compare particular aspects of their relative performance. It is therefore the objective of this work to conduct an extensive study that contemplates the leading strategies for online batch process monitoring, in order to provide practitioners with a clear and updated vision of the advantages and disadvantages of each method.
In order to conduct the proposed task, normal and faulty batches generated from simulated batch processes were monitored by the above methods. The respective control limits were computed using current and leave-one-out approaches. For the residual statistics, different window sizes using k observations before and after the current residuals were also considered. The assessment of the studied monitoring methods, as well as the effects of the different in-fillings and control limits, was made taking into account two distinct dimensions of performance: detection strength and detection speed. Detection strength was evaluated through a performance index based on the receiver operation curve (ROC) of the methods in each trial, which provides a robust measure of their ability to discriminate among normal and faulty batches. Detection speed was assessed through the analysis of the Conditional Expected Delay versus Probability of False Alarms curve.
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