Fault Detection and Diagnosis for Non-Gaussian Processes with

Article pubs.acs.org/IECR

Fault Detection and Diagnosis for Non-Gaussian Processes with Periodic Disturbance Based on AMRA-ICA Ying Tian, Wenli Du,* and Feng Qian Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, East China University of Science and Technology, 130 MeiLong Road, Shanghai, 200237, China ABSTRACT: Fault detection and diagnosis is important in ensuring the stability and safety of chemical processes. However, limited studies have focused on strong periodic disturbance and non-Gaussian process monitoring. By utilizing the data-driven monitoring method, we have proposed the residual analysis independent component analysis based on average multivariate cumulative sum (AMRA-ICA) method to avoid the influence of periodic disturbance in non-Gaussian chemical processes with periodic disturbance. Average multivariate cumulative sum (AM) is introduced in the AMRA-ICA method for disturbance cycle synchronization. Residual analysis (RA) is employed to remove the disturbance in the data set and to obtain the normal residual. The independent component analysis (ICA) method is then utilized to monitor the residual, and an improved contribution histogram method is proposed to identify the cause of the fault. The proposed method has been applied to the classic benchmark Tennessee Eastman process with and without periodic disturbance and to an ethylene compressor which is periodically affected by ambient temperature. Simulation results illustrate that the proposed AMRA-ICA method could solve the monitoring problem of non-Gaussian processes with periodic disturbance more effectively and accurately compared with the residual analysis PCA (RA-PCA) and the local tangent space alignment-ICA (LTSA-ICA). The AMRA-ICA method can also manage conventional processes without periodic disturbance.

■

INTRODUCTION Many chemical processes are complex, and small operational changes during critical periods may degrade the quality and yield of final products. Online monitoring allows for the faults detection during process operations, thereby enabling the prompt identification and correction of faults. Thus, using online monitoring during process operations significantly improves product quality. Several techniques based on multivariate statistical analysis have been proposed for online monitoring and fault detection. For example, principal component analysis (PCA) and partial least-squares (PLS) have been applied to process monitoring successfully.1 However, these methods have strict data requirements; specifically, these methods require processes to have linear and Gaussian distribution. Most industrial processes have difficulty meeting these prerequisites, which results in underreporting and misreporting in process monitoring.2 As a result, nonlinear and non-Gaussian process monitoring methods have become the focus of much research. The periodic change that process variables undergo because of the influence of periodic disturbances in the ambient environment is considered a special type of nonlinear process. Therefore, this paper discusses the monitoring of non-Gaussian processes with periodic disturbances. Hiden and Dong proposed improved PCA methods based on genetic programming3 and neural network,4 respectively, to monitor the nonlinear process. Although their methods can solve nonlinear problems, the accuracy of their methods is greatly affected by the completeness and representation of data. The fault data of industrial processes are difficult to obtain, thereby making these methods difficult to use in industrial processes. Scholkopf proposed the support vector machine (SVM) method, which requires fewer samples compared with the neural network method.5 However, applying SVM to large-scale data © 2013 American Chemical Society

process monitoring is difficult. Scholkopf proposed kernel principal component analysis (KPCA).6 Consequently, Cho applied the KPCA method to process monitoring.7 Ge proposed improved KPCA to monitor nonlinear process.8 And a probabilistic KPCA method and a Gaussian PPCA are proposed for process monitoring by Ge and Song.9,10 Different from the traditional PCA method, the probabilistic KPCA approach can successfully extract the nonlinear relationship between process variables; the Gaussian PPCA method can exhibit more detailed information of uncertainty for process data, through which the operation condition and the fault behavior can be interpreted more easily. Zhang proposed fault diagnosis of a nonlinear process using multiscale KPCA and multiscale KPLS, which can capture process variable correlations occurring at different scales.11 Multiblock KPLS(MBKPLS) was proposed by Zhang to monitor large-scale processes.12 Compared to PLS, the MBKPLS can capture more useful information between and within blocks; compared to MBPLS, MBKPLS gives nonlinear interpretation. Other kernel-based methods such as kernel dissimilarity analysis are also proposed.13 Moreover, a novel linear subspace and Bayesian inference-based monitoring method for nonlinear processes is proposed.14 In this method, monitoring results are first generated in each subspace and then transferred to fault probabilities by the Bayesian inference strategy. Those improved algorithms usually use kernel function. However, the kernel-based method has a long computing time because a uniform standard for the selection of kernel function is unavailable. Considering this, Received: Revised: Accepted: Published: 12082

March 4, 2013 June 19, 2013 July 24, 2013 July 24, 2013 dx.doi.org/10.1021/ie400712h | Ind. Eng. Chem. Res. 2013, 52, 12082−12107

Industrial & Engineering Chemistry Research

Article

Figure 1. The steps of the particle swarm optimization algorithm.

method based on ICA.16 The ICA algorithm is a linear method, whereas the actual industrial process is a nonlinear method. Considering this, Bach proposed an improved ICA for nonlinear process monitoring.17 Many improved algorithms based on the ICA have also been proposed.18,19For example, Zhang proposed modified ICA based on particle swarm optimization (PSO) (PSO-ICA).20 The basic idea of the approach is to use the PSO-ICA algorithm to extract some dominant independent components from normal operating process data. Zhao proposed enhanced process comprehension and statistical analysis method for slow-varying batch processes21 as well as phasebased KICA-PCA for nonlinear batch process monitoring.22 In those methods, the variability along batch direction is addressed and analyzed in the difference subspace using two-step modeling strategy, the ICA-PCA for the former and the KICA-PCA for the later. An adaptive monitoring method based on multiphase ICA

Rotem proposed the improved model-based principal component analysis (MBPCA) method and applied it to ethylene compressor process monitoring.15 The MBPCA method utilizes the structure, function, and behavior information of processes to eliminate periodic disturbances effectively. However, nonlinear processes have difficulty establishing process models, which makes the MBPCA inapplicable to nonlinear processes. PCA etc. methods are only appropriate for identical and independent Gaussian variables, which restricts the practical application of this method because variables usually follow nonGaussian distribution in actual industrial processes. Therefore, a method that can deal with non-Gaussian processes is necessary. The independent component analysis (ICA), which is an expansion of the PCA, can extract hidden independent components from observed data, thus enabling the accurate identification of process characteristics. Kano proposed a process monitoring 12083

dx.doi.org/10.1021/ie400712h | Ind. Eng. Chem. Res. 2013, 52, 12082−12107


Article

where β ∈ {β/1 ≤ β ≤ T, β ∈ Z}, and

is also proposed by Zhao.23 In this method, the local process correlation structures are explored and thus multiple phasespecific models are developed, then an adaptive updating strategy is adopted to accommodate more underlying process information and normal batch-to-batch slow-varying behaviors with the accumulation of new batch data. Zhang proposed enhanced statistical analysis of nonlinear processes using KPCA, KICA, and SVM,24 which combine the advantages of KPCA and KICA to develop a nonlinear dynamic approach to detect a fault online and use SVM for classifying faults. In this study, residual analysis ICA with average multivariate cumulative sum (AMRA-ICA) is proposed to handle nonGaussian problems with periodic disturbance. In the proposed AMRA-ICA, the disturbance cycle is initially synchronized through the average multivariate cumulative sum (average MCUSUM, AM). The periodic disturbance in the data is then eliminated, and the residual under normal conditions is obtained through residual analysis (RA) between two data sets under normal conditions. Second, the AMRA-ICA employs the ICA method to handle residuals and constructs T2 and SPE statistics to determine the appropriate statistical control limits. Third, AMRA-ICA executes online monitoring to obtain the residual between the monitoring data and the normal condition data to establish the statistics for process monitoring. Finally, the improved variable contribution histogram, which is proposed according to AMRA-ICA, is used to determine the cause of the fault. The proposed method is applied to the Tennessee Eastman (TE) process with and without periodic disturbance and to an ethylene compressor affected by ambient temperature. Results show that the proposed method is more effective in removing the influence of periodic disturbance compared with residual analysis PCA (RA-PCA) and the local tangent space alignment-ICA (LTSA-ICA). The proposed method is also suitable for conventional processes without periodic disturbance and has relatively good monitoring performance compared with ICA and LTSA-ICA.

⎡ b11 ⎢ ⎢ b21 ⎢ ⎢ ⋮ A2 = ⎢ b ⎢ (T − 1)1 ⎢ ⋮ ⎢ ⎣⎢ bm1

··· ⋱ ··· ⋱ ···

··· a1n ⎤ ⎥ a 2k + 2r ··· a 2n ⎥ ⋮ ⋱ ⋮ ⎥⎥ aTk + Tr ··· aTn ⎥ ⎥ ⋮ ⋱ ⋮ ⎥ ⎥ amk + βr ··· amn ⎦

···

···

b2k + 3r

···

⋱

⋮

⋱

··· b(T − 1)k + Tr ··· ⋱

⋮

⋱

···

bmk + αr

···

b1n ⎤ ⎥ b2n ⎥ ⎥ ⋮ ⎥ b(T − 1)n ⎥ ⎥ ⋮ ⎥ ⎥ bmn ⎥⎦

(2)

Usually we want to eliminate the periodic disturbance completely. However, periodic disturbances cannot be excluded if the residuals between A1 and A2 are obtained directly because the data disturbance cycles cannot be automatched, as shown in eq 3. ⎡ b11 ⎢ ⎢ b21 ⎢ ⎢ ⋱ ⎢ b(T − 1)1 M * = A 2 − A1 = ⎢ ⎢ bT1 ⎢ ⎢ b(T + 1)1 ⎢ ⎢ ⋱ ⎢ b ⎣ m1 ⎡ a11 ⎢ ⎢ a 21 ⎢ ⋱ ⎢ ⎢ a(T − 1)1 −⎢ ⎢ aT1 ⎢ ⎢ a(T + 1)1 ⎢ ⋱ ⎢ ⎢⎣ am1

RESIDUAL ANALYSIS WITH AVERAGE MULTIVARIATE CUMULATIVE SUM (AMRA) 2.1. Residual Analysis with Average Multivariate Cumulative Sum (AMRA). The data collected during a process generally undergo pretreatment to extract useful information from the data and to simplify the problem. This section discusses the relationship between data sets and finds a suitable pretreatment method to eliminate periodic disturbance. Suppose that there have two observed variable data sets A1 and A2 with the same sampling rate and same observed variables under normal working conditions. In each data set, there have n variables sampled m times, and the kth variable is affected by the periodic disturbance R = [r 2r ... Tr], where T is the disturbance cycle. Since the initial sampling time is stochastic, so the disturbance value in initial sampling time is uncertain. In this paper, we suppose that the initial disturbance value in A1 is r, and that in A2 is 2r. Thus, under normal conditions, the sample data with periodic disturbance is expressed as ···

b1k + 2r

where α ∈ {α/1 ≤ α ≤ T, α ∈ Z}. The kth variable is a nonlinear data with periodic disturbance.

■

⎡ a11 ⎢ ⎢ a 21 ⎢ ⋮ A1 = ⎢ ⎢ aT1 ⎢ ⎢ ⋮ ⎢ ⎣ am1

···

⎡ e11 ⎢ ⎢ e 21 ⎢ ⋱ ⎢ ⎢ e(T − 1)1 =⎢ ⎢ eT1 ⎢e ⎢ (T + 1)1 ⎢ ⋱ ⎢ ⎣ em1

···

b1k + 2r

···

···

b2k + 3r

···

···

⋱

···

··· b(T − 1)k + Tr ··· ···

bTk + 2r

···

··· b(T + 1)k + Tr ··· ···

⋱

···

···

bmk + αr

···

···

a1k + r

···

···

a 2k + 2r

···

···

⋱

···

··· a(T − 1)k + (T − 1)r ··· ···

aTk + Tr

···

···

a(T + 1)k + r

···

··· ···

⋱ amk + βr

··· ···

···

e1k + r

···

···

e 2k + r

···

··· ···

⋱ e(T − 1)k + r

··· ···

··· eTk + (T − 1)r ··· ···

e(T + 1)k + r

···

··· ···

⋱ emk + r

··· ···

b1n ⎤ ⎥ b2n ⎥ ⎥ ⋱ ⎥ b(T − 1)n ⎥ ⎥ bTn ⎥ ⎥ b(T + 1)n ⎥ ⎥ ⋱ ⎥ bmn ⎦⎥

a1n ⎤ ⎥ a 2n ⎥ ⋱ ⎥ ⎥ a(T − 1)n ⎥ ⎥ aTn ⎥ ⎥ a(T + 1)n ⎥ ⋱ ⎥ ⎥ amn ⎦⎥

e1n ⎤ ⎥ e 2n ⎥ ⋱ ⎥ ⎥ e(T − 1)n ⎥ ⎥ eTn ⎥ e(T + 1)n ⎥⎥ ⋱ ⎥ ⎥ emn ⎦

(3)

Moreover, if the periodic disturbance in the initial sample time becomes 3r in A2, then eq 2 can be expressed as eq 2* ⎡ b11 ⎢ ⎢ b21 ⎢ ⎢ ⋮ A2 = ⎢ b ⎢ (T − 2)1 ⎢ ⋮ ⎢ ⎣⎢ bm1

a1k + r

(1) 12084

···

b1k + 3r

···

···

b2k + 4r

···

⋱

⋮

⋱

··· b(T − 2)k + Tr ··· ⋱

⋮

⋱

···

bmk + αr

···

b1n ⎤ ⎥ b2n ⎥ ⎥ ⋮ ⎥ b(T − 2)n ⎥ ⎥ ⋮ ⎥ ⎥ bmn ⎥⎦

(2*)



Article

⎧ t ⎪∑ xi ; t ≤ d ⎪ i yt = ⎨ ⎪ t ∑ xi ; d ≤ t ≤ m ⎪ ⎪ ⎩i=t−d+1

then M* becomes a different value, as shown below. ⎡ b11 ⎢ ⎢ b21 ⎢ ⎢ ⋱ ⎢ b(T − 2)1 ⎢ M * = A 2 − A1 = ⎢ b(T − 1)1 ⎢ ⎢ bT1 ⎢ ⎢ b(T + 1)1 ⎢ ⎢ ⋱ ⎢ b ⎣ m1

···

b1k + 3r

···

···

b2k + 4r

···

···

⋱

···

··· b(T − 2)k + Tr ··· ··· b(T − 1)k + r ···

bTk + 2r

··· ···

··· b(T + 1)k + 3r ··· ···

⋱

···

···

bmk + αr

···

⎡ a11 ⎢ ⎢ a 21 ⎢ ⋱ ⎢ ⎢ a(T − 2)2 ⎢ − ⎢ a(T − 1)1 ⎢ ⎢ aT1 ⎢ ⎢ a(T + 1)1 ⎢ ⋱ ⎢ ⎢⎣ am1

···

a1k + r

···

···

a 2k + 2r

···

···

⋱

···

⎡ e11 ⎢ ⎢ e 21 ⎢ ⋱ ⎢ ⎢ e(T − 2)1 ⎢ = ⎢ e(T − 1)1 ⎢ ⎢ eT1 ⎢ ⎢ e(T + 1)1 ⎢ ⎢ ⋱ ⎢⎣ em1

··· a(T − 2)k + (T − 2)r ··· ··· a(T − 1)k + (T − 1)r ··· ···

aTk + Tr

···

···

a(T + 1)k + r

···

··· ···

⋱ amk + βr

··· ···

···

e1k + 2r

···

···

e 2k + 2r

···

··· ···

⋱

··· ···

e(T − 2)k + 2r

··· e(T − 1)k + (2 − T )r ··· ···

eTk + (2 − T )r

···

···

e(T + 1)k + 2r

···

··· ···

⋱ emk + 2r

··· ···

b1n ⎤ ⎥ b2n ⎥ ⎥ ⋱ ⎥ b(T − 2)n ⎥ ⎥ b(T − 1)n ⎥ ⎥ bTn ⎥ ⎥ b(T + 1)n ⎥ ⎥ ⋱ ⎥ bmn ⎥⎦

Where, d is the accumulated steps, i is the ith sample, and m is the number of samples. Thus, the variable after average MCUSUM is ⎧ t ⎪(∑ xi)/t ; t ≤ d ⎪ i yt′ = ⎨ t ⎪ ( ∑ xi)/d ; d ≤ t ≤ m ⎪ ⎪ ⎩ i=t−d+1

a1n ⎤ ⎥ a 2n ⎥ ⋱ ⎥ ⎥ a(T − 2)n ⎥ ⎥ a(T − 1)n ⎥ ⎥ aTn ⎥ ⎥ a(T + 1)n ⎥ ⋱ ⎥⎥ amn ⎥⎦ e1n ⎤ ⎥ e 2n ⎥ ⋱ ⎥⎥ e(T − 2)n ⎥ ⎥ e(T − 1)n ⎥ ⎥ eTn ⎥ ⎥ e(T + 1)n ⎥ ⎥ ⋱ ⎥ emn ⎥⎦

(7)

The influence of d to the periodic disturbance is analyzed and divided into two cases. Case 1. In Case 1, d = T, that is, the number of accumulated steps is equal to the disturbance cycle. Assume that the mean value of the ith variable without periodic disturbance is ai, and the disturbance of initial moment is r. When the sampling time t < d, t

atk′ =

∑1 aik

t a1k + a 2k + ··· + atk r + 2r + ··· tr = + t t (1 + t ) = ak + r 2

(4)

(8)

when the disturbance of the initial moment has a different value in [r 2r ... Tr], a′tk is similar to the situation that the disturbance of initial moment is r. When the sampling time t ≥ d,

When the initial periodic disturbance in A1 is r and that in A2 is Tr, the periodic disturbance after directed residual analysis is (T −1)r, which is a maximum. When the initial periodic disturbance in A1 is Tr and that in A2 is r, the periodic disturbance after directed residual analysis is (1 − T)r, which is a minimum. In summary, the range of periodic disturbance after directed residual analysisis is MR* ∈ [(1 − T )r , (T − 1)r ]

(6)

t

atk′ = =

(5)

∑i = t − d + 1 aik d a(t − d + 1)k + a(t − d + 2)k + ··· + atk

= ak +

Note that M*R can only be some discrete value in the interval and not be a continuous value. From the above equations, we can conclude that the periodic disturbance cannot be excluded completely. In other words, the residual is still nonlinear. Given that ICA is a linear method, analyzing the residual using ICA will lead to inaccurate monitoring results, such as fault underreporting and misreporting. So it is hard to monitor when the residuals between A1 and A2 are directly obtained. Thus, the average multivariate cumulative sum (average MCUSUM, AM) is adopted to solve this problem. With the assumption that the collected data Xm×n have been normalized, where m is the number of samples and n is the number of variable dimensions, the variable at time t after MCUSUM is then

d

+

(1 + T ) r 2

r + 2r + ··· Tr d (9)

Therefore, the disturbance is stationary after the sampling time T. Thus, the sample variables data set A1 after residual analysis with average MCUSUM are expressed as ⎡ a11 ′ ⎢ ⎢ a 21 ′ ⎢ ⎢ ⋮ ⎢ A1′ = ⎢ a ′ ⎢ T1 ⎢ ⎢ ⋮ ⎢ ⎢ am′ 1 ⎣ 12085

··· a1′n ⎤ ⎥ a 2′k + 1.5r ··· ··· a 2′n ⎥ ⎥ ⋱ ⋮ ⋱ ⋮ ⎥ ⎥ (1 + T ) ′ + ′ ⎥ r ··· aTn ··· aTk ⎥ 2 ⎥ ⋱ ⋮ ⋱ ⋮ ⎥ ⎥ (1 + T ) ′ + ′ ⎥ r ··· amn ··· amk ⎦ 2 ···

a1′k + r

(10)



Article

where δr is the disturbance of sample time t − d + 1. When T − d + 1 < δ ≤ T, a′tk follows eq 14; when 1 < δ ≤ T − d + 1, a′tk follows eq 15; when δ = 1, θ = (1 + d)/2 is minimum; when δ = T − d + 1, θ = (2T − d + 1)/2 is maximum. In summary, θ ∈ [(1 + d)/2, (2T − d + 1)/2]. θ is some discrete value in the interval. After performing residual analysis with average MCUSUM for two data sets under normal conditions, the residual range of periodic disturbance changes to

Similarly, the sample variables data set A2 through AMRA are ⎡ b11 ′ ⎢ ⎢ b21 ′ ⎢ ⎢⋮ ⎢ A 2′ = ⎢ b ′ ⎢ T1 ⎢ ⎢⋮ ⎢ ⎢ bm′ 1 ⎣

··· b1′n ⎤ ⎥ b2′k + 2.5r ··· ··· b2′n ⎥ ⎥ ⋱ ⋮ ⋱ ⋮⎥ ⎥ (1 + T ) ′ + ′ ⎥ r ··· bTn ··· bTk ⎥ 2 ⎥ ⋱ ⋮ ⋱ ⋮⎥ ⎥ (1 + T ) ′ + ′ ⎥ ··· bmk r ··· bmn ⎦ 2 ···

b1′k + 2r

MR ∈ [(d − T )r , (T − d)r ]

(11)

where MRmin = θmin − θmax and MRmax = θmax − θmin. MR is some discrete value in the interval. The above residual range reduces significantlycompared with that of the periodic disturbance without AMCUSUM (eq 5). When d → T, MR → 0. When d = nT + l, n ∈ N, the value of M is similar to d = l. Therefore, periodic disturbance can be reduced or completely removed through residual analysis with average MCUSUM. The key to elimination of the periodic disturbance in processes is the accurate selection of the accumulated steps d. 2.2. Selection for the Number of Accumulated Steps d. In an industrial process, we usually know which variable is affected by periodic disturbance according to operator’s experience and then we can obtain the disturbance cycle easily through our experience. For example, in the ethylene compressor case study, through empirical analysis, we know that the temperature related variable in the ethylene compressor is affected significantly by ambient temperature change. The ambient temperature changes between day and night, so it is a periodic disturbance. Thus we can conclude that the disturbance cycle is 24 h. Since the sampling frequency is once per 5 min, 288 samples are collected per 24 h for the ethylene compressor case study and the accumulated steps d is 288. For the TE case study, we add the periodic disturbance to the TE model, so we know the disturbance cycle, and the accumulated steps d is the same for the disturbance cycle we added. For a new system, if only one variable is affected or if all affected variables have the same disturbance cycle, and the disturbance cycle can be obtained from experience, then this cycle is chosen as accumulated steps d. If more than one variable is affected by periodic disturbances and the disturbance cycles are inconsistent, or the disturbance cycle cannot be obtained by experience, then the optimization algorithm can be used to calculate the optimal accumulated steps d to eliminate the periodic disturbance as much as possible. For example, taking particle swarm optimization (PSO) as the optimization algorithm to choose the optimal accumulated steps d, the fitness function is as follows.

The residual under normal condition after residual analysis with average MCUSUM is ⎡ e11 ⎢ ⎢ e 21 ⎢⋮ M = A 2′ − A1′ = ⎢ ⎢ eT1 ⎢ ⎢⋮ ⎢⎣ em1

··· e1k + r ··· e1n ⎤ ⎥ ··· e 2k + r ··· e 2n ⎥ ⋱ ⋮ ⋱ ⋮⎥ ⎥ ··· eTk ··· eTn ⎥ ⎥ ⋱ ⋮ ⋱ ⋮⎥ ··· emk ··· emn ⎥⎦

(12)

Periodic disturbance is successfully removed through residual analysis with average MCUSUM, except in several initial samples. In this process, average MCUSUM guarantees the cycle synchronization between two data sets. If d = nT, n ∈ N+, the value of M is similar to d = T. Case 2. In Case 2, d = 1 < T, that is, the number of accumulated steps is not equal to the disturbance cycle. Assume that the mean value of the ith variable without periodic disturbance is ai and the disturbance of initial moment is r. When the sampling time t ≥ d, t

atk′ =

∑1 aik

t a1k + a 2k + ··· + atk r + 2r + ··· tr = + t t (1 + t ) = ak + r 2

(13)

When the sampling time t ≥ d, t

atk′ =

∑i = t − d + 1 aik t

=(a(t − d + 1)k + a(t − d + 2)k + ··· + atk )/d + (δr + (δ + 1)r + ··· + Tr + r + ··· + [d − (T − δ + 1)]r )/d =ak + (δr + (δ + 1)r + ··· + Tr + r + ··· + [d − (T − δ + 1)]r )/d =ak + θr

(14)

or atk′ = =

Fitness(di) = || M(di)T M(di)||

t ∑i = t − d + 1 aik

d δr + (δ + 1)r + ··· + (δ + d − 1)]r + d δr + (δ + 1)r + ··· + (δ + d − 1)]r d

=ak + θr

(17)

where, di is the ith accumulated steps d, M(di) is the residual after the AMRA change with the accumulated steps di. The flowchart is as shown in Figure 1. 2.3. The Sensitivity to the Sampling Frequency f and the Number of Accumulated Steps d. The accuracy of fault detection depends on the elimination degree for the periodic disturbance. So the sensitivity of AMRA-ICA with respect to the sampling frequency depends on the sensitivity of AMRA with respect to the sampling frequency.

t a(t − d + 1)k + a(t − d + 2)k + ··· + atk

=ak +

(16)

(15) 12086



Article

Figure 2. Extended infomax ICA algorithm block diagram.

Figure 3. Algorithm flowchart of AMRA-ICA.

Figure 4. Flow diagram of the TE process.

Case 1. When the sampling frequency f decreases, while accumulated steps d is kept constant, it means that in one accumulated steps cycle, the periodic disturbance changes more than one cycle. Thus, the situation is similar to d = nT, n ∈ N+, or d = nT + l, n ∈ N+, l ∈ N+ which is discussed in section 2.1.

For the former situation, the value of M is similar to the situation d = T, and the periodic disturbance can be removed completely; for the latter situation, the value of M is similar to situation d = l, and the periodic disturbance can be removed partially. 12087



Article

Table 1. Process Measurements

Table 3. Manipulated Variables

variables

process measurements

unit

variable

description

XMEAS(1) XMEAS(2) XMEAS(3) XMEAS(4) XMEAS(5) XMEAS(6) XMEAS(7) XMEAS(8) XMEAS(9) XMEAS(10) XMEAS(11) XMEAS(12) XMEAS(13) XMEAS(14) XMEAS(15) XMEAS(16) XMEAS(17) XMEAS(18) XMEAS(19) XMEAS(20) XMEAS(21) XMEAS(22)

A feed (stream 1) D feed (stream 2) E feed (stream 3) A and C feed (stream 4) recycle flow (stream 8) reactor feed rate (stream 6) reactor pressure reactor level reactor temperature purge rate (stream 9) product separator temperature product separator level product separator pressure product separator underflow stripper level stripper pressure stripper underflow (stream 11) stripper temperature stripper steam flow compress work reactor cooling water outlet temp separator cooling water outlet temp

kscmh kg/h kg/h kscmh kscmh kscmh kPa gauge % °C kscmh °C % kPa gauge m3/h % kPa gauge m3/h °C kg/h kW °C °C

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

D feed flow (stream 2) E feed flow (stream 3) A feed flow (stream 1) A and C feed flow (stream 4) compressor recycle value purge valve (stream 9) separator pot liquid flow (stream 10) stripper liquid product flow (stream 11) stripper steam valve reactor cooling water valve condenser cooling water flow stirring rate

Table 4. Process Faults variable

state

XMEAS(23) XMEAS(24) XMEAS(25) XMEAS(26) XMEAS(27) XMEAS(28) XMEAS(29) XMEAS(30) XMEAS(31) XMEAS(32) XMEAS(33) XMEAS(34) XMEAS(35) XMEAS(36) XMEAS(37) XMEAS(38) XMEAS(39) XMEAS(40) XMEAS(41)

composition composition composition composition composition composition composition composition composition composition composition composition composition composition composition composition composition composition composition

stream

sample time/min

6 6 6 6 6 6 9 9 9 9 9 9 9 9 11 11 11 11 11

6 6 6 6 6 6 6 6 6 6 6 6 6 6 15 15 15 15 15

A B C D E F A B C D E F G H D E F G H

⎡ a11 ⎢ ⎢ a 21 ⎢ ⋮ ⎢ B1 = ⎢ ⎢ aT1 ⎢ ⎢ ⋮ ⎢ ⎣ am1

··· ··· ⋱ ··· ⋱ ···

··· a1n ⎤ ⎥ ··· a 2n ⎥ a 2k + 1.5r ⋮ ⋱ ⋮ ⎥⎥ ⎥ T+1 aTk + r ··· aTn ⎥ 2 ⎥ ⋮ ⋱ ⋮ ⎥ ⎥ ··· amn ⎦ amk + β′r

IDV (8) IDV (9)

D feed temperature (stream 2)

IDV (10)

C feed temperature (stream 4)

IDV (11)

reactor cooling water inlet temperature

IDV (12)

condenser cooling water inlet temperature

IDV IDV IDV IDV IDV IDV IDV IDV IDV

reaction kinetics reactor cooling water valve condenser cooling water valve unknown unknown unknown unknown unknown the valve for stream 4 was fixed at the steady state position

(2) (3) (4) (5) (6) (7)

(13) (14) (15) (16) (17) (18) (19) (20) (21)

⎡ b11 ⎢ ⎢ b21 ⎢ ⎢ ⋮ B2 = ⎢ ⎢ b(T − 1)1 ⎢ ⎢ ⋮ ⎢ ⎢⎣ bm1

Case 2. When the frequency f increases, while accumulated steps d is kept constant, it means that in one accumulated steps cycle, the periodic disturbance changes less than one cycle. For example, f new = 2fold,then the eq 1 and eq 2 in the paper can be re-expressed as

type

A/C feed ratio, B composition constant (stream 4) B composition, A/C ratio constant (stream 4) D feed temperature (stream 2) reactor cooling water inlet temperature condenser cooling water inlet temperature A feed loss (stream 1) C header pressure loss-reduced availability (stream 4) A, B, C feed composition (stream 4)

IDV IDV IDV IDV IDV IDV

Table 2. Composition Measurements variables

description

IDV (1)

···

b1k + 2r

···

···

b2k + 2.5r

···

⋱

⋮

⋱

··· b(T − 1)k ⋱ ···

T+1 + r ··· 2 ⋮ ⋱

bmk + α′r

···

b1n ⎤ ⎥ b2n ⎥ ⎥ ⋮ ⎥ ⎥ b(T − 1)n ⎥ ⎥ ⋮ ⎥ ⎥ bmn ⎥⎦

step step step step step step step random variation random variation random variation random variation random variation slow drift sticking sticking unknown unknown unknown unknown unknown constant position

(19)

When the sampling time t < d,

a1k + r

t

atk′ =

∑1 aik

t t+1 r + 1.5r + ··· 2 r a1k + a 2k + ··· + atk = + t t t+3 r = ak + 4

(18) 12088

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Figure 5. TE process data distribution and Gaussian distribution.

Figure 6. Periodic disturbance.

When the sampling time t ≥ d,

In summary, no matter what change happens to the sampling frequency f and accumulated steps d, as long as the relationship between f and d still obey d = nG, n ∈ N+, G is the sampling number in one disturbance cycle, then the periodic disturbance can be removed completely. In the other situation, the periodic disturbance only can be eliminated partially. In this paper, for the ethylene compressor case study, the sample frequency is one per 5 min. No matter what change happens to the sampling frequency, if the relationship between f and d still obeys d = nG, n ∈ N+ (G is the sampling number in a disturbance cycle), then the accumulated steps is a multiple of the times of the sampling number in a disturbance cycle, and the periodic disturbance in the ethylene compressor can be removed completely. In the other situation, the periodic disturbance only can be eliminated partially.

t

atk′ = =

∑i = t − d + 1 aik t a(t − d + 1)k + a(t − d + 2)k + ··· + atk d

= ak + θ′r

+ θ′r (21)

where ⎡ T + 3 3T + 3 ⎤ , θ′ ∈ ⎢ ⎥ ⎣ 4 2 ⎦

(21)

Therefore, the disturbance is not stationary, and the disturbance can only be removed partially. Case 3. When d changes with the change of the sampling frequency f, for example, when the sampling frequency decreases, the sample number in one disturbance cycle decreases, so the accumulated steps d decreases; when the sampling frequency increases, the change is verse vice. In this situation, the value of M is the same as has been discussed in section 2.1.

■

INDEPENDENT COMPONENT ANALYSIS

The periodic disturbance in certain variable has been removed through AMRA. However, the residual generally follows nonGaussian distribution; thus, this paper uses the ICA method to analyze the residual. 12089



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Figure 7. (a) Data distribution of XMEAS (9) without periodic disturbance. (b) Data distribution of XMEAS (9) under the influence of periodic disturbance.

The ICA method originates from blind source separation, which is proposed by Herault et al.25−27 and described systematically by Comon.28 ICA can separate non-Gaussian source signals from mixed signals. ICA handles non-Gaussian problems under the principle that the mixed signal of several independent components is more likely to obey Gaussian distribution than any source signal. Therefore, the non-Gaussianity of the separated signal is taken as standard to test the independence of separated signals. That means, the stronger is the non-Gaussianity of a separated signal, the more likely that this signal is a source signal. When nonGaussianity of all the separated signals reaches the maximum, the separated signals are most likely to be non-Gaussian source signals. Then the separation process finishes. Several different algorithms for ICA to separate the nonGaussian source signals have been proposed. The following algorithms are widely used for ICA: information maximization (infomax) and extended infomax,29 natural gradient algorithm, which combines minimization of mutual information with artificial neural network,30,31 and fixed point algorithm.32−34 A review of the infomax and extended infomax for ICA shows that35 n residual variables e1, e2, ..., en are assumed to be expressed as linear combinations of m unknown independent variables s1, s2, ..., sm, where m ≤ n (Figure 2). If random column vectors are represented as M = [e1, e2, ..., en]T and

Table 5. Detection Rates of the Three Methods for the TE Process with Periodic Disturbance (%) RA-PCA 2

LTSA-ICA 2

AMRA-ICA 2

fault

T

SPE

I

T

SPE

1 2 4 5 6 7 8 10 11 12 13 14 16 17 18 19 20 21

95.43 39.14 8.86 5.43 87.43 97.14 64.86 27.14 6 5.71 5.14 5.71 11.14 24.57 5.14 87.43 26.57 5.14

95.71 79.43 13.14 9.14 97.43 97.43 67.71 66.29 16 48.86 14 8.29 72 41.14 6.29 92.86 56.57 6.26

97.65 78.24 98.53 3.24 99.71 99.71 70.88 66.76 42.06 14.41 2.94 9.41 80.29 47.35 4.41 92.65 70.88 2.06

95 82.65 95 1.47 99.41 99.12 67.05 70.88 64.12 27.94 6.47 0 83.82 77.94 0 90.29 70.88 0

87.14 66.47 93.24 0 97.54 99.12 60.59 66.47 67.15 0 0 68.53 79.41 74.41 0 88.24 65.59 0

s = [s1 s2 ..., sm]T, then the relationship between these two vectors is given by 12090


Industrial & Engineering Chemistry Research M = As

Article

(22)

the reconstructed matrix y are independent from each other. Thus, y is the best estimation of the original variable s, which is given by

where A = [a1, ..., am] ∈ Rn×m is the unknown mixing matrix. The basic problem of ICA is estimating the mixing matrix A and the independent components s from M alone. This solution finds a demixing matrix w ∈ Rn×m, wherein the elements of

y = wM

(23)

Figure 8. continued 12091



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Figure 8. (a) T2 monitoring of RA-PCA for fault 11 in the TE process with periodic disturbance. (b) SPE monitoring of RA-PCA for fault 11 in the TE process with periodic disturbance. (c) I2 monitoring of LTSA-ICA for fault 11 in the TE process with periodic disturbance. (d) T2 monitoring of AMRA-ICA for fault 11 in the TE process with periodic disturbance. (e) SPE monitoring of AMRA-ICA for fault 11 in the TE process with periodic disturbance.

⎧[I ⎪ ⎪ ∇w(k) ∝ ⎨ ⎪[I ⎪ ⎩

The calculation method of w in the infomax algorithm is first proposed according to the conventional stochastic gradient algorithm. When g(u) = 1/(1 + e−u), ∇w(k) ∝ [w T(k)]−1 + [1 − 2y(k)]x T(k)

(24)

(25)

w(k + 1) = w(k) + μ[I + [1 − 2y(k)u T(k)]]w(k)

(27)

(28)

∇w(k) ∝ [I − K tanh(u(k))u(k)T − u(k)u(k)T ]w(k) ⎧ kii = −1 super‐Gaussian ⎨ ⎩ kii = 1 sub‐Gaussian (29)

where μ is the learning rate and k is the iteration algebra. The natural gradient is introduced to replace the conventional stochastic gradient algorithm and is given by (26)

+ tanh(u(k))u(k)T − u(k)u(k)T ]w(k) sub‐Gaussian

Equation 26 can be simplified as

w(k + 1) = w(k) + μ[[w T(k)]−1 + (1 − 2y(k))x T(k)]

∇w(k) ∝ [I + [1 − 2y(k)u T(k)]]w(k)

− tanh(u(k))u(k)T − u(k)u(k)T ]w(k) super‐Gaussian

⎪

⎪

where, kii is the main diagonal element of the diagonal matrix K. The adaptation of the source density parameters kii is defined as kii = sin{E[sec h2(ui)]E[ui2] − E{[tanh(ui)]ui}}

However, the above formulas predefine the distribution of the source signal and can only separate mixed signals with positive peaks. To use the algorithm for super-Gaussian and sub-Gaussian signals, g(u) is defined as g(u) = tanh (u) and the following extended infomax algorithm is proposed:

(30)

The source distribution of the extended infomax algorithm is super-Gaussian when kii = 1 and sub-Gaussian when kii = −1. w(k + 1) = w(k) + μ∇w(k) 12092

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The steps of the extended infomax ICA algorithm are as follows: (1) Data normalization. (2) Initialization of the initial value of separating matrix w(0) = 0.1·I. (3) Iteration of the separation matrix

w(k + 1) = w(k) + μ∇w(k) = w(k) + μ[BI − K tanh(u(k))u(k)T − u(k)u(k)T ]w(k)

(32)




Article

Figure 9. (a) T2 monitoring of RA-PCA for fault 17 in the TE process with periodic disturbance. (b) SPE monitoring of RA-PCA for fault 17 in the TE process with periodic disturbance. (c) I2 monitoring of LTSA-ICA for fault 17 in the TE process with periodic disturbance. (d) T2 monitoring of AMRA-ICA for fault 17 in the TE process with periodic disturbance. (e) SPE monitoring of AMRA-ICA for fault 17 in the TE process with periodic disturbance.

where I is a unit matrix, μ = 0.005 is the learning rate, μ(0) is the product of w(0) and the block measurement variable matrix, L = 0.0001 is the learning step length, and B = 100 is the block. The algorithm will iterate for 50 times. (4) Obtain the estimated value of the source signal, y = wM. Independent components are sorted according to the negative entropy value of each individual component. Finally, the appropriate independent components are selected as the main part sd. The corresponding main part separation matrix wd is also selected. The remaining independent components are denoted as remnants se, and the corresponding remnants separation matrix as we.

The Hotelling T2 statistic is given by T 2(i) = sd̂ (i)T sd̂ (i)

(33)

where, ŝd(i) can be calculated as

sd̂ (i) = wdM(i)

(34)

Here, M(i) = [ei1 ... eik ... ein]T and n is the number of observed variables. The SPE statistic for the non systematic part of the common cause variations can be visualized in a chart with control limits. SPE statistic is expressed as

■

PROCESS MONITORING BASED ON AMRA-ICA After calculating the separation matrix w and extracting the information of the independent components, the statistics and the corresponding control limits need to be established in the monitoring process. The Hotelling T2 and the squared prediction error (SPE) are then used.36

SPE(i) = e(i)T e(i)

(35)

where, e(i) = M(i) − M̂ (i) and M̂ (i) can be calculated as M̂ (i) = wdTwdM(i) 12094

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Article

Figure 10. (a) T2 monitoring of AMRA-ICA for multiple (simultaneous) faults 2 and 4 in the TE process with periodic disturbance. (b) SPE monitoring of AMRA-ICA for multiple (simultaneous) faults 2 and 4 in the TE process with periodic disturbance.

Considering that s is non-Gaussian, T2 and SPE control limits cannot be estimated by approximate distribution. Zhang adopted the kernel density estimation method to calculate T2 and SPE control limits.37 However, this method has complex computation. Therefore, this paper uses a simpler method to determine the control limits, that is, estimating the control limits according to T2 and SPE statistics under normal conditions. Fault occurs in the process when T2 or SPE statistics exceeds its control limit. Fault diagnosis can be achieved through a contribution histogram once the fault is detected. Through interrogation of the underlying process where the abnormality is detected, we can reveal the group of process variables that are mostly influencing the processor residuals. The contributionbased approach easily identifies faults and does not require prior fault knowledge. In the AMRA-ICA, the contribution of T2and SPE statistics for the ith sampling time can be represented by38 MT 2 =

w TwM(i) || wM(i)|| || w TwM(i)||

MSPE(i) = abs(M(i) − w TwM(i))

where, M(i) = [ei1 ... eik ... ein]T and n is the number of observed variables. When fault occurs, the variable with the largest contribution is considered as the most probable cause of the fault. Considering that the result of this method is not efficient, this paper proposes a simpler contribution histogram method according to AMRA-ICA to produce a better effect compared with the conventional contribution histogram. Given that M is already residual, the contribution of the jth variable in the ith sampling time can be expressed as Cbt(i , j) = M(i , j)*M(i , j)

(39)

Accordingly, the contribution histogram of all the variables in the jth sample time can be obtained from eq 34. The fault variable has a large contribution when the fault occurs. However, other relevant variables are also affected and become abnormal as time lapses. Therefore, only collecting the contributions of variables at a certain moment after the fault occurs will result in the inaccuracy of the contribution histogram. This study proposes an improved contribution histogram method that sums the contribution of every variable from the

(37) (38) 12095



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Figure 11. (a) The conventional variable contribution histogram for fault 6 in the TE process with periodic disturbance. (b) Improved variable contribution histogram for fault 6 in the TE process with periodic disturbance.

moment the system detects the fault until the operator notices the fault, which is expressed as

(1) Acquire two operating data sets during normal operation. The normal data which include N samples should be obtained and then partitioned to two data sets, the first N/2 samples consist of the first data set and the later N/2 samples make up the second data set. The two data sets need to contain an equal number of samples. (2) Calculate the average MCUSUM of the two data sets by using eq 7, and then calculate the residuals by using eq 12. As for the selection of accumulated steps d, which is the most important parameter for the ethylene compressor case study, through empirical analysis, we can conclude that the effected variable is the suction temperature of Stage I; the disturbance cycle is 24 h and we monitor this variable. The results verify our judgment. So the sampling number of 288 in 24 h is considered as accumulated steps d. For the TE case study, we add the periodic disturbance, so we know the disturbance cycle, and the accumulated steps d is the same as the disturbance cycle. For a new system, if the disturbance cycle cannot be obtained by experience, then the PSO can be chosen to calculate the optimal accumulated steps d, which is discussed in section 2.2.

n

CBT (j) =

∑ Cbt(i , j) i=m

(40)

where m is the moment when the system detects the fault and n is the moment when the operator notices the fault. Given that the fault variable has been faulted since the fault occurred, the summed contribution value should be greater than the summed contribution of other variables which are later affected by the fault. Thus, eq 40 is better than eq 39. Figure 3 shows that the entire monitoring process is divided into offline modeling and online monitoring. In the offline modeling stage, the improved ICA model is established by two normal historical data sets. Consequently, in the online monitoring stage, the process variables are collected to monitor the process. 4.1. Developing Normal Operating Conditions (NOCs). The NOCs are developed via the following steps: 12096



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Figure 12. Improved variable contribution histogram for multiple (simultaneous) faults 2 and 4 in the TE process with periodic disturbance.

Table 6. Detection Rates of the Three Methods for the TE Process without Periodic Disturbance (%) ICA 2

fault

T

1 2 4 5 6 7 8 10 11 12 13 14 16 17 18 19 20 21

100 98 64 100 100 100 91 67.4 97.96 97.74 94 100 76 84.5 89 24.2 69 51.5

LTSA-ICA

and the standard deviation are the ones under normal operating condition. (4) Handle the new residuals by using extended infomax ICA. (5) Calculate T2 and SPE statistics. (6) Determine whether T2 and SPE statistics are beyond the control limits. If their values exceed the control limit, identify the cause of the fault by using eq 40.

AMRA-ICA

SPE

2

2

I

T

SPE

100 98 95 100 100 100 97 63.9 63.9 98.75 92 100 73 93.4 93.5 19.1 66 18

100 97.74 83.53 100 100 100 95.29 88.25 71.76 100 97.94 100 91.76 92.65 77.94 77.06 79.41 7.06

97.94 95 98.24 97.94 100 98.24 98.24 70 92.65 97.35 90.29 0 82.35 97.94 84.12 10.29 80.29 0

95.59 94.12 100 98.82 100 96.18 98.18 65.59 75.29 89.41 89.41 3.24 57.65 96.47 82.94 9.12 79.71 20.29

■

SIMULATION OF TE PROCESS WITH PERIODIC DISTURBANCE The AMRA-ICA method is applied to the TE process with periodic disturbances to illustrate the monitoring performance of the proposed method. In this section, three methods are employed for monitoring: RA-PCA, the local tangent space alignment-ICA (LTSA-ICA)39 and AMRA-ICA. In addition, two methods are employed for fault diagnosis: conventional contribution histogram and improved contribution histogram. The purpose is to show the performance of different methods and to demonstrate the advantages of AMRA-ICA and the superiority of the improved contribution histogram over the simple contribution histogram. 5.1. The TE Process. The TE process, which is created by the Eastman Chemical Company, is a realistic industrial process that has been widely used in fault diagnosis and monitoring.40 This process has five main units: a reactor, a condenser, a compressor, a separator, and a stripper. The corresponding differential equations are established for each unit on the basis of material balance, energy balance, and vapor−liquid equilibrium. The entire process contains eight types of components: A, B, C, D, E, F, G, and H. Here, A, C, D, and E are the reactants, B is the catalyst, F is the byproduct, and G and H are the final products. The flow diagram of the process is shown in Figure 4. In the TE process, the discrete proportional−integral−derivative control algorithm41 is used. The TE process has 22 continuous process measurements, 19 composition measurements, and 12 manipulated variables, as shown in Tables 1−3, respectively. In this paper, all 52 variables except for the agitation speed of the reactor stirrer (XMV (12)) are used to show the monitoring performance of

(3) Normalize the residuals using the mean and standard deviation of each variable. (4) Compute the separation matrix w using eq 32. (5) Select the appropriate number of independent components. (6) Calculate T2 and SPE monitoring statistics according to eq 33 and 35, respectively. (7) Determine T2 and SPE statistics control limits. 4.2. Online monitoring and diagnosis. Online monitoring involves the following steps: 1) Collect new process data and calculate the average MCUSUM value of the collected data using eq 7. (2) Compute the residual between the treated new process data and the normal condition data. In this stage, one normal data set is used as reference data set and the residuals between new measurement data and the reference data is obtained through AMRA. (3) Normalize the new residuals. Note that when normalization is applied to the new process data, the mean value 12097



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5.2. The Non-Gaussian Distribution and Periodicity of the TE Process. Most industrial processes exhibit nonGaussian behavior. Gaussian processes can be described by skewness and kurtosis. Skewness measures the deviation between the data distribution and the Gaussian distribution. The variable whose skewness is not zero must obey the non-Gaussian

the residual ICA. The TE process contains 21 preprogrammed faults (Table 4), in which 16 are known and 5 are unknown. Some faults lead to large and easy-to-detect changes in most variables, whereas others cause little deviation of variables from normal behavior. In this section, we consider all faults except IDV(3), IDV(9), and IDV(15).




Article

Figure 13. (a) T2 monitoring of ICA for fault 11in the TE process without periodic disturbance. (b) SPE monitoring of ICA for fault 11 in the TE process without periodic disturbance. (c) I2 monitoring of LTSA-ICA for fault 11in the TE process without periodic disturbance. (d) T2 monitoring of AMRA-ICA for fault 11in the TE process without periodic disturbance. (e) SPE monitoring of AMRA-ICA for fault 11 in the TE process without periodic disturbance.

distribution. Kurtosis is used to distinguish non-Gaussian signal from Gaussian data, as well as to identify sub-Gaussian (whose kurtosis value is less than zero) and super-Gaussian signal (whose kurtosis value is greater than zero). For example, the skewness and kurtosis of the XMEAS (1) in the TE process is about 0.7714 and 6.1141, respectively. Figure 5 illustrates the Gaussian distribution and XMEAS (1) distribution. It shows that XMEAS (1) follows the nonGaussian distribution. Many real-world data sets have superGaussian distributions, which have probability densities with heavy tails that peak at the middle. Therefore, the statistics that is based on the assumption that data follow the Gaussian distribution may result in inaccurate and incomplete reporting. In addition, given that the periodicity of the TE process is not obvious, a periodic disturbance (Figure 6) is added to the reactor temperature (XMEAS(9)) to simulate the influence of ambient temperature to the measure reactor temperature. Since the TE process is a strong coupling system, every variable are correlated with the others. So when periodic disturbance is added to XMEAS (9), the change in XMEAS (9) will affect other variables, then the other variables will continue affect XMEAS (9) in return.

In this paper, periodic disturbance (Figure 6) is added to XMEAS (9) (Figure 7a), then the change in XMEAS (9) affects other variables and the other variables continue to affect XMEAS (9). All these make the measurement value of XMEAS (9) as shown in Figure 7b. Moreover, the whole system is under a time delay, so the change in XMEAS (9) is nearly a half cycle later than the change of periodic disturbance. 5.3. Fault Monitoring Performance of RA-PCA, LTSAICA, and AMRA-ICA for TE Processes with Periodic Disturbance. In the monitoring process, periodic disturbance (Figure 6) is added to the XMEAS (9) to imitate the influence of ambient temperature (Figure 7b). A total of 1500 normal samples are gathered, which includes 500 samples for the first normal condition data set, 500 for the second normal condition data set, and 500 for the test. Fault data are gathered to verify the fault monitoring performance of the proposed method. For each fault pattern, a data set comprising 500 samples is generated. Faults are introduced from the 161th sample and are continuously introduced until the end of the process. This paper compares the AMRA-ICA with the PCA and RA-PCA. This paper uses 100 as accumulated steps d, which is the same as the number of disturbance cycle. 12099



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Figure 14. (a) Conventional variable contribution histogram for fault 1 in TE process without periodic disturbance. (b) Improved variablecontribution histogram for fault 1 in the TE process without periodic disturbance.

The average fault-detection rates of the three methods are summarized in Table 5. Given that fault 3, 9, and 15 are quite small and have almost no effect on the overall process, the detection rates of all methods for these faults are lower than 5%. Therefore, these faults will not be considered in this research. Considering that a periodic disturbance is added to XMEAS(9), some faults that can be detected easily by RA-PCA and LTSAICA when no periodic disturbance exists can no longer be detected by those methods. Table 5 denotes that AMRA-PCA has significant better monitoring performance compared with RA-PCA and LTSA-ICA in faults 10, 11 and 17, because the former introduces average MCUSUM and eliminates periodic disturbance thoroughly. For faults 2, 4, 16, and 20, AMRA-ICA and LTSA-ICA have the similar monitoring performance that is better than that of RA-PCA. All in all, the simulation results demonstrate that the AMRA-ICA has the best monitoring performance among these methods. The monitoring chart of faults 11 and 17 are respectively shown in Figures 8 and 9 to illustrate the advantages of AMRAICA over the other methods.

Figure 8 shows the monitoring results for fault 11. Fault 11 is caused by the random variation of the inlet temperature of the reactor cooling water. The RA-PCA cannot detect fault 11, thus providing the process operator an incorrect message of the process status. The LTSA-ICA can detect the fault in relatively low detection rate. In contrast to RA-PCA and LTSA-ICA, the AMRA-ICA monitoring charts show that both T2 and SPE statistics have successfully detected the fault until the end of process time, with the exemption of some samples. These results indicate that the proposed method can effectively detect faults that are usually hard to detect by the conventional methods. Figure 9 shows the monitoring performance of the three methods for fault 17, which is an unknown fault. For the RAPCA and LTSA-ICA, a lot of points fall under the control limits after the fault occurred, whereas only a few points fall under the control limit for the AMRA-ICA. Therefore, this fault has been detected more efficiently via AMRA-ICA monitoring compared with other methods of monitoring. The T2 and SPE statistics of the AMRA-ICA have 77.94% and 74.41% detection rates, respectively, whereas that of RA-PCA have 24.57% and 41.14%, respectively, and that of LTSA-ICA have 47.35%. 12100



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Figure 15. Flowchart of ethylene compressor refrigeration system.

Table 7. Technological Parameters of Ethylene Compressor stage I

stage II

stage III

parameters

inlet

outlet

inlet

outlet

inlet

outlet

flow rate, kg/b pressure, MPa temp,°C molecular weight Cp/Cv (avg) rated power, kw

8459 0.104 −102 28.052 1.286 278

8459 0.407 −12.2 28.052 1.286 278

13532 0.407 −36 28.052 1.269 205

13532 0.682 5 28.052 1.269 205

30786 0.682 −33 28.052 1.255 905

30786 1.82 45 28.052 1.255 905

AMRA-ICA is an ICA-based fault detection and diagnosis method. In an industrial process, changes in data happen no matter if a single fault or multiple faults occur, and as long as the data changes data-driven approaches, such as ICA and ICAbased methods, can detect the change and warn the operator that faults are happening in the system. So the proposed method in this paper can effectively detect multiple faults. This is shown in Figure 10. Figure 10 shows the monitoring performance of AMRA-ICA for multiple (simultaneous) faults 2 and 4 in the TE process with periodic disturbance. Faults 2 and 4 are introduced to the disturbance affected TE process at the same time from the 161th sample. Fault 2 is caused by the step change of B composition, while the A/C ratio is retained. Fault 4 is caused by the step change of reactor cooling water inlet temperature. It is obvious that the proposed method can detect the multiple faults in a timely manner with a considerable high detection rate. 5.4. Fault Diagnosis of the AMRA-ICA for TE Processes with Periodic Disturbance. The improved variable contribution histogram is used to analyze the cause of the fault for fault 6, as shown in Figure 11b. Fault 6 is caused by the loss of

Table 8. Ethylene Compressor Variables (Faults) no.

process variable (fault)

1 2 3 4 5 6 7 8 9 10

suction temperature of stage I added temperature of stage II added temperature of stage III suction pressure of stage I added pressure of stage II added pressure of stage III discharge pressure of stage III suction flow of stage I added flow of stage II added flow of stage III

A feed. When the fault occurs, the A feed (variable 44, XMV (3)) has the highest contribution; thus, this variable is the most likely cause of the fault. However, the traditional contribution histogram (Figure 11a) shows that the compressor recycle value (variable 46, XMV (5)) and the stripper steam value (variable 50, XMV (9)) are the most likely causes of the fault, which may confuse the operator. 12101



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Figure 16. Ethylene compressor data distribution and standard Gaussian distribution.

Figure 17. Data distribution of the suction temperature of stage I.

Most of the existing ICA-based algorithms for performing fault diagnosis assume the presence of a single root cause for the given abnormal situation. The assumption of a single fault has its own reason. From the view of engineering, this is reasonable and multiple faults seldom appear at the same time. From the view of mathematics, this can be strictly proved through probability calculation. However, in the industry process, the situation of multiple faults indeed exists. The reason is that many faults are not critical for continued operation. The consequence of this is that the number of faulty components increases over time, and when the system is finally repaired, the number of faulty components may be substantially larger than one. The disadvantage of the ICA-based approach for performing multiple faults diagnosis is that this information is not enough to aid the operator in identifying the root cause. This is because, for a large process with many variables, the interpretation of measured variable contributions is difficult. Figure 12 shows the diagnosis performance of an improved variable contribution histogram for multiple (simultaneous) faults 2 (idv (2)) and 4 (idv (4)) in the TE process with periodic

disturbance. The six biggest contribution variables are variable 7 reactor pressure, variable 13 product separator pressure, variable 16 stripper pressure, variable 30 composition B, variable 45 A and C feed flow (stream 4), variable 51 reactor cooling water valve. Two of the six are related to the reactor, so we can conclude that the fault happens in reactor. But even the contribution chart shows that variable 30 composition B is wrong; we cannot really confirm that another fault reason is the reactor cooling water inlet temperature (idv(2)), since other three biggest contribution variables orient to other fault reasons. Therefore, the proposed method can well detect the multiple faults, but cannot find the reason for faults effectively.

■

SIMULATION OF TE PROCESS WITHOUT PERIODIC DISTURBANCE In a real industrial process, the periodic disturbance does not always exist. For example, in an ethylene compressor, the apparent periodic disturbance only exists in winter. It means that, during the whole monitoring period, the process is affected from periodic disturbance at sometime and is not affected from 12102



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In this paper, the simulation that focuses on the TE process without periodic disturbance illustrates that the proposed method can well handle processes without periodic disturbance. To illustrate the applicability of the proposed method in processes without periodic disturbance, AMRA-ICA is applied

periodic disturbance at other times. So a method which can monitor the conventional process as well as the periodicdisturbance-affected process is needed. And a new method which can only well manage the process with periodic disturbance but cannot deal with the conventional process is less useful in reality.




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Figure 18. (a) T2 monitoring of RA-PCA for fault 2 in the ethylene compressor. (b) SPE monitoring of RA-PCA for fault 2 in the ethylene compressor. (c) I2 monitoring of LTSA-ICA for fault 2 in the ethylene compressor. (d) T2 monitoring of AMRA-ICA for fault 2 in the ethylene compressor. (e) SPE monitoring of AMRA-ICA for fault 2 in the ethylene compressor.

The detection rates under several monitoring methods are computed and tabulated in Table 6. All methods have produced high detection rates for faults 1, 2, 5, 7, 6, 8, 12, 13, and 17. The AMRA-ICA has the best monitoring performance among the three methods for faults 4, 11, and 20. For faults 10, 16, and 19, the LTSA-ICA method performs better than AMRA-ICA. But for fault 14, both ICA and LTSA-ICA can detect the fault effectively, while the AMRA-ICA cannot detect it. It may be because the fault has a similar cycle as the accumulated step d used in AMRA-ICA, and the fault has been eliminated from the observed data through AMRA. Overall, the monitoring performance of the AMRA-ICA for TE processes without periodic disturbance is similar to that of ICA and LTSA-ICA in terms of detection rate. Thus AMRA-ICA is suitable to monitor the processes with and without periodic disturbance. Figure 13 shows the monitoring charts of fault 11 to illustrate the monitoring performance of the AMRA-ICA. Fault 11 is caused by the random variation of the inlet temperature of the reactor cooling water. All of the three methods can detect the fault with a similar detect rate, so it proves again that AMRA-ICA is suitable to monitor the processes without periodic disturbance.

Table 9. Detection Rates of the Three Methods for the Ethylene Compressor (%) RA-PCA (T2/SPE) LTSA (I2) AMRA-ICA (T2/SPE)

false alarm rate

detection rate

2.14/1.40 6.07 1.07/0.13

55.8/82.8 78.2 97/95.2

to the TE process without periodic disturbance. In this section, three methods are employed for monitoring: ICA, LTSA-ICA, and AMRA-ICA. Two methods, namely, the conventional contribution histogram and the improved contribution histogram, are employed for fault diagnosis. 6.1. The Monitoring Performance of ICA, LTSA-ICA, and AMRA-ICA for TE Processes without Periodic Disturbance. In this section, the conventional TE model with the same monitoring variables and data collection method that are same to that in the previous section is used. We consider all faults except IDV(3), IDV(9), and IDV(15). The monitoring result of ICA used in this section are partially obtained from refs 42−45. 12104



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Figure 19. (a) Conventional variable contribution histogram for fault 2 in the ethylene compressor. (b) Improved variable contribution histogram for fault 2 in the ethylene compressor.

6.2. Fault Diagnosis of the AMRA-ICA for TE Processes without Periodic Disturbance. The modified variable contribution histogram analyzes the cause of fault 1, as shown in Figure 14b. Fault 1 is caused by the step change of the A/C ratio while the B composition remains constant. When the fault occurs, the A feed of variable 1(XMEWAS (1)) and the A feed flow of variable 44(XMV (3)) have the highest contribution. The A and C feed (variable 4, XMEWAS (4)) and the A and C feed flow (variable 45,XMV (4)) have the second highest contribution. Therefore, the changed A/C feed ratio is the most likely cause of fault 1. However, the traditional contribution diagram (Figure 14a) denotes that the product separator, stripper level, separator pot liquid flow, and stripper steam values are incorrect.

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This paper takes ethylene compressor as an example. In the process of ethylene production, the ethylene compressor is used to obtain the high level cold quantity which is used for cryogenic separation process. The ethylene refrigeration compressor is a three centrifugal compressor, and the refrigeration medium is ethylene. The ethylene compressor refrigeration system and propylene refrigeration constitute a cascade refrigeration system. The compressed ethylene is cooling and condensing by propylene. Then, the low temperatures of −102 °C, −75 °C, and −55 °C are obtained, respectively. The process and parameters are shown in Figures 15 and Table 7. The compressor operates under a significant periodic disturbance caused by ambient temperature change in winter. Therefore, this compressor cannot be monitored effectively by using conventional methods. The monitoring variables used in this paper include the temperature, pressure, and flow of the three reciprocating compression processes (Table 8). The sampling frequency is once per 5 min, and 288 samples are collected per day. The AMRA-ICA is used to monitor the process, and the improved contribution histogram is adopted to identify the cause of the fault. Figure 16 shows that the data of ethylene compressor still do not obey the Gaussian distribution after normalization.

ETHYLENE COMPRESSOR SIMULATION

Gas compression is an important part in the ethylene separation processes, such as cracked gas compression, ethylene compression, propylene compression, methane compression, and air compression. These processes determine the energy consumption level of a separation unit, as well as the stability of the ethylene plant. The stable operation of a separation unit is guaranteed once the stable operation of the compressor is ensured. Therefore, the detection and diagnosis of the compressor fault is significant. 12105



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Selecting the appropriate accumulated steps d is important because this parameter determines the removal extent of the periodic disturbance. In this paper, we choose this parameter through our experience as well as the monitoring the variables that are influenced by periodic disturbance. In our future research, we will aim to select the appropriate accumulated steps d through an optimization algorithm such as PSO when the affected variable is not known or when the disturbance cycle is hard to obtain.

Figure 17 shows that the data of the stage I suction temperature changes periodically because this stage is influenced by ambient temperature, which is different during day and night. The complete cycle lasts about one day, which is equivalent to some 288 samples. 7.1. Monitoring Performance of RA-PCA, LTSA-ICA, and AMRA-ICA for the Ethylene Compressor. Figure 18 and Table 9 illustrate the monitoring performance for fault 2 in the ethylene compressor based on RA-PCA, LTSA-ICA, and AMRA-ICA. Fault 2 is caused by the change of added temperature on the second compression process. Fault 2 starts to abort during the 1500th sample, which continues for about 40 h. An attempt to monitor directly, the compressor via RAPCA or LTSA-ICA will lead to unavoidable difficulties because of the oscillatory measured data, as illustrated in Table 9. However, given that the AMRA-ICA can eliminate the periodic disturbance thoroughly, this method is significantly better than other methods both in SPE and T2 monitoring. 7.2. The Fault Diagnosis of the AMRA-ICA for the Ethylene Compressor. The ultimate goal of fault monitoring is to determine the cause of a fault and to eliminate this cause to ensure production safety. This paper uses the variable contribution histogram to identify the cause of the fault in the ethylene compressor. Figure 19 shows the variable contribution histogram for fault 2 in the ethylene compressor. The improved variable contribution histogram obtains the direct cause of fault 2 (Figure 19b). By contrast, the conventional variable contribution histogram fails to point the real cause of the fault (Figure 19a).

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AUTHOR INFORMATION

Corresponding Author

*Tel: +86-021 64252060. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Basic Research Program of China (No. 2012CB720500), the National Natural Science Foundation of China (Key Program: 61134007), the National Science Fund for Outstanding Young Scholars (No.61222303), and the National Natural Science Foundation of China (No.21276078, 61174118).

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REFERENCES

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CONCLUSION This paper proposes the AMRA-ICA, as well as its corresponding improved contribution histogram method, to monitor chemical processes with periodic disturbance and non-Gaussian distribution. The AMRA-ICA introduces the average MCUSUM (AM) to synchronize the disturbance cycle and employs residual analysis (RA) to remove the periodic disturbance in data and to obtain normal residual. Given that the residual still follows non-Gaussian distribution, ICA is adopted to monitor the residual and to obtain the independent components. ICA obtains statistically independent components to extract useful information. Consequently, the improved contribution histogram is applied to analyze the cause of faults. The AMRA-ICA method is applied to the Tennessee Eastman process with and without periodic disturbance and to an ethylene compressor which is affected by ambient temperature. The simulation results show that the proposed method can more effectively detect and diagnose faults than RA-PCA and LTSA-ICA for non-Gaussian processes with periodic disturbance both in the simulation platform and in the actual industry; it is also suitable for conventional processes without periodic disturbance and has relatively good monitoring performance when compared with ICA and LTSA-ICA. The proposed method can be extended to other ICA algorithms, such as kernel ICA, multiway-ICA, and so forth. This work shows the superiority of AMRA-ICA over conventional methods. However, there are still some disadvantage in the proposed algorithm. The biggest disadvantage is that if the fault is also a periodic one and a the similar cycle as the periodic disturbance, it is likely to be eliminated through AMRA. And then the system cannot detect the fault. Further improvements are needed to the proposed method. The major challenge is the selection for accumulated steps d. 12106



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Fault Detection and Diagnosis for Non-Gaussian Processes with

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