Robust Statistical Process Monitoring for Biological Nutrient Removal Plants

Transcription

1 Robust Statistical Process Monitoring for Biological Nutrient Removal Plants Nabila Heloulou and Messaoud Ramdani Laboratoire d Automatique et Signaux de Annaba (LASA), Universite Badji-Mokhtar de Annaba. BP. 12, Annaba 23000, Algeria hel.nabila@yahoo.com, messaoud.ramdani@univ-annaba.dz Abstract. This paper presents an approach by combining robust fuzzy principal component analysis (RFPCA) technique with the multiscale principal component analysis (MSPCA) methodology. Thus the two typical issues of industrial data, outliers and changing process conditions are solved by resulting MS-RFPCA methodology. The RFPCA is proved to be effective in mitigating the impact of noise, and MSPCA has become necessary due to the nature of complex systems in which operations occur at different scales. The efficiency of the proposed technique is illustrated on a simulated benchmark of biological nitrogen removal process. Keywords: Process monitoring, robust fuzzy PCA, multiscale PCA, fault diagnosis, wavelet analysis, water treatment plant. 1 Introduction The monitoring techniques allows to monitor continuously any changes in processes. For this purpose a statistical techniques have been implemented, it aims to achieve and maintain process under control. The first ideas of SPM for quality improvement go back as far as the beginning of the century. Where the principal components analysis is the most widely accepted technique to this day, the PCA technique can be seen as a projection method which allows to project the observations since space with p variable dimensions towards a space with k dimensions (k p) such as a maximum of information is preserved. This fully take the nature of modern WWTPs characterized by a multitude of correlated variables [1]. Tomita et al [2] have shown the possibility of reducing the analysis from 12 variables of an activated sludge wastewater treatment down to 3 principal components which are more relevant to the process, deviation of measurements is then detected. This work has shown that the PCA is an adequate tool for representation and extracting of information. Several others recent applications of this approach and adaptation of it to wastewater treatment operations have found their way, [3], [4], [5]. Despite its success in this field, one of the most important obstacle faced is the sensitivity to outliers, also the fact that the majority of collected data from industrial processes are normally contaminated by noise makes N. Heloulou and M. Ramdani are with the Department of Electronics, University Badji-Mokhtar of Annaba, Algeria. A. Laurent et al. (Eds.): IPMU 2014, Part I, CCIS 442, pp , c Springer International Publishing Switzerland 2014

2 428 N. Heloulou and M. Ramdani it unreliable in some cases. Also, in order to circumvent this difficulty, several approaches of PCA have been proposed, among the variants, robust fuzzy PCA (RF-PCA) showed promising results. In this approach, fuzzy variant of PCA uses fuzzy membership and diminish the effect of outliers by assigning small membership values to outliers in order to make it robust. Another shortcoming of conventional PCA is that modeling by PCA is done at a single scale where the actual industrial process may include events and disturbances that occur at different time-frequency range, the waste water treatment plant (WWTP) is exemplary in this respect. To solve this problem the PCA is extended for single scale to multiscale (MSPCA) modeling approach. MSPCA uses wavelet decomposition to approximately decorrelate variables autocorrelation, and also capture the linear variable correlation by PCA to extract features. In this work, we propose the combination of MSPCA approach with a RFPCA algorithm, we thus aim to solve the problem of noise and provide a solution to the problem of monitoring during changing process conditions. The result is an effective monitoring methodology. 2 PCA Statistical Monitoring Principles Principal component analysis is a multivariate statistical projection method, it is presented as a search of the subspace that maximizes the variance of the projected points, ie. the subspace that best represents the diversity of individuals through the variance-covariance structure. The first step of this method is the construction of data matrix X, containing all the available data obtained by collected measurements when the process is in control. Denote the correlation matrix of X as Σ = X T X / (N 1), and performing singular value decomposition (SVD) to the matrix Σ yields Σ = UΛU T where U n n is a unitary matrix and Λ = diag(λ 1,...,λ n ) is a diagonal matrix formed by the eigenvalues of the covariance matrix in decreasing magnitude (λ 1 λ 1...,λ n ). The column vectors in the matrix U =[u 1,u 2,...,u n ] forms a new orthogonal base of space R n, For dimension reduction, only the eigenvalues λ q n are used for projection of output data in the new space. It is quite useful to consider all the directions which covers a significant portion of a total variance, these directions also be called as new basis vectors for the subspace, so the desired transformation matrix consisting of the first q columns of the matrix U, ie. the eigenvalues belonging to the largest eigenvalues of Λ. So the first q (< n) linear independence vectors ˆP U q =[u 1,u 2,...,u q ]ofu spans the principal component subspace Ŝ. The other n q vectors P U n q = [u q+1,u q+2,...,u n ]ofu spans the residual space S. The data vector x R n can be decomposed as x =ˆx + x = Ĉx + Cx where ˆx Ŝ and x S are projection of x on the subspaces Ŝ and S, respectively. The matrix Ĉ = UÛT and C = UÛT. The score vector in the space model

3 Robust Statistical Process Monitoring 429 t = ˆP T x R q is a reduced, q dimensional representation of the observed vector x. On the other hand, the residual e =(I ˆP ˆP T )x R n,representsthe portion not explained by the PCA model. In genaral, a PCA based statistical process monitoring scheme utilizes two monitoring statistics, Hotelling s T 2 and Q statistics (SPE), which are most frequently used in the industrial processes. Typically, these indices are used to detect faults respectively in the principal component subspace Ŝ and the residual subspace S. In this work only the Q statistics is used. The SPE index is defined as a measure of the squared norm of the residual vector x. Box[6]shownthat the confidence limit for SPE from a PCA model can be calculated as : where θ i = m λ i j j=q+1 index shall be satisfied : SPE δ 2 α. 3 Robust Fuzzy PCA δ 2 α = gχ 2 h,α (1) g = θ 2 /θ 1 (2) h = θ1 2 /θ 2 (3) for i =1, 2. Under normal operating conditions, the SPE In actual industrial process modeling, data were often contain outliers problem. RFPCA addresses this limitation. It uses robust rules in order to replace traditional PCA and create robust fuzzy PCA. Then the influence of outliers will be reduced and consequently defects will accurately detected. The RFPCA algorithms used here were introduced in [7]. These algorithms are based on Xu and Yuille algorithms [8]. Xu and Yuille proposed an optimization function with an energy measure e (x i ) subject to the membership set u i {0, 1} given as : E (U, w) = n u i e (x i )+η i=1 n (1 u i ) (4) The goal is to minimize E(U, w) with respect to u i and w. Where X = {x 1,x 2,.., x n } is the data set, U = {u i i =1,..., n} is the membership set and η is the threshold. The variable u i serves to decide whether x i is an outlier or a sample. When u i = 1 the portion of energy contributed by the sample x i is taken into consideration; otherwise x i is considered as an outlier [8]. Since u i is the binary variable and w is the continuous variable, the optimization with gradient descent approach is hard to solve using gradient descent. To overcome the problem the fuzzy variant of the objective function is proposed in [7]. E = n u m i e (x i)+η i=1 i=1 n (1 u i ) m (5) subject to u i [0, 1] and m [0, 1). Now u i being the membership of x i belonging to data cluster and (1 u i )isthemembershipofx i belonging to noise i=1

4 430 N. Heloulou and M. Ramdani cluster. m is the so-called fuzziness variable. In this case, e (x i )measuresthe error between x i and the class center. This idea is similar to the C-means algorithm [9]. Since u i is now a continuous variable the difficulty of a mixture of discrete and continuous optimization can be avoided and the gradient descent approach can be used. Firstly, the gradient of equation (2) is computed respect to u i and equaled to zero, therefore : u i = 1+ ( e(xi) η 1 ) 1/((m 1)) (6) Using this result in the objective function and simplifying, we obtain E = n 1+ i=1 ( e(xi) η 1 ) 1/((m 1)) m 1 e (x i ) (7) The gradient with respect to w is ( ) δe δe δw = β (x (xi ) i) δw (8) where, β (x i )= 1+ 1 ( e(xi) η ) 1/(m 1) and m is the fuzziness variable. if m = 1, the fuzzy membership reduced to the hard membership and can be determined by following rule: m (9) u i = { 1if (e (xi )) η 0otherwise (10) Now η is a hard threshold in this situation. There is no general rule for the setting of m, but most papers set m = 2. In [7], authors derived three RFPCA algorithms, these ones are slightly different, for each algorithm the same procedure is followed except step 6 and 7. We have applied the first one in this work. FRPCA1 algorithm : Step 1: Initially set the iteration count t = 1, iteration bound T, learning coefficient α 0 (0, 1] soft threshold η to a small positive value and randomly initialize the weight w. Step 2: While t is less than T, perform the next steps 3 to 9. Step 3: Compute α t = α 0 (1 t T ), set i =1andσ =0. Step 4: While i is less than i, do steps 5-8. step 5: Compute y = w T x i, u = yw and v = w T u.

5 Robust Statistical Process Monitoring 431 step 6: Update the weight: w new = w old + α T β (x i )[y (x i u)+(y v) x i ] step 7: Update the temporary count δ = δ + e 1 (x i ). step 8: Add 1 to i. step 9: Compute η = ( δ n) and add 1 to t. The weight w in the updating rules converges to the principal component vector almost surely [10], [11] 4 Multiscale PCA Methodology Multiscale Principal Components Analysis (MSPCA) was introduced by Bakshi (1998) that combines the merits of wavelet analysis and PCA. In particular, PCA is used to extract linear relations among variables, whereas wavelets have the ability to extract deterministic features in the measurements and decorrelate the autocorrelation among the measurements. Using wavelet, methodology involves choosing the mother wavelet from a large library of admissible functions, a selected family of wavelets with the decomposition level L is applied to the signal s, yielding detail coefficients D is(i=1...s) and approximation coefficient A Ls. Next, principal component analysis is performed on each matrix of the detail scales and the matrix of approximation AL. The goal is to extract the correlation across the sensors. PCA control charts such as Q statistics can be monitored the resulting coefficient at each scale. The wavelet coefficients of data representing normal operation are beforehand calculated, also the detection limit for Q statistics are determined from them. Applying the Q statistics at each scale permits identification and selection of the scales that contain the significant features representing the abnormal operation. Fig. 1. MSPCA Methodology

6 432 N. Heloulou and M. Ramdani 5 MS-RFPCA Methodology Multiscale RFPCA algorithm here intgrates RFPCA with multiscale analysis of wavelet. A decision that is crucial for the performance of the MS-RFPCA model is when the choice of the depth or number of scales of the wavelet decomposition does not take an important part of consideration. Usually, the choice of this number is an important factor in the MSPCA methodology. It should be selected to provide maximum separation between the deterministic and stochastic components of signals [12]. Therefore if we select a very small number of depth, then the last scaled signal will have a significant amount of noise that will be retained in the result of MSPCA. In the case of MSRFPCA and sight that is based on FRPCA method, thus this problem will not be asked, otherwise and when the depth is too large, the matrix of coefficients at coarser scales will have very few rows due to the dyadic down sampling, and this will affect the accuracy of the PCA at that scale. In our version (MSRFPCA), we choose the number of depth so that will not be very large to avoid the latter case. We give here the detailed algorithm including fault identification. Setup MSPCA Reference Model: 1. Get the reference data when a process is under normal condition. For each variable in the reference data matrix, compute the wavelet decomposition and get the reference wavelet coefficients. 2. For each scale, put the reference wavelet coefficients from all variables together and apply FRPCA to get the reference RFPCA model (including mean, standard deviation and PC loadings) and control limits for T 2 or Q statistics. Repeat this procedure for all scales. 3. Define reconstruction scenarios based on the number of decomposition level. For each reconstruction scenario, assign the selected significant scales with the corresponding reference wavelet coefficients and the insignificant scales with zeros (hard thresholding). Reconstruct the signal from the selected and thresholded coefficients for each variable. Put the reconstructed signal of all variables together and apply RFPCA to get the reference RFPCA model (including mean, standard deviation, PC loadings, and control limits) for this reconstruction scenario. Repeat the same procedure to all reconstruction scenarios. Online Process Monitoring: 1. Determine the size of the moving window of dyadic length, w. Generate a data window with w samples from the real-time data by moving the time window. For each variable in the data window, compute the wavelet coefficients. 2. For each scale, calculate T 2 and Q scores based on the reference RFPCA model in I.(2). 3. Compare the T 2 and Q scores with the control limits in I.(2), retain wavelet coefficients that violate the control limits and assign those within the control limits to zero. 4. Reconstruct the signal in the moving window variable-by-variable.

7 Robust Statistical Process Monitoring Since only the most recent sample (the last data in the moving window) is of interest, determine the reconstruction scenario based on the last T 2 and Q scores of each scale and get the detection limits for the last reconstructed signal. Calculate T 2 and Q scores of the last reconstructed signal based on the reference RFPCA model in I.(3). 6. If T 2 and/or Q scores of the last reconstructed signal exceed the detection limits, fire an alarm. 6 Application on Water Treatment Plant In this section an activated sludge model No.1 (ASM1) for nitrogen removal is presented. The basic design of this plant is shown in Fig. 2. In general, nitrogen removal proceeds in two steps. The first step is nitrification, i.e. the biological oxidation of ammonium to nitrate, this process requires an aerobic environment. During this phase ammonia nitrogen is converted into nitrite by Nitrosommas and subsequently into nitrate by Nitrobacters, for the process this phase is the crucial step. In the next step, the produced nitrate is subjected to anoxic conditions in denitrification reactor, where it is converted into harmless nitrogen gas. Anoxic/anaerobic processes operate alternately to enhance the nitrogen removal. As illustrated in Fig 2, before it enters the aeration reactor, raw wastewater Q in is passed by the anoxic zone, afterward the influent flow Q out isfedintoasettler to separate the stream into the clean water and sludge, the major part of it is recycled to reactor Q r, and a small part is wasted Q w. The actual process model is based on the activated model sludge No.1 (ASM1) by [13]. It was adopted with two modifications: (i) the nitrification is modeled by a two step processes (the conversion of nitrite to nitrate by the nitrosoma bacteria and the conversion of nitrite to nitrate by the nitrobacters) and (ii) the hydrolysis of rapidly biodegradable substrate is included. Fig. 2. Schematic of a typical wastewater tretment plant Then the resulting biodegradation model consists of 18 state variables (particles and soluble concentrations) and 30 model parameters. However it is possible to reduce the model, such model is proposed by [14]. This model consists of 8 states variables : dissolved oxygen ( S p, ) ( O2 Sn O2,nitrate p ) S NO 3, SNO n 3, ammonia ( p ) S NH 4, SNH n 4, and biodegradable substrate concentrations (S p S, Sn S ), for each reactor zone (p and n denote pre-denitrification and nitrification respectively).

8 434 N. Heloulou and M. Ramdani So, this model consists of eight state variables : x =[x 1,..., x 8 ] T = [ S p NO 3, S p O2, Sp NH 4, S p S, Sn NO 3, S n O2, Sn NH 4, S n S] T. More information about parameters and mathematical model can be consulted in [13]. Validation results for the developed model are shown and discussed in the next for simulated data. Fig. 3. The SPE plot of linear PCA model in faulty state Multiscale RFPCA is firstly applied to the normal operation data. Seven measured variables are constituted the measurement vector z (k) available to be monitored, it is given as : z (k) = [ S p NO 3 (k),s p O 2 (k),s n NO 3 (k),s n O 2 (k),u 1 (k),u 2 (k),u 3 (k) ] T where u 1 = Q in (influent flow rate), u 2 = Q r (internal recirculation rate), u 3 = q air (aeration rate). Fig. 4. The SPE of MS-RFPCA model in faulty state

9 Robust Statistical Process Monitoring 435 Matrix Z then consists of N observations of the vector z (k). Also, this step includes determination of MS-FRPCA reference model : reference wavelt coefficient, reference FRPCA model, statistical limitation (Q). Now, we test the fault of offset type (brusque fault) created on the level of the third sensor at at 46 sample time. Fig. 3 and Fig. 4 are the SPE plot of the classical PCA and the MS-FRPCA approach respectively. According to Fig. 3, the SPE plot shows the distinct change only after a delay of 5 day, whereas the multi-scale SPE results (Fig. 4) shows that the SPE violates 95% SPE confidence limit before the 5 th day. So, the MS-FRPCA can detect the ramp fault earlier than traditional PCA which ensure plant safety. Scale Q charts can help to determine the nature of a disturbance. The SPE plot in Fig. 4 shows a fault detected in the wavelet approximation model of MS-RFPCA, that results in change which occur in low frequencies. Fig. 5. Fault isolation using contribution to SPE (fault in the third sensor) Our study was not only dedicated to the detection of fault to a certain level but also to the detection of faulty sensor. To identify the faulty sensor, it was exploited the contributions approach to detection index SPE, Fig. 5 indicates clearly that the fault sensor is s x =3. In this work, only the sensor faults have been considered. Althought, it is important to model fault process due to different conditions, such as the toxicity shock fault caused by a reduction in the normal growth of heterotrophic organisms, there is also an inhabitation fault produced by hospital waste that can contain bacteria, another fault process that can be considered is bulking fault produced by the growth of filamentous microorganisms in the active sludge. In the other side, the method has been tested with drift fault, other different kind of sensor faults (drift, bias, precision degradation,...) can be handled by the proposed monitoring scheme. We will take these points into considerations in the next work.

10 436 N. Heloulou and M. Ramdani 7 Conclusions WWTP posed an interesting challenge from the point of changing process conditions. MS-RFPCA methodology is used to monitor WWTP data during changing operational conditions. It is based firstly on time-scale decomposition in terms of increasing the sensivity of the monitoring, and secondly on RFPCA in terms of reducing noise sensivity. The results showed the advantages of the proposed monitoring method for continuous wastewater treatment plant. References 1. Lee, J.-M., Yoo, C., Lee, I.-B.: Statistical monitoring of dynamic processes based on dynamic independent component analysis. Chem. Engng. Sci. 59, (2004) 2. Tomita, R.K., Park, S.W., Sotomayor, O.A.Z.: Analysis of activated sludge process using Multivariate statistical tools-a PCA approach. Chem. Engng. J. 90, (2002) 3. Lee, J.-M., Yoo, C., Choi, S.W., Vanrolleghem, P.A., Lee, I.-B.: Nonlinear process monitoring using kernel principal component analysis. Chemical Eng. Sci. 59, (2004) 4. Aguado, D., Rosen, C.: Multivariate statistical monitoring of continuous wastewater treatment plants. Engng. Applications of Artificial Intelligence 21, (2008) 5. Corona, F., Mulas, M., Haimi, H., Sundell, L., Heinonen, M., Vahala, R.: Monitoring nitrate concentrations in the denitrifying post-filtration unit of a municipal wastewater treatment plant. J. Proc. Cont. 23, (2013) 6. Box, G.E.P.: Some theorems on quadratic forms applied in the study of analysis of variance problems: Effect of inequality of variance in one-way classification. The Annals of Mathematical Statistics 25, (1954) 7. Yang, T.N., Wang, S.D.: Robust algorithms for principal component analysis. Pattern Recognition Letters 20, (1999) 8. Xu, L., Yuille, L.: Robust principal compoenent analysis by self-organizing rules based on statistical physics approach. IEEE Trans. Neural Networks 6(1), (1995) 9. Oja, E.: The Nonlinear PCA Learning Rule and Signal Separation - Mathematical Analysis. Technical Report, Helsinki University of Technology (1995) 10. Oja, E.: A simplified neuron model as a principal component analyzer. J. Math. Biol. 15, (1982) 11. Oja, E., Karhunen, J.: On stochastic approximation of the eigenvectors and eigenvalues of the expectation of a random matrix. J. Math. Analysis and Appl. 106, (1985) 12. Yang, Q.: Model-based and data driven fault diagnosis methods with applications to process monitoring. Ph.D. thesis. Case Western Reserve University, Electrical Engineering and Computer Sciences (2004) 13. Lopez-Arenas, T., Pulis, A., Baratti, R.: On-line monitoring of a biological process for wastewater treatment. AMIDIQ 3, (2004) 14. Gomez-Quintero, C., Queinnec, I., Babary, J.P.: A reduced nonlinear model for an activated sludge process. In: Proceeding of ADCHEM, vol. 2, pp (2000)