Monitoring and Control of Biological Wastewater Treatment Process ChangKyoo Yoo

Transcription

1 Monitoring and Control of Biological Wastewater Treatment Process ChangKyoo Yoo Department of Chemical Engineering (Process Control and Environmental Engineering Program) Pohang University of Science and Technology

2 Monitoring and Control of Biological Wastewater Treatment Process by Chang Kyoo Yoo Department of Chemical Engineering (Process Control and Environmental Engineering Program) Pohang University of Science and Technology A thesis submitted to the faculty of Pohang University of Science and Technology in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the department of Chemical Engineering (Process Control and Environmental Engineering program) Pohang, Korea Approved by Major Advisor

3 DCE 유창규 Chang Kyoo Yoo Monitoring and Control of Biological Wastewater Treatment Process, 생물학적폐수처리공정의모니터링및제어 Department of Chemical Engineering (Process Control and Environmental Engineering Program) 2002, 195 pages Advisor: In-Beum Lee, Text in English. ABSTRACT Increasingly stringent demands are being placed on nutrient removal from wastewater. These stricter demands increase the complexity of the treatment process and necessitate the upgrading of treatment plants. As a consequence, there is a need to optimize plant operation as well as to maximize plant efficiency and reliability. Process monitoring and advanced control systems are generally considered to be an important means of achieving stable operation under large load variations. Recent advances in modeling and sensor technology have motivated significant research aimed at constructing better process monitoring and advanced control systems. In the present study we aim to design a process control and monitoring algorithm appropriate for the wastewater treatment process (WWTP). In Section III, entitled Autotuning and Supervisory DO Control in Fullscale WWTP, we introduce the autotuning of the PID controller to the dissolved oxygen (DO) control system in the WWTP and propose a simple supervisory control law to suggest its set point. Process identification for autotuning approximates the DO dynamics to a high-order model using the integral transform method and reduces it to the first-order plus time delay (FOPTD) or second-order plus time delay (SOPTD) model for the PID controller tuning. Simultaneously, a simple algorithm for the supervisory control of set point decision is proposed to decide a proper DO set point for the current operation condition of the aeration basin. The key idea in this method i

4 is that the DO set point is proportional to the respiration rate, which is the indicator of the biologically degradable load. The full-scale experimental results showed good identification performance and good tracking ability. As a result of the improved control performance, the fluctuation of the variation of the dissolved oxygen process decreased and 15% of the electrical power was saved. In Section IV, entitled Generalized Damped Least Squares method, we propose a generalized damped least squares (GDLS) method to systematically remove an estimation windup problem in the adaptive control and self-tuning control system. The key element of the proposed method is the addition of a penalty of parameter variations to the objective function of the normal least squares algorithm to prevent the singularity problem. Mathematical analysis shows that the proposed method has almost equivalent properties to the normal least squares method and guarantees that no estimation problem will be encountered for poorly excited situations. The proposed method was applied to estimate the parameters of a first-order system under closed-loop control and to estimate the respiration rate (R) and oxygen transfer rate (K L a) of the DO control system in the WWTP, which was used to derive an adaptive model-based DO control law. Simulation results show that a GDLS algorithm gives excellent estimation performance under closed-loop control and can be used in adaptive model-based DO control in WWTP. In Section V, entitled Disturbance Detection and Isolation in WWTP, we propose a new fault detection and isolation (FDI) method. The proposed method monitors the distribution of process data and detects changes in this distribution, which reflect changes in the corresponding operating condition. A modified dissimilarity index and an FDI technique are defined to quantitatively evaluate the difference between successive data sets. This technique considers the importance of each transformed variable in the multivariate system. The proposed FDI technique is applied to a benchmark simulation and to data from a real WWTP. In addition, we ii

5 investigate the kind of disturbance and various scenarios that frequently occur in the WWTP. Simulation results show that the proposed method could immediately detect disturbances and automatically distinguish between serious and minor anomalies for various scales of fault by facilitating the interpretation of the disturbance scales. In particular, the simulations confirmed that the proposed method is efficient in adaptive and nonstationary processes, such as the WWTP. In Section VI, entitled Modeling and Multiresolution Analysis in WWTP, modeling and multiresolution analysis (MRA) are described for the full-scale WWTP. The proposed method is based on the modeling by partial least squares (PLS) regression method and multiscale monitoring by application of a generic dissimilarity measure (GDM) to PLS score values. PLS score values are normally distributed as a consequence of the central limit theorem, regardless of the distribution of the original variables; hence, the proposed monitoring method is suitable for non-stationary and non-normal data sets. Experimental results show that the PLS method gives good modeling performance and is a powerful tool for analyzing the full-scale WWTP. MRA also certified the detection and isolation capability of the proposed method. In particular, the MRA indicated that the proposed strategy was appropriate for the detection and isolation of various faults and events in biological treatment, that is, the proposed method could cope with multiscale process changes in non-stationary signals with non-normal characteristics. In Section VII, entitled Process Monitoring for a Continuous Process with Cyclic Operation, we propose a method for monitoring a continuous process with diurnal cyclic characteristics in the domestic WWTP. A subspace identification method is used to extract within cycle and between-cycle correlation information from historical data in the form of a state-space model. This method is designed to describe the variations from the mean behavior of a periodically timevarying state-space model. The method can also incorporate the concept of iii

6 inferential sensing to predict the quality variables and to enhance process monitoring. In the simulation results, the proposed method could detect small mean shifts and abnormalities in the slowly decreasing nitrification rate that were difficult to detect using the conventional PCA method. In Section VIII, entitled Simultaneous Prediction and Classification in the Secondary Settling Tank, we propose a method of prediction of solid volume index (SVI) and a simultaneous classification of the current state of a secondary settler. Adaptive modeling scheme is implemented as recursive least squares (RLS) method to update the model parameters adaptively and neural network (NN) classifier is used as a process classifier. The basic idea is that RLS model parameters have good features to classify the current state of a secondary settler, thus secondary clarifier can be detected by monitoring the variations of RLS parameters during the SVI prediction. Experiments and theoretical analysis shows that the RLS method can predict SVI of a secondary settler well and parameters of RLS method can be a good feature for monitoring the state of secondary settler, which could be verified through the power spectrum analysis. In Section IX, entitled Nonlinear Fuzzy PLS Modeling, we propose a new nonlinear partial least squares (NLPLS) algorithm that embeds the Takagi-Sugeno- Kang (TSK) fuzzy model into the regression framework of the partial least squares (PLS) method. The proposed method applies the TSK fuzzy model to the PLS inner regression. Using this approach, the interpretability of the TSK fuzzy model overcomes some of the handicaps of previous NLPLS algorithms. The proposed method uses the PLS method to solve the problems of high dimensionality and collinearity and the TSK fuzzy model is used to capture the nonlinearity and to increase the use of experts knowledge. As a result, the FPLS model gives a more favorable modeling environment in which the knowledge of experts can be easily applied. Simulation results showed good modeling performance of the FPLS model iv

7 in a simulation benchmark and a full-scale WWTP. v

8 Contents I. Introduction Research Motivation Research Objective... 4 II. Biological Wastewater Treatment Process Activated Sludge Process Simulation Benchmark Fullscale WWTP III. Autotuning and Supervisory DO Control in Fullscale WWTP Introduction Method Autotuning Method Supervisory Control Experimental Results Conclusions IV. Generalized Damped Least Squares Method Introduction Theory Generalized Damped Least Squares Algorithm Theoretical Analysis Soft Sensor of Oxygen Transfer Rate and Respiration Rate Adaptive Model-based DO Control Simulation Study Conclusions V. Disturbance Detection and Isolation in WWTP Introduction Theory Modified Dissimilarity Measur Fault Detection and Isolation (FDI) vi

9 5.3 Simulation Studies Simulation of Benchmark Plant Fullscale WWTP Conclusions VI. Modeling and Multiresolution Analysis in WWTP Introduction Theory Partial Least Squares (PLS) Generic Dissimilarity Measure (GDM) Multiresolution Analysis Result and Discussion Conclusions VII. Process Monitoring for Continuous Process with Cyclic Operation Introduction Theory Simulation Study Conclusions VIII. Simultaneous Prediction and Classification in the Secondary Settling Tank Introduction Theory Simulation Study Conclusions IX. Nonlinear Fuzzy PLS modeling Introduction Theory PLS Modeling Method TSK Fuzzy Modeling Nonlinear FPLS Modeling Result and Discussion Conclusions Appendix vii

10 Summary in Korean X. References viii

11 List of Figures Figure 2.1 A basic activated sludge process with an aerated tank and a settler Figure 2.2 A simplified process scheme of an activated sludge process using predenitrification (layout of simulation benchmark) Figure 2.3 Plant layout of coke WWTP, Korea Figure 3.1 Identification and autotuning procedure for PID controller Figure 3.2 Supervisory DO control scheme Figure 3.3 Schematic diagram of full-scale WWTP Figure 3.4 Experimental result during identification phase Figure 3.5 Bode plots of the identified FOPTD and SOPTD models Figure 3.6 The validation test: real data and model prediction value Figure 3.7 Estimated respiration rate during identification phase Figure 3.8 DO control result using autotuning and supervisory control algorithm.. 35 Figure 4.1 Robust estimator of K L a(u(t)) and R(t) with a GDLS method Figure 4.2 Adaptive model-based DO control strategy with GDLS algorithm Figure 4.3 Flow chart of soft sensor and the model-based DO control algorithm Figure 4.4 Process output and input during the simulation (a) process output (b) control input Figure 4.5 Parameter estimates a ˆ( t) and b ˆ( t ) of RLS with exponential data weighting Figure 4.6 Trace of P with RLS estimation method Figure 4.7 Parameter estimates a ˆ( t) and b ˆ( t ) of RLS with constant trace ix

12 Figure 4.8 Parameter estimates aˆ ( t) and b ˆ( t ) with GDLS Figure 4.9 Estimation comparisons of RLS and GDLS method under PID control 62 Figure 4.10 Model-based DO control result with a GDLS method (a) process output, (b) control input Figure 5.1 Moving windows between successive two datasets Figure 5.2 Measured variables during the storm weeks Figure 5.3 Monitoring performances under an external disturbance (a) dissimilarity index, (b-d) individual eigenvalue plots Figure 5.4 Monitoring performances under internal disturbances caused by decreasing nitrification (left plot) and settler bulking (right plot) (a) dissimilarity index, (b-d) individual eigenvalue plots Figure 5.5 Monitoring performances of sensor faults (left plot) and setpoint change (right plot) (a) dissimilarity index, (b-d) individual eigenvalue plots Figure 5.6 PCA monitoring performances (a) Hotteling s T 2 chart, (b) SPE plot Figure 5.7 FDI monitoring performances (a) dissimilarity index, (b-f) the 1, 2, 3, 4, 5 th eigenvalues Figure 6.1 Multiresolution analysis for PLS monitoring Figure 6.2 Normal probability plot and histogram of original data and PCA score values (a) normal probability plot of original data (b) probability plot of score values (c) histogram of original data (d) histogram of score values Figure 6.3 Prediction results of PLS model with real Y value (solid line with squares) and predicted value (dotted line) (a) SVI (b) reduction of CN (c) reduction of COD (d) residual error of Y variables (SPE Y ) x

13 Figure 6.4 The second PLS weight vector plotted against the first for PLS model Figure 6.5 Variable influence on projection (VIP) for the predictor variables Figure 6.6 Monitoring performances based on T 2 and SPE X statistics with 95% confidence limits Figure 6.7 Monitoring performances of MRA for the PLS score values with 95% confidence limits (a) GDM (b) EV 1 (c) EV 2 (d) EV Figure 6.8 Contribution plot of the PLS score value for the first event Figure 7.1 Measured variables of the first 10 days of normal data set Figure 7.2 Conventional PCA monitoring result during the first 10 days of normal data set Figure 7.3 Conventional PCA monitoring result for nitrification linear decrease: T 2 and SPE plot Figure 7.4 PCA monitoring result with periodic removal for nitrification linear decrease: T 2 and SPE plot Figure 7.5 Monitoring result of the proposed method for nitrification linear decrease Figure 7.6 Conventional PCA monitoring result for nitrification step decrease: T 2 and SPE plot Figure 7.7 PCA monitoring result with periodic removal for nitrification step decrease: T 2 and SPE plot Figure 7.8 Monitoring result of the proposed method for nitrification step decrease xi

14 Figure 7.9 Prediction results of S NH,e and S NO,e for validation data set with static PLS method Figure 7.10 Prediction results of S NH,e and S NO,e for validation data set with static PLS method (after periodic removal) Figure 7.11 Prediction results of S NH,e and S NO,e for validation data set with the proposed method Figure 8.1 Schematic diagram of the proposed hierarchy structure Figure 8.2 One-step ahead prediction value of SVI using RLS method Figure 8.3 Sensitivity of the ARX model parameters of each state Figure 8.4 Power spectrum in each state (a) normal (b) bad (c) bulking state Figure 9.1 Block diagram of the TSK fuzzy model Figure 9.2 Block diagram of the FPLS method Figure 9.3 Scatter plots and firing strength plots of FPLS model in benchmark (a) first LV (b) second LV (c) third LV (d) fourth LV Figure 9.4 Comparisons of LPLS and FPLS for the predicted and actual S NHe in benchmark (a) Time series plot (b) Scatter plot Figure 9.5 Comparisons of LPLS and FPLS for the predicted and actual S NOe in benchmark (a) Time series plot (b) Scatter plot Figure 9.6 Scatter plots and firing strength plots of FPLS model in BET (a) first LV (b) second LV (c) third LV (d) fourth LV Figure 9.7 Time series plots of predicted and actual output in BET (a) SVI with LPLS and FPLS (b) CN with LPLS and FPLS (c) COD with LPLS and FPLS xii

15 Figure 9.8 Scatter plots of predicted and actual output in BET (a) SVI with LPLS and FPLS (b) CN with LPLS and FPLS (c) COD with LPLS and FPLS xiii

16 List of Tables Table 2.1 Process Input/Output Variables in full-scale WWTP Table 5.1 Fault (disturbance) sources in the simulation benchmark Table 5.2 Process variables in full-scale WWTP Table 6.1 Process Input/Output Variables in WWTP Table 6.2 Variations explained by the PLS model of four latent variables Table 7.1 Two disturbances in the benchmark Table 7.2 Percent variance captured by PCA model Table 7.3 MSE of two PLS methods and the proposed method Table 8.1 Confusion matrix of the test data Table 9.1 Percent variance captured (%) and MSE of several PLS models in benchmark Table 9.2 Percent variance captured (%) and MSE of several PLS models in BET xiv

17 I. Introduction 1.1 Research Motivation The requirements imposed on the wastewater treatment process (WWTP) in regard to effluent quality have become increasingly stringent, and existing plants are subject to increasing loads. To meet these stricter guidelines requires the development of an efficient wastewater treatment methodology. One way to improve process efficiency is to build new and larger treatment plants; however, this option is expensive and in many cases impossible due to lack of a suitable site. Another way to improve efficiency is to introduce advanced control techniques and to optimize operating conditions. This approach may improve the effluent water quality, decrease the use of chemicals, save energy, and reduce operating costs. Any sustainable solution to the current problems confronting wastewater treatment will require the development of an adequate information system for the control and supervision of WWTP. Close inspection of the current operation of WWTP reveals that instrumentation, control and automation (ICA) technology is minimal (Olsson and Newell, 1999). Few plants are equipped with more than a few elementary sensing elements and control loops, which are mostly used for flow metering and control, and for monitoring the basic plant performance over relatively long periods of time. Little progress has been made since the early 1970s, when a major leap forward was made by the widespread introduction of dissolved oxygen (DO) control. The introduction of ICA technology has been slow due to the lack of reliable instrumentation and the harsh environment in which the computer and automation devices are housed and operated. However, this situation is rapidly changing due to advances in sensor technology and the introduction of smart sensors capable of selfcleaning, self-calibration and self-reconfiguration. The current trend is towards 1

18 integrated systems that control and monitor the process from the wastewater sources right through to the receiving waters and sludge disposal. The primary purpose of ICA is to facilitate efficient operation of the WWTP, allowing effluent standards to be met for the lowest possible operational and capital costs. The main bottlenecks for the implementation of ICA technology within the WWTP are related to the following (Olsson and Newell, 1999; Jeppsson et al., 2001): Poor legislation Inadequate education, training and understanding Lack of confidence and acceptance within WWTP industries Lack of collaboration between stakeholders/organizations Economy and time to develop solution in practice and making sure that they work Unreliable measuring devices Plant constraints and inadequate sewer systems Lack of transparency Lack of software and instrument standardization The increase in public awareness about wastewater disposal over the past decade, as reflected in more stringent effluent regulation, has considerably increased the requirements imposed on treatment plants. The treatment process must now eliminate not only organic carbon pollution from wastewater, but also nutrients (e.g., nitrogen and phosphorus). The introduction of biological nutrient removal, the most economical method for removing nutrients, has significantly increased the complexity of process configurations. The main driving forces for ICA are related to : Stricter effluent quality standards Demand for lower sludge production 2

19 Economic incentive Reduce energy consumption and increase energy production Increased plant complexity (co-ordination of processes and loops, monitoring etc.) New treatment concepts e.g. more compact plants and water reuse New and cheaper technical solutions computers and communication At present, many WWTPs are operated according to predetermined schemes with very little consideration given to variations in influent load. The use of on-line sensors for on-line control of plant operation could enhance the ability of WWTP to comply with assigned effluent standards. Application of modern control theory in combination with new on-line sensors and appropriate models have great potential to improve effluent water quality, decrease the use of chemicals, and to save energy and money. In particular, the input/output behavior of these processes can be such that they appear highly stable right up to the time at which a gross process failure occurs, apparently significant input disturbances do not excite any significant output response, while very significant responses may occur in the absence of any corresponding input disturbance. This distinctive feature of the WWTP has long challenged control engineers. (Lindberg, 1997; Jeppsson, 1996; Lukasse, 1999; Olsson and Newell, 1999; Singman, 1999; Steffens and Lant, 1999; Lee, 2000; Sotomayor et al., 2000). In contrast to the situation for the WWTP, multivariate statistical monitoring and diagnosis of the process operating performance are extremely important aspects of plant safety and economical viability in most other process industries, for example the petrochemical and pharmaceutical industries. However, wastewater treatment industries are not among the most diligent and systematic users of statistical monitoring methods. To date, monitoring in wastewater treatment has mostly focused on a few key effluent quantities that are subject to regulations 3

20 enforced by government or other authorities. However, the ongoing tightening of environmental restrictions requires increased efforts to improve effluent quality from the WWTP using advanced monitoring technology. To effectively monitor process behavior statistically, important information must be extracted from the large number of measured variables, and this information must be presented in a form that is readily interpreted. The concept of sustainability, which entails minimizing the use of resources such as energy, chemicals and manpower, has also become an important issue in the design and modification of WWTPs (Rosen, 1998; Olsson and Newell, 1999; Teppola, 1999). However, there are further difficulties to overcome before a monitoring system can successfully be applied to WWTP (Rosen, 2001). Non-stationary data The conditions in which WWTP are operated are normally of a varying nature. Diurnal, weekly and seasonal patterns are normally found in the influent wastewater characteristics. These disturbances must be considered as normal and is in practice seen as state of things rather than disturbances. It is often difficult to discern other process disturbances from those caused by the varying influent conditions, which tend to have a dominant effect on the process behavior. Multiscale data A difficulty related to the dynamic properties of the disturbances as well as of the process is that disturbances occur in many different time scales. It means that some disturbances affect the process in a short time frame, whereas others have a much slower response. Apart from that this fact complicates the discernment of disturbances in a similar way to that of non-stationarity, it also deteriorates the performance of many monitoring techniques. Moreover, information on the time scale of a disturbance may prove crucial for a decision on counteractive actions. The multiscale nature of data is, however, not only a problem; it can also be used to decouple the 4

21 process time. Nonlinearities Wastewater processes display a nonlinear behavior and relationships between variables cannot always be approximated by a linear function. Consequently, if this is the case, nonlinearities must be taken into account when developing a monitoring system. Dynamic data Almost all data form dynamic process are autocorrelated, which means that each observation is not independent of the previous observation. This may have a great impact on statistical properties of the monitoring output and consequently caution must be taken when interpreting the result. 1.2 Research Objectives The principal goal of this research is to develop advanced control and monitoring systems that improve the operation of the WWTP. This work was undertaken with the following detailed objectives. Objective 1: To apply autotuning and supervisory control in the fullscale WWTP In the WWTP, PID controllers are familiar to process operators and very popular because of their simplicity, ease of operation and robustness to modeling error. However, it is well known that DO dynamics cannot be effectively controlled by PID controllers with fixed gain parameters. Moreover, manual tuning of PID controllers is tedious and laborious. In the present study, closed-loop identification and auto-tuned PID controller are applied to the DO control system in the full-scale WWTP. In addition, we propose a method for deciding on a proper DO set point for the current operation condition of the aeration basin. The full-scale experimental results showed good identification performance and good control performance. 5

22 Objective 2: To develop a robust estimation and control algorithm We present a generalized damped least squares (GDLS) algorithm to systematically remove an estimation windup problem in the adaptive control and self-tuning control system. The key element of the proposed method is the addition of a penalty of parameter variations to the objective function of the normal least squares algorithm to prevent the singularity problem. Mathematical analysis shows that the proposed method has almost equivalent properties to the normal least squares method and guarantees that no estimation problem will be encountered for poorly excited situations. Simulation results show that the proposed method gives better estimation performance than previous methods. We applied this method in the simultaneous control and estimation of important variables in the WWTP. Objective 3 and 4: To develop a disturbance detection and isolation algorithm The biological nutrient removal process alters gradually over time, indicating that the process is nonstationary. This represents a problem for developing a conventional multivariate statistical analysis because such an analysis must be developed from a set of "normal" operating data. Moreover, monitoring of the biological treatment process is very important because recovery from failures is time-consuming and expensive. Hence, a reliable detection procedure is needed. We propose an on-line fault detection and isolation algorithm, which uses a dissimilarity measure to evaluate the difference between successive data sets and to discriminate between serious and minor abnormalities. In addition, to cope with the nonstationary and multiscale process changes in WWTP, we propose a modeling and multiresolution analysis (MRA) method. Objective 5: To develop a process monitoring for continuous process 6

23 with cyclic operation Most WWTPs are subject to large diurnal fluctuations in the flow rate and composition of the feed stream. Consequently, WWTPs exhibit daily periodic characteristics, with strong diurnal fluctuations in the process input and output variables. Although these processes are non-stationary, their behavior tends to repeat from cycle to cycle and hence their cycle-to-cycle behavior may be assumed stationary. We propose a method for monitoring a continuous process with diurnal cyclic characteristics in the domestic WWTP. The proposed method uses a statespace model to capture and utilize the cycle-to-cycle correlation structure. The method can also incorporate the concept of inferential sensing to predict the quality variables and enhance process monitoring. Objective 6: To develop a simultaneous prediction and classification in the secondary settling tank An efficient operation of the secondary settler is very important since it separates the biomass from the treated wastewater and is a key mechanism of determining the effluent quality in a biological WWTP. Simultaneous prediction and classification in the secondary settling tank is proposed. Adaptive modeling scheme is implemented as recursive least squares (RLS) method to update the model parameters adaptively and neural network (NN) classifier is used as a process classifier. Experimental results shows that the prediction model describes the dynamics of the secondary settler well and neural network classifier combined with an adaptive scheme is quite adequate for the monitoring of the secondary settler in the WWTP. Objective 7: To develop a nonlinear fuzzy partial least squares method Fuzzy modeling has proved an efficient alternative for describing nonlinear 7

24 biological processes. Recently, the Takagi-Sugeno-Kang (TSK) fuzzy model has received considerable attention because of its prediction ability and suitability for continuous process modeling. The TSK model allows us to combine a set of linearized models into a global model to approximate the complex nonlinear system with less complexity. We propose a new nonlinear fuzzy partial least squares (FPLS) algorithm that embeds the TSK fuzzy model into the regression framework of the partial least squares (PLS) method. The proposed method applies the TSK fuzzy model to the PLS inner regression. It uses the PLS method to solve the problems of high dimensionality and collinearity and the TSK fuzzy model to capture the nonlinearity and to increase the use of experts knowledge. We applied this method in the simulation benchmark and full-scale WWTP. 8

25 II. Biological Wastewater Treatment Process 2.1 Activated sludge process The activated sludge process with its many variations is the basis for the treatment of wastewater almost everywhere in the United States. Especially, nearly 99% of municipal WWTPs in Korea use the traditional activated sludge process (Lee, 2000). Figure 2.1 shows the basic layout of an activated sludge process (Lindberg, 1997). The activated sludge process is a biological process in which microorganisms oxidize and mineralize organic matter. All microorganisms enter the system with the influent wastewater. The composition of the species depends not only on the influent wastewater but also on the design and operation of WWTP. The activated sludge is kept suspended in water by stirring and aeration. The microorganisms to oxidize organic matter use oxygen. To maintain the microorganism concentration, the sludge from the settler is recycled to the aeration tank. The growth of the microorganisms and influent particulate inert matter are removed from the process as excess sludge. Microorganism concentration is controlled by the excess sludge flow rate. Biological nitrogen removal Nitrogen materials can enter the aquatic environment from either natural or human caused sources. Excessive accumulation of various forms of nitrogen in surface and ground waters can lead to adverse ecological and human health effects. One of the major effects has been the direct and indirect depletion of oxygen in receiving water. Other impacts can be of major importance in particular situations. These include ammonia toxicity to aquatic animal life, adverse public health effects, and a reduction in the suitability of water for reuse. Since the early 1970s significant developments have taken place in the activated sludge method of treating 9

26 wastewater, as nutrient removal has become a very important factor in the WWTP. Nitrogen materials are present in several forms in wastewater, e.g. as ammonium (NH + 4 ), nitrate (NO - 3 ), nitrite (NO - 2 ) and organic compounds. Nitrogen is an essential nutrient for biological growth and is one of the main constituents in all-living organisms. When untreated wastewater arrives to the wastewater treatment most nitrogen is present in the form of ammonium. Nitrogen can be removed by a two-step procedure. In the first step, ammonium is oxidized to nitrate in aerated zones (nitrification). The microorganisms carrying out this process are generally considered to be nitrosomonas and nitrobacter. The aerobic growth of autotrophs consumes soluble carbon, ammonia and dissolved oxygen to produce extra biomass and nitrate in solution. This step can be further divided into two, one producing nitrites and the second further oxidizing nitrites to nitrates. NH NO O + 0.5O 2 2 NO NO H + + H O 2 (2.1) The second major step is the anoxic growth of heterotrophs, which use nitrates as oxidizer and produces extra biomass and nitrogen gas (denitrification). This process takes place in an anaerobic environment where the bacteria responsible for denitrification respire with nitrate instead of oxygen (anoxic). + 4NO 3 + 5" CH 2O" + 4H 2N 2 + 5CO2 + 7H 2O (2.2) where CH 2 O stands for diverse the organic matter (Lindberg, 1997). By using these two bacterial processes, nitrogen is removed from wastewater biologically. Anoxic zones in the activated sludge process are necessary for denitrification. Anoxic zones can be placed either in the beginning of the tank (predenitrification) or in the end of the tank (post-denitrification). In a pre-denitrifying system, an extra recirculation flow is usually introduced to transport the nitrate rich water back to the anoxic zone. For successful denitrification, a sufficiently high influent carbon:nitrogen ratio is required. When this requirement is not met, an 10

27 external carbon source has to be added. The dosing rate of that carbon is important. Dosing an insufficient amount will result in a high effluent nitrate concentration. Dosing too much will increase the costs considerably due to a high external carbon use, a high sludge production, and an increased oxygen demand. The strong variation in influent flow and composition, which if typical for WWTPs, generates a demand for on-line control of the denitrification process in order to guarantee a sufficiently low effluent nitrate concentration. Two variables can be manipulated to achieve this objective: (1) the external carbon dosage, to guarantee that almost all the recirculated nitrate is removed in the anoxic zone; and (2) the nitrate recirculation flow rate, to control the amount of nitrate that is recirculated. For optimal control of the process, the two variables should be controlled simultaneously of form a multivariable control system (Jeppsson, 1996; Lindberg, 1997; Steffens and Lant, 1999; Sotomayor et al., 2000). 2.2 Simulation Benchmark WWTPs are large non-linear systems subject to perturbations in flow and load, together with uncertainties concerning the composition of the incoming wastewater. Nevertheless these plants have to be operated continuously, meeting stricter and stricter regulations. Many control strategies have been proposed in the literature but their evaluation and comparison, either in real-life applications or based on simulations, is difficult. This is partly due to the variability of the influent, the complexity of the biological and hydrodynamic phenomena and the large range of time constants (from a few minutes to several days, even weeks), but also to the lack of standard evaluation criteria. It is difficult to judge the particular influence of the applied control strategy on reported performance increase, because the reference situation is often not optimal. Due to the complexity of the systems the effort to develop alternative control approaches is so high that a fair comparison between 11

28 different options is very rarely made. Then it remains difficult to conclude to what extent the proposed solution is process or location specific. To enhance the acceptance of innovating control strategies the evaluation should be based on a rigorous methodology including a simulation model, plant layout, controllers, performance criteria and test procedures. To this end, there has been a recent effort to develop a standardized simulation protocol simulation benchmark. The COST 682 Working Group No.2 has developed a benchmark for evaluation of control strategies by simulation (COST-624). The benchmark is a simulation environment defining a plant layout, a simulation model, influent loads, test procedures and evaluation criteria. For each of these items, compromises were persued to combine plainness with realism and accepted standards. Once the user has validated the simulation code, any control strategy can be applied and the performance can be evaluated according to certain criteria (Alex et. al, 1999; Pons et al., 1999; Copp et. al, 2000). A relatively simple layout was selected in simulation benchmark. It combines nitrification with pre-denitrification, which is most commonly used for nitrogen removal. Figure 2.2 shows a schematic representation of the layout. The plant consists of a 5-compartment bioreactor (6000 m 3 ) and a secondary settler (6000 m 3 ). It combines nitrification with predenitrification, which is most commonly used for nitrogen removal. The first two compartments of the bioreactor are not aerated whereas the others are aerated. The IAWQ model No. 1 (Henze et al., 1987) and a ten-layer one-dimensional settler model (Takács et al., 1991) are used to simulate the biological reactions and the settling process, respectively. Influent data developed by a working group on benchmarking of WWTP, COST 624, are used in the simulation. The return sludge flow rate (Q r ) is set to 100% of the influent flow rate and internal recirculation (Q a ) is controlled using a setpoint (S NO, ref ) of 1.0 mg N/l for the nitrate concentration in the second aerator. The aeration (K L a) in the 12

29 aerator 3 and 4 is set to a constant value, 240 day -1. The DO concentration in the aerator 5 is controlled to a set point 2.0 mg /l. Simulated influent data are available in three two-week files derived form real operating data. The files were generated to simulate three weather situations representing dry weather, storm weather (dry weather + 2 storm events), and rain weather (dry weather + long rain period). The file exhibits characteristic diurnal variations inflow and component concentrations. Each of data contains 14 days of influent data at 15 minutes sampling intervals. Any control strategy should be tested using each of these weather files. 2.3 Full-scale WWTP The process data were collected from a WWTP that treated the coke plant wastewater of the iron and steel making plant in Korea. It is a general activated sludge process that has five aeration basins (each 900 m 3 ) and a secondary clarifier (1200 m 3 ). The plant layout of the studied activated sludge plant is presented in Figure 2.3. It has two wastewater sources, where one directly comes from a coke making plant (called BET3) and the other comes from a pretreated wastewater of upstream WWTP at other coke making plant (called BET2). The coke-oven plant wastewater is produced during the conversion process of coal to coke in the steel making industries. It is extremely difficult to treat the coke wastewater because it is highly polluted and most of the chemical oxygen demand (COD) originated from large quantities of toxic, inhibitory compounds and coal-derived liquors (e.g. phenolics, thiocyanate, cyanides, poly-hydrocarbons and ammonium). In particular, cyanide (CN) concentration occupies the most important thing among the influent load of the coke wastewater. The influent flow rate is m 3 /hr, influent COD is mg/l, influent cyanide is 5 30 mg/l, influent temperature is C, temperature in the aeration basin is C and operation cost was 0.08 $/ton in

30 Twelve process and manipulated variables, X blocks, were used to model three process output variables, Y blocks. Y blocks consist of the solid volume index (SVI), the reduction of cyanide, and the reduction of COD. Table 2.1 describes the process variables and presents the mean and standard deviation (SD) values of X and Y blocks. The process data consisted of daily mean values from 1 January, 1998 to 9 November, 2000 with a total number of 1034 observations. The first 720 observations were used for the training data. And the remaining 314 observations were used as a test data set in order to verify the proposed methods. 14

31 Figure 2.1 A basic activated sludge process with an aerated tank and a settler 15

32 Wastewater Biological reactor Clarifier To river Q e, Z e Q 0, Z 0 Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 k L a PI m = 10 m = 6 Anoxic section PI Nitrate Aerated section Dissolved oxygen Internal recycle Q a, Z a Q f, Z f Q u, Z u m = 1 k L a = oxygen transfer coefficient Q r, Z r External recycle Q w, Z w Wastage Figure 2.2 A simplified process scheme of an activated sludge process using predenitrification (layout of simulation benchmark) 16

33 Pretreated WWTP BET2 Cokes plant BET3 Eq. T/K A B C D E Settler Final WWTP Aeration basin Recycle sludge Waste sludge Figure 2.3 Plant layout of coke WWTP, Korea 17

34 Table 2.1 Process Input/Output Variables in full-scale WWTP No Variable Description Unit Mean SD X 1 Q 2 Flow rate from BET2 m 3 /h X 2 Q 3 Flow rate from BET2 m 3 /h X 3 CN 2 Cyanide from BET2 mg/l X 4 CN 3 Cyanide from BET3 mg/l X 5 COD 2 COD from BET2 mg/l X 6 COD 3 COD from BET3 mg/l X 7 MLVSS at final aeration MLSS_%E mg/l basin X 8 MLSS _ R MLSS in recycle mg/l X 9 DO aerator DO at final aeration basin mg/l X 10 T influent Influent temperature C X 11 T aerator Temperature at final C aerator X 12 ph AT ph at final aeration basin mg/l Y 1 SVI settler Solid volume index at mg/l settler Y 2 CN red Cyanide reduction mg/l Y 3 COD red COD reduction mg/l

35 III. Autotuning and Supervisory DO Control in Fullscale WWTP 3.1 Introduction The dissolved oxygen (DO) concentration in WWTP has been recognized as an important variable to be controlled both for economical and process efficiency purpose. The proper control of DO could achieve improved process performances and there is an economic incentive to minimize excess oxygenation by supplying only necessary air to meet the time-varying oxygen demand of the mixed liquors. Despite the relatively simple dynamics of the DO mass balance, the control may be known difficult because of time-varying influent wastewater conditions, nonlinearity, time delay, sensor noise and slow sensor dynamics. To overcome these problems, several adaptive control strategies have been suggested recently to the control of DO concentration in the aeration basin (Holmberg et al., 1989; Carlsson, 1993; Carlsson et al., 1994, 1996; Lindberg, 1997). The previous works with advanced control algorithm require detailed process information such as oxygen transfer rate, respiration rate, reactor volume, wastewater flow rate and use a mathematically complex algorithm that is difficult to be implemented in on-line manner. Moreover, these cannot be implemented with the PID controller that is the most common controller in the real WWTP. Therefore, a method of improving the PID controller performance is required to use only the process input-output data without requiring any complicated algorithm. It is an automatic tuning of PID controller (autotuning). In WWTP, PID controllers are familiar to process operators and very popular because of its simplicity, easiness in operation and robustness to modeling error. But it is well known that the DO concentration cannot be controlled effectively by using the PID controller with fixed gain parameters. And the manual tuning of PID 19

36 controller is tedious and laborious. To overcome the time-consumed manual tuning procedures of PID controller, many on-line identification methods have been proposed to obtain the process information (Åström and Hägglund, 1985; Sung et al., 1998a, 1999). In WWTP, Carlsson et al. (1994) used an autotuning controller suggested by Åström-Hägglund to control DO concentration in WWTP. And Diue et al. (1995) used relay feedback method in order to tune PID loop controller parameters of PLC in the chlorination and dechlorination process. The first objective of this research is to apply an autotuning to actual DO control system in the full-scale WWTP. It approximates the dissolved oxygen dynamics to a high order model using the integral transform method and reduces it to the first-order plus time delay (FOPTD) or second-order plus time delay (SOPTD) for the PID controller tuning. And then PID controller is tuned based on the reduced method. The second objective is to suggest a simple supervisory control algorithm which decides a proper DO set point in the aeration basin s current operating. The key idea is that DO set point is determined in proportion to the respiration rate that is the indicator of biologically degradable load. Because we cannot have the real-time respirometry in the full-scale plant, we have used the well-known respiration rate estimation algorithm suitable to the surface aerator type of WWTP by using the recursive parameter estimation approach. The proposed methods have been evaluated in the full-scale WWTP. 3.2 Method Autotuning Method Nowadays, system identification under the closed-loop condition has been a special issue in industrial and environmental applications since the process output can go away from the normal steady state using the open-loop identification method. 20

37 This section explains a closed-loop identification method using the integral transform as the identification method (Whitfield and Messali, 1987; Sung et al., 1998a, b, 1999). It can utilize the process output and input activated by any test signal generator (e.g. controller itself, relay, P controller, simple set point change of PID control, pulse or step response) only if the process is activated sufficiently. So, the operator can activate the process in different ways according to his taste. The identification has the following steps. First, the process is activated sufficiently to guarantee that the process output and the process input include required information. Second, the differential equation of the parametric model is converted to the corresponding linear algebraic equation by using the integral transform. Third, the model parameters are estimated by using a least squares method based on the measured process data. Consider a general high order process model in the time domain. a n n 1 m m 1 d n n 1 + n 0 d y + a n dt dt y dy d u d u du + L + a1 + y = bm + bm b b u 1 m 1 + L+ m 1 1 (3.1) dt dt dt dt The above model (3.1) can approximate usual processes as accurately as desired, even though the processes include time delay or non-minimum phase zeroes. To convert the differential equation to an algebraic equation, the following integral transform (3.2) is applied to both sides of equation (3.1). tf j I _ y( i, tf j ) = L y( τ1) dτ1 L dτ i 1dτ i (3.2) i and as a result, equation (3.3) is obtained. a I y(0, tf ) + a I _ y(1, tf ) + L + a I _ y( n 1, tf ) + I _ y( n, tf ) = n _ j n 1 j 1 j j b I u( n m, tf ) + b I _ u( n m + 1, tf ) + L + b I _ u( n, tf ) (3.3) m _ j m 1 j 0 j The objective of the identification is to estimate the coefficients of a k and b k. In equation (3.3), all integrated values can be calculated numerically for various tf i 21

38 values. Then a k and b k are obtained by least squares algorithm. Here, it should be pointed out that the identification method using the integral transform does not care the types of the signal generators only if the signal can activated the process sufficiently. And it uses only the least squares method to estimate the parameters of the process model. The identified high order transfer function model can be used as the process model for other adaptive control, a Smith predictor or other model-based controller. On the other hand, we should reduce the identified model to the FOPTD or SOPTD model to tune the PID controller automatically because many developed on-line PID tuning methods such as internal model control (IMC), the Integral of time-weighted absolute value of the error (ITAE) and Cohen-Coon (C-C) are based on these models. The PID controller can be tuned at a number of operating conditions such as high/low respiration rate or load. Because we can separately use the proposed method for each operation condition, the proposed method can effectively compensate for the operation condition change by different PID controller parameters. But, how can we suggest its set point? The next idea is that DO set point is determined in proportion to the respiration rate that is the indicator of biologically degradable load Supervisory Control Supervisory control which recommends a proper DO set point in the aeration basin s current operation condition has been problematic in WWTP. The proper aeration is crucial to treatment efficiency since an insufficient DO level impairs the oxidation process and eventually leads to biomass death. Whereas too high DO may cause the sludge to settle poorly and excessive aeration is also undesirable from an economic point of view since the oxygen in excess is lost to the atmosphere. Therefore, the proper DO set point gives or may give the following advantages such 22

39 as better control of effluent and energy saving from the lower DO level. However, there have been few guidelines for the proper supervisory control of DO set point until now (Lindberg, 1997). Lindberg (1997) suggested a set point controller which utilizes measurements of ammonia concentration in the aeration basin. The key idea is that DO set point is determined in proportion to the respiration rate and influent loading because the respiration rate is the important variable that characterizes the DO process and the associated removal and degradation of biodegradable matter and is the only indicator of biologically degradable load. That is, if toxic matter enters the plant, for example, this can be detected as a decrease in the respiration rate, since the microorganisms degrade their activity or some of them die. Then, a rapid decrease in the respiration rate may hence be used as a warning that toxic matter has entered the plant. In this case, we should increase the DO set point. Therefore, we can suggest the following decision rule that The higher respiration rate, the lower DO set point. The lower respiration rate, the higher DO set point. Figure 3.2 shows the scheme of the supervisory control to decide the set point of DO controller. Because we didn t have the real-time respiration rate meter in the waste loading state, we used the well-known respiration rate estimation algorithm with Kalman filter approach suitable to the surface aerator type of full-scale WWTP. And the estimated parameter is also used to give judgment of the present operation states and process load. There are several different approaches such as recursive method in order to estimate oxygen transfer rate (K L a) and respiration rate (R(t)) from measurements of DO and airflow rate (Holmberg, 1989; Carlsson et al., 1994; Lindberg, 1997; Joanquin et al., 1998). Here we used the Lindberg s method (Lindberg, 1997). The K L a and the respiration rate are tracked by a Kalman filter by using measurements of DO and air flow rate, u(t). During the autotuning phase, the airflow rate or aerator speed variation is given a high excitation both in amplitude 23

40 and frequency. The estimation procedure is performed on a relatively short date set, in our case, autotuning s identification time. Then, the estimated models of the respiration rate and oxygen transfer rate could be used to the other controller design. The estimated value of the respiration rate would be used as a base rule in the set point decision. 3.3 Experimental Results In this work, the experiment was performed in the industrial coke wastewater treatment facility of the iron and steel making plant, Korea. Figure 3.3 shows a schematic diagram of the WWTP considered in this research. The plant consists of two parts: one is the biological process made up of the activated sludge process. The other is the chemical treatment process. As shown in Figure 3.3, WWTP has five aeration basins and one settling tank in the biological process. Each aeration basin is equipped with sensors (ph, DO, ORP, MLSS) and a speed controllable surface aerator in order to supply the oxygen. The automatic control system has a PC/PLC structure, which is based on a number of tag points for supervision, data acquisition, data storage and analysis. It was designed as the user-friendly control system using the commercial man machine interface (MMI) software known as FIX DMACS 7.0. The PID control algorithm was been installed in the MMI and autotuning program was implemented with the visual basic 6.0. The closed-loop identification methods were experimented using various input signal in the real plant from Feb to Feb In this research, a simple set point change of PID controller itself was chosen as the activation signal without any control mode change. It is simple, stable and easy to implement the proposed on-line identification method. The tested aeration basin was the last basin which was the most important in the total wastewater treatment process. The DO set point was increased from 1.6 to 2.0 mg/l at 0.05hour for identification. Figure 3.4 shows the 24

41 variations of DO concentration and aerator speed during the closed-loop identification. Using the acquired data, a high order model is identified using the explained identification method. The system order is chosen as n = 4 and m = 3. The equation (3.4) represents the identification result. 3 y( s) s s 0.001s G p ( s) = = (3.4) u( s) s s s s + 1 For the PID controller tuning, the high order model is reduced to the FOPTD and SOPTD model using the model reduction technique. The reduced models are as follows and the time unit is hour. 2 G FOPTD 0.19 s 0.2e ( s) 0.35s + 1, G SOPTD 0.17 s 0.2e ( s) s s + 1 (3.5) In Figure 3.5, Bode plots of the identified models are presented for the model selection. If a high control performance is required, SOPTD model is recommended. If just a stabilizing controller is the main objective, FOPTD model is sufficient. In this research, FOPTD model is selected for simplicity because the reduced FOPTD and SOPTD models show the similar results in the Bode plots. From the identification result and theoretical analysis, one can know that its time constant is 21 min, the time delay is 11.4 min, and the steady state gain is 0.2. For the model validation, the aeration speed is increased from 50 to 60 RPM as a step input. Figure 3.6 shows the step response of the real data and the prediction values of FOPTD model, where the obtained model approximates the behavior of the real plant successfully in spite of the experimental error and the identified model shows robustness to measurement noises. During the identification phase of autotuning, we estimated the respiration rate using the previous estimation techniques. We experimented on the following process conditions. Influent flow rate is 280 m 3 /hr, the temperature of aeration basic is 38 C 25

42 and DO sat is 6.5 mg/l. The estimation result is showed in the Figure 3.7 and the estimated respiration rate converges 54 mg/l/h. In the supervisory control, we determined the previous simple set point decision rule, The higher respiration rate, the lower DO set point. The lower respiration rate, the higher DO set point. To avoid the DO set point becomes too high or too low, it should be only be allowed to vary in an interval, mg/l in our case. And the respiration rate range is mg/l/h. In our coke WWTP, we suggested following simple set point decision rule. DO s = Rˆ( t) (3.6) Figure 3.8 shows the control results with PID parameter tuned using the acquired FOPTD model during 26 days. As a tuning rule, the ITAE disturbance rejection rule was selected because it is appropriate for the step input disturbance that occurs frequently in WWTP. With the set point set of 1.4 mg/l, it shows the PID control result during the first 13 days. Then, abnormal process changes in the influent load variations occurred at about 14 th day after the first identification step. Here, the identification method was used again to acquire a new process model by the set point change of about 1.5 mg/l. Because the experimental result showed the estimated respiration rate was low at about 25.0 mg/l/h, we recalculated DO set point on 2.1 mg/l based on the proposed supervisory control law. And then we changed the PID control parameters based on the refreshed process model. The experimental result showed good control performance in spite of the frequent load variations, abrupt upstream transition or influent toxicity. Since it considers all measured data sets to estimate several adjustable parameters, it represents robustness to measurement noises in particular. As a result of autotuning and supervisory control, it has achieved the overall improvement of effluent quality and have reduced 15 % of the electric power cost than the fixed gain PID controller. 26

43 3.4 Conclusions Autotuning and supervisory control algorithm for DO control in the full-scale WWTP were evaluated and proposed. Though the proposed method are concise and doesn t require any complicated numerical techniques, experimental results confirmed that overall improvement of effluent quality and 15% reduction of the electricity cost had been accompanied by autotuning and supervisory control. Autotuning method has been applied to other control variables in full-scale WWTP, such as ph, polymer addition control and sludge recycle rate control. 27

44 Activated Process Data Identification Model reduction Tuning Rule k c τ i τ d Figure 3.1 Identification and autotuning procedure for PID controller 28

45 Input R(t) Setpoint Controller DO s PID(DO) Controller AFR ASP DO Figure 3.2 Supervisory DO control scheme 29

46 Influent Wastewater ph, Temp, flowrate Equalization Basin ph, DO, ORP, MLSS #A #B #C #D #E Aerators Biological Treatment (Activated Sludge) PLC 1st Settling Tank MMI (VB) Effluent Flocculator 2nd Settling Tank Thickening Tank Filter Press Chemical Treatment Figure 3.3 Schematic diagram of full-scale WWTP 30

47 DO RPM 2.5 aerator speed [RPM] DO [mg/l] time [hour] Figure 3.4 Experimental result during identification phase 31

48 0.1 HighOrder SOPTD FOPTD 0 High Order SOPTD FOPTD -30 AR φ (deg) ω (rad/hour) ω (rad/hour) Figure 3.5 Bode plots of the identified FOPTD and SOPTD models 32

49 3 2.8 DO [mg/l] ` Real FOPTD Time [min] Figure 3.6 The validation test: real data and model prediction value 33

50 Respiration rate [mg/l/h] time [hour] Figure 3.7 Estimated respiration rate during identification phase 34

51 DO [mg/l] time [day] Figure 3.8 DO control result using autotuning and supervisory control algorithm 35

52 IV. Generalized Damped Least Squares Method 4.1 Introduction Since the main difficult in controlling biological processes arises from the variability of kinetic parameters and the limited amount of on-line information, an adaptive controller is best suited for this purpose. Adaptive control or self-tuning control is usually based upon simultaneous model identification and control and thus requires on-line updating of the model parameters rather than off-line processing of the process data. Recursive Least Squares (RLS) method is the most popular method used in the on-line recursive parameter estimation algorithm (Lambert, 1987; Ljung, 1987; Söderström & Stoica, 1989; Åström, 1995; Gau & Stadther, 2000). The recursive parameter estimation is a useful tool in WWTP, since the system have often time varying characteristics and parameters (Marsili-Libelli, 1990). In general, RLS method generally works well only if the process is properly excited all the time. But there are problems with exponential forgetting when the exciting is poor, that is, reveal a good control performance in the view of the control. A closed-loop control introduces linear dependencies among the process information. And it produces a singular problem when the signals are not activated sufficiently under the closed-loop system. It is known as an estimation windup or blowup, which the gain of the estimator grows exponentially. This is easiest to see which the process has a good control performance and is operated in the steady state. It is a main bottleneck in applying adaptive controllers (Åström, 1995). On the other hand, the estimation of the oxygen transfer rate K L a(u) and the respiration rate R(t) in WWTP is needed to monitor the biological activity and process control system performance or construct a nonlinear controller for controlling DO concentration more effectively. Knowledge of the two variables is of interest in both process diagnosis and process control. In particular, the respiration 36

53 rate is the key variable that characterizes the DO process and the associated removal and degradation of biodegradable matter. It is the only true indicator of biologically degradable load. If toxic matter enters the plant, this can be detected as a decrease in the respiration rate, since the microorganisms slow down their activity or some of them die. A rapid decrease in the respiration rate may hence be used as a warning that toxic mater has entered the plant and by-pass action may be taken to save the microorganisms (Lindberg, 1997). For on-line purposes, it is crucial to be able to estimate both R(t) and K L a simultaneously. In this case, the control strategy is dual in the sense that the control signal is used both to control the dissolve oxygen (DO) concentration and to excite the DO dynamics sufficiently to allow the parameters to be estimated. That is, in considering the combined mechanism of the two algorithms (DO control and R(t)/K L a estimation), a conflict arises, because controller keeps the DO constant, consistent R(t) and K L a estimation requires this quantity to vary. These contrasting requirements can be reconciled by adding a specific input such as relay or pseudo random binary signal (PRBS) to control signal (Holmberg et al., 1989; Marsili- Libelli and Vaggi, 1997). And it is then suggested to stop the estimation as soon as convergence is obtained. It can be detected by low diagonal values in the covariance matrix of RLS algorithm. The limit of this procedure is that time varying parameters cannot be estimated unless the algorithm is periodically reinitialized. But under feedback control, the conventional recursive estimation methods such as RLS always show the estimation windup. Several solutions have been suggested and can indeed reduce the parameter drift but are not a systematic solution of estimation windup problem in the adaptive controller. To overcome shortcomings of the previous methods, we will propose a generalized damped least squares method as the fundamental solution of estimation windup problem in the adaptive controller. And then an adaptive model-based DO 37

54 control algorithm will be proposed, which can simultaneously estimate the two key parameters, oxygen transfer rate and respiration rate. 4.2 Theory Generalized Damped Least Squares Algorithm Numerous papers have appeared to avoid the estimation windup which often occur in the practical application, where are normalization of the regression vector, matrix regularization, constant trace with a variable forgetting factor, information measurements for turning adaptation on/off, parameter variations and maximum limit. There have been few previous methods which can solve the estimation windup problem fundamentally. The interesting two papers have been reported. Lambert (1987) introduced a modification to classical RLS with exponential forgetting factor. His basic idea is to weight the estimated parameter vector in only one sampled interval. However, his idea lacks a multi-step approach in the estimated parameter vector. Tu (1990) suggested regularized least squares algorithm with a smoothing constraint and a selfadaptation of regularization parameter. This method is suitable for the illconditioned least squares which regressor vector has a high condition number. This method is originated for numerical stability property remedy. So, this approach is not adequate for the system identification and control fields. To overcome shortcomings of the previous methods, we extend Lambert s idea and propose a generalized damped least squares (GDLS) method. Consider a dynamic system with input signal {u(t)} and output signal {y(t)}. Suppose that these signals are sampled in discrete time t=1, 2, 3, and that the sampled values can be related through the Auto-Regressive with an exogeneous input (ARX). 38

55 39 ) ( ) ( 1) ( ) ( 1) ( ) ( 1 1 t e m t u b t u b n t y a t y a t y m n = L L (4.1) where y(t), u(t) are deviation variable or incremental mode and e(t) is white noise. Model (4.1) describes the dynamic relationship between the input and output signals. equation (4.1) can be rewritten as ) ( ) ( 1) ( ) ( 1) ( ) ( 1 1 t e m t u t u n t y t y t y m n n n = + + θ θ θ θ L L (4.2) [ ] m n T b b a a L L L 1 1 = θ (4.3) To solve the estimation windup and the drift problem in the closed-loop control system, we modify a normal least squares algorithm. We add the supplementary exponential weighted parameter variation restriction in the objective function of the least squares method. The objective function follows as: + = = = ) ( ) ( ) ( ) ( MIN ) ( N t t N N t t N N t N t w V θ θ δ ε λ θ θ (4.4) where λ and δ are the exponential forgetting factor for the model error and parameter variation, respectively, w is the weighting factor between the modeling error and parameter variation and ε(t) is one step ahead prediction error. The problem is to obtain the parameter estimates, θˆ which minimize the quadratic objective function (4.4). This can be rewritten as follows. ( ) ) ( ) ( ) ( ) ( ) ( N Y N wy N N N w add ls add ls + = + Φ Φ θ (4.5) = Φ = = = = = = = = = N t t N N t t N N t t N N t t N N t t N N t t N N t t N N t t N N t t N ls m t m u t u t y m t u t y m t u m t u t y t y t y t y t y m t u t y t y t y t y t y N ) ( ) ( 2) ( ) ( 1) ( ) ( ) ( 2) ( 2) ( 2) ( 1) ( 2) ( ) ( 1) ( 2) ( 1) ( 1) ( 1) ( ) ( λ λ λ λ λ λ λ λ λ L M L L (4.6)

56 40 = Φ = ) ( L O L L N t t N add N δ (4.7) [ ] T 2 1 ) ( ) ( ) ( ) ( N N N N m = θ n+ θ θ θ L (4.8) T ) ( ) ( 2) ( ) ( 1) ( ) ( ) ( = = = = N t N t t N N t t N t N ls m t u t y t y t y t y t y N Y λ λ λ L (4.9) T ) ( ) ( ) ( ) ( = = + = = N t m n t N N t t N N t t N add t t t N Y θ δ θ δ θ δ L (4.10) where Φ ls and Y ls have the same meaning as the least squares method, Y add and Φ add of the additional information vector and matrix come from the penalty of parameter variation. Then the solution vector is as follows: ( ) ( ) ) ( ) ( ) ( ) ( ) ( 1 N Y N wy N N w N add ls add ls + + Φ Φ = θ (4.11) This equation is the solution of the batch GDLS algorithm. The matrix wφ ls (N) + Φ add (N) of the batch GDLS solution is invertible unlike the least squares method even though eth signal is not excited enough. It is desirable to make the computations recursively to save computation in an adaptive control because the process data are obtained successively in real-time. The following equations are required to derive the recursive robust estimation algorithm. [ ] ) ( ) ( ) ( ) ( ) ( N Y N wy N N N w add ls add ls + = + Φ Φ θ (4.12) + Φ = Φ ) ( ) ( ) ( 1) ( ) ( 2) ( 1) ( 2) ( ) ( 1) ( 1) ( 1) ( 1) ( ) ( m N m u N u m N u N y m N u N y N y N y m N u N y N y N y N N ls ls L M L L λ (4.13)

57 ls L L 0 Φ = Φ + add ( N) λ add ( N 1) (4.14) 0 0 O L 1 ls [ y( N) y( N 1) y( N) y( N 2) y( N) u( N m) ] T Y ( N) = λ Y ( N 1) + L (4.15) Y add ( N) = δ Y ( N 1) + θ ( N 1) (4.16) add 1 ( wφ ( N) + Φ ( N) ) ( wy ( N) Y ( N) ) θ ( N) = (4.17) ls add ls + In these formulae, we call Equation (4.12) as the normal equation of recursive GDLS algorithm. Equation (4.17) has strong intuitive and meaningful appeals. Because the Φ add (N) term can make the matrix wφ ls (N) + Φ add (N) invertible, it does not suffer the possibility of an numerical ill conditioning in spite that y(t) and u(t) has zero value in the closed loop control or steady state data set Theoretical Analysis In this section, we will prove that the proposed GDLS algorithm guarantees the theoretical properties of least squares algorithm like unbiased, consistent, minimum variance and an exponential rate of convergence and more robust numerical properties. Property 1. When the weight goes infinite (w ), GDLS parameter estimate converges to that of least squares method ( lim ˆ θ = ˆ θ ). w GDLS This may easily be verified. If w is infinite, GDLS equation becomes as w ls add ls w ls ls add add lim w Φ + Φ = wφ, lim wy + Y = wy which gives a parameter estimate, ls lim ˆ θ w GDLS = lim w 1 1 ( wφls ( N) + Φ add ( N) ) ( wyls ( N) + Yadd ( N) ) = ( wφls ( N) ) wyls ( N) = ˆ θ ls lim ˆ θ = ˆ θ w GDLS ls (4.18) 41

58 Property 2. For large number of observation data (N ), GDLS objective function becomes same as least squares method with exponential forgetting ( V GDLS = Vls N lim ). For a large N value, there exists such a small M (<N) as δ N-M 0 and assume a stable convergence. Then θ(n) = θ(n-1) = = θ(n-m) and the objective function becomes as follows. lim V N GDLS N N = Vls ( θ ) = MIN w λ θ ( N ) t= 0 t ε( t) 2 (4.19) From this, we can know that its objective function is equal to the objective function of least squares method. We can assume that statistical properties of the normal least squares estimation can be established asymptotically for large number of observations in the proposed method. Property 3. In the absence of exciting data, the present parameter estimate is equal to a previous estimate value and thus is free from covariance windup problem for even steady- state data sets or under the closed loop control. This may be certified that an introduction of additional diagonal matrix of parameter variation becomes a bound and makes the singular matrix nonsingular. Under closed loop control or for steady state data sets, y(t) and u(t) become zero, and then the matrix becomes, L 0 0 L 0 Φ ( ) = 1 ls N M, add N L 0 1 Φ ( ) = = I 1 δ O 1 δ 0 L 0 0 L 1 Y ( N) = [ 0 L 0] T 1 ls, Y add ( N) = θ ( N 1) 1 δ (4.20) (4.21) Φ ( N) = wφ ( N) + Φ ( N) = Φ ( N) (4.22) aug ls add add 42

59 lim y( t), u( t) 0 1 ˆ θ 1 1 GDLS = I ˆ( θ N 1) = ˆ( θ N 1) 1 δ 1 δ (4.23) Thus the current solution is equivalent to a previous estimate value ˆ θ ( N 1). It changes the invertible term of the augmented matrix nonsingular and avoids the blow-up associated with exponential weighted least squares. Moreover, it is expected to reduce undesirably large variation in the estimated parameters for the abrupt measurement noises. Property 4. The proposed method has better numerical property than the least squares method. Estimation algorithms are normally implemented on digital computers and hence there is the possibility of numerical ill conditioning. Since, in an ill-posed problem where ( Φ T 1 ls Φls ) takes a large condition number in the normal least squares, least squares solution may becomes basically very sensitive to the perturbation in data y(t). This implies that the norm of such solution in significant greater that the norm of exact solution. In our algorithm, we overcome this problem N fundamentally by considering additional constraint, 1 δ t= 0 N 1 t θ ( N) θ ( t) 2 2, on the objective function. So, we can easily solve the ill-posed inverse problem efficiently and do not suffer from the possibility of the numerical ill-conditioning for the steady state data set or closed loop control. This property is the same as that of the regularized least squares method. In the analysis, we conclude that the proposed GDLS algorithm keeps the theoretical properties of least squares unlike other modified least squares method. Other modifications such as covariance resetting alter the geometry and convergence of the true least squares properties. So, it has supplementary robust properties and retains properties of least squares without the desirable theoretical properties. In 43

60 addition, GDLS method can also use merits of other variants, e.g. low pass filtering, conditional updating and variable forgetting factor Soft Sensor of Oxygen Transfer Rate and Respiration Rate Two general approaches of the estimation of respiration rate, soft sensor, have been developed during the last years. The first approach is based on the respirometer that estimated the respiration rate from DO mass balance in a respiration chamber (Spanjers et al., 1998; Olsson and Newell, 1999). The second approach is to estimate the respiration rate directly from the DO sensor and airflow rate measurement in the aeration basin. In this research, the latter approach is used. There are several different approaches of the estimation of R(t) and K L a based on simple from measurements of DO sensor and airflow rate in the real aeration basin. Holmberg et al. (1989) estimated the linear oxygen transfer rate model in a recursive way. Here, the excitation of the process was improved by invoke a small relay which increases the excitation. Carlsson (1993) developed a novel approach to estimate the respiration rate by the constrained piecewise linear model. Lindberg (1997) developed the nonlinear controller using the estimated oxygen transfer rate during the identification step and presented a systematic estimation method for oxygen transfer rate and respiration rate. Marsili-Libelli and Voggi (1997) summarized various estimation methods about respirometric activities in the bioprocess. Holmberg et al.(1989) and Marsili-Libelli and Vaggi (1997) described a simultaneous estimation scheme for K L a and R(t) based on the conventional RLS method, taking advantage of the differing time scale of the two variables. We select this approach for estimating the respiration rate and oxygen transfer rate. In this approach, dissolved oxygen deficit (D) in mass balance equation was introduced. dy( t) Q( t) = ( yin ( t) y( t)) + K La( u( t))( ysat y( t)) R( t) (4.24) dt V 44

61 D( t) = y y( t) (4.25) sat where y(t) is the DO concentration in the aeration basin, y in (t) is the DO concentration of the input flow, y sat is the saturated value of the DO concentration, Q(t) is the influent wastewater flow rate, V is the volume of the aerator, K L a(u(t)) is the oxygen transfer rate, u(t) is the airflow rate into the aeration tank from the air production system, R(t) is the respiration rate, respectively. Due to K L a and R(t)'s variations, DO dynamics show time varying behavior. The discrete-time equation with sampling time h is D( t + h) = e hkla 1 + (1 e K a L 1 D( t) + (1 e K a hkla L hkla ) R( t) ) Q / V (1 + r) y( t) (4.26) After some manipulations, equation (4.26) can be put in the standard estimation form. z( t + h) = y( t + h) = K L [ 1 hq / V (1 + r) ] a( u( t)) hd( t) hr( t) y( t) (4.27) where z(t+h) is the updating information at each sampling instant. The oxygen transfer rate K L a(u(t)) can be structured as a function of the airflow rate. K a( u( t)) K + K U K U (4.28) L = 0 1 It is easily seen that this model can be written as air a air T zˆ ( t + h) = ϕ ( t) θ ( t) where ϕ T (t)= [D(t)hU air h] is the regressors, θ(t) = [K a R(t)] T (4.29) is and e ( t) = z( t+ h) - ẑ( t+ h) is error vector. From the on-line measurements, a soft sensor of K L a(u) and R(t) will be designed and constituted by a recursive state estimator that uses the influent flow rate, airflow rate and DO measurements. The estimated parameters θ(t) can be updated according to the RLS method but are always experienced an estimation 45

62 windup problem under feedback control. In this research, we use a GDLS method as an estimation algorithm because under closed-loop control. A schematic figure of the robust estimator is shown in Figure Adaptive Model-based DO Control Despite of the relatively simple dynamics of DO process, DO control may not sufficiently be satisfied by the operator in the biological treatment process since the DO process dynamics has a time varying characteristics. This means that high control and estimation performance for all operating conditions may be hard to be achieved with a conventional method. After estimating two key variables, we can design an adaptive model-based DO control in order to correct the process/model mismatch and to estimate the unmeasured state variables. Using the available process input and output measurements, model-based control adaptively updates the estimated parameters θ. In this research, GDLS method is used for the estimation windup problem under closed-loop control. And then updated model is used by an adaptive generic model control (AGMC) for the control input. In AGMC, nonlinear process models can be imbedded into the controller directly without any linearization. AGMC is very simple and robust nonlinear control algorithm in single-input and single-output (SISO) process. Although the proposed control parameters are constant, the updated model can compensate the process/model mismatch because of its adaptive and model-based characteristics (Lee and Sullivan, 1988; Signal and Lee, 1992). The proposed method in the DO control is as follows. DO Process: dy( t) Q( t) = ( yin ( t) y( t)) + KLa( u( t))( ysat y( t)) R( t) (4.30) dt V DO model: 46

63 dy( t) Q( t) = ( y ( ) ( )) ˆ in t y t + KLa( u( t))( ysat y( t)) Rˆ( t) (4.31) dt V Desired trajectory: dy t t desired ( ) = K1 ( ys y( t)) + K 2 ys y t dt dt ( ( )) 0 = PI (4.32) Here, we made the desired trajectory as Proportional Integral controller (PI) without a state observer. Kˆ L a and R ˆ( t ) are estimated by the previous described GDLS estimation algorithm. Using equations (4.31) and (4.32), we can derive the following equation in order to obtain the control input u(t). ˆ PI + Rˆ( t) + Q( t) V y( t) Q( t) V yin ( t) K La( u( t)) = (4.33) y y( t) Using the linear model ( Kˆ ( ( )) ˆ L a u t = k u( ) ), u(t) can be easily obtained by 1 t sat PI u( t) = + Rˆ( t) Q / V ( y ( ) ( )) in t y t (4.34) kˆ 1 ( y sat y( t)) with the constraint u min u t) ( u max. In equation (4.34), u(t) is explicitly shown and the function of the estimated values and the measured process values. The control input u(t) has the nonlinear gain and all variables in numerator except PI are the sum of the bias of steady state term and feed-forward compensation of the respiration rate. Because we can update the estimated parameters of the oxygen transfer rate K L a and respiration rate R(t) under the proposed controller, we can easily compute the control action from equation (4.34). Figure 4.2 shows the structure of the model-based DO control scheme. This model-based DO controller shows no offset about the modeling error since it contains the integral action by the external input in the structure itself. As an ideal case, if the estimated values are equal to the true ones, Kˆ L a( t) = K La( t) and R ˆ( t) = R( t), the model-based control algorithm makes the offset free. Combining the 47

64 DO dynamics (4.30) with control input (4.34) gives the following error equation. dy( t) t = K e( t) + K 2 e t dt ( 0 1 ) dt = PI (4.35) So, the error signal, e(t) will approach to zero exponentially. Moreover, it has the special robustness to the disturbance of DO process, that is, respiration rate because it contains the estimated respiration rate and oxygen transfer rate in the controller structure. Adaptive model-based DO control with the proposed GDLS algorithm need not require any specific estimation phase and can acquire both estimation of respiration rate and DO control. The estimated value of respiration rate can give the information about the biological activity and can be used monitoring index. In Figure 4.3, we represent the proposed procedure for the soft sensor and model-based DO control structure. 4.3 Simulation Study In this section, we will see the performance of a GDLS algorithm. First, we will apply a GDLS method to the estimation problem of first-order system under the closed loop control and discuss about simultaneous estimation and control problem. Second, a GDLS method will be applied in DO process dynamics. Examples illustrate what happens when RLS is used versus the improvements obtained by using the proposed algorithm. First-Order System under Closed Loop Control The following first-order process is simulated and controlled by a proportional control law. Through the section, data is generated by y ( t) = a y( t 1) + b u( t 1) (4.36) where 48

65 a = 0.1 t (1, 750) = 0.9 t (750, 2000) = 0.5 t (2000,3000) b = 1.0 t (1, 750) = 0.5 t (750, 2000) = 0.1 t (2000,3000) where y(t) is process output, u(t) is control input. During the simulation, the two parameters (a, b) were changed two times. Signal to noise ratio changes in the same way. Notice that for t > 2000, the process was experienced the steady state gain s sign change. It is very large change in the system. The process excited by a Proportional controller activation signal of the following structure, u(t)=0.15(y s (t)- y(t)). We generated a random set point every 10 sampling time unit during t < 1000 and the fixed set point, 1.0 during 1000 < t < 3000 for the closed loop control. The corresponding input and output response are shown in the Figure 4.4. The input and output sequences generated for 3000 sample intervals are used to illustrate the comparisons with RLS and GDLS algorithms for the two parameter estimation problem (a, b) in the closed loop control. The RLS initial conditions (λ= 0.95, θ ( 0) = 0, P(0) = 10 9 ) were used in the simulation runs. Values used in the simulation have deviation form. The RLS algorithm with deadband update was used to show the estimation and investigate the corresponding estimated parameter windup in the closed loop control. We have observed that continued identification during periods of low excitation leads to parameter drifting and bad estimates in Figure 4.5. During the random set point change (0, 1000), the estimated parameters are accurate and bounded more or less. But in the absence of any input excitation in the interval (1000, 3000), the estimated parameters escape from real value. The noise in the process and insufficient exciting signal then cause drifting of the parameter estimates (e.g. a ˆ( t) and b ˆ( t ) ). This increases the probability of bursting and results in deterioration of 49

66 subsequent set point changes. During the closed loop, the P matrix grows unbounded whenever the system excitation is insufficient. Figure 4.6 represent the trace of P(t) in closed loop control. The covariance matrix blows up and the trace of P(t) increased rapidly in the absence of any input excitation in the interval (1000, 3000). The blowup usually occurs between set point change, unmeasured disturbance and measurement noises. In order to compare the performance of variants of RLS algorithm, the previous experiment was repeated for constant-trace algorithm. This scheme is to scale it in such a way that the trace of the matrix is constant. An additional refinement is to also add a small a unit matrix. This gives the so-called regularized constant-trance algorithm (Åström, 1995). ˆ θ ( t) = ˆ( θ t 1) + K( t)( y( t) ϕ( t) T ˆ( θ t 1)) T ( λi + ϕ ( t) P( t 1) ϕ( )) 1 K( t) = P( t 1) ϕ ( t) t T 1 P( t 1) ϕ( t) ϕ ( t) P( t 1) P( t) = P( t 1) T λ 1+ ϕ ( t) P( t 1) ϕ( t) P( t) ( t) = c1 + c I (4.37) tr P 2 ( P( t) ) where c 1 > 0, c2 0. Its result is shown in Figure 4.7, for constant trace RLS combined with conditional updating. We used the following parameters: c 1 is 100, c 2 is 1 and its estimation deadband condition is 0.1. The estimate of control input, b ˆ( t ) has comparatively accurate value, while estimated value of process output, aˆ ( t) cannot tract its correct value and shows the slow and poor estimation result once excitation is removed during t > We simulated the proposed GDLS upon the same process data for the comparison of RLS. The following conditions are used in a GDLS simulation (λ= 0.95, δ=0.95, w=1000, θ(0) =0). Through many simulations, we could know that 50

67 weighting factor, w was adequate around 1000 in the closed loop control, forgetting factor of least squares, λ could have value between 0.9 and 1.0 and forgetting factor of parameter variation, δ could have value between 0.0 and 1.0. We can select other values in the other process. Figure 4.8 shows the identification result of GDLS. Despite periods of exciting and non-exciting data, the result shows good estimation performance and indicates that the estimated parameters exactly keep track of the real value under closed loop control. Even process gain change, GDLS can correctly track the parameter variations. DO Process In the simulations, the following DO process is setup. dy( t) Q( t) = ( yin( t) y( t)) + KLa( u( t))( ysat y( t)) R( t) (4.38) dt V where Q(t) = l/h, V = 1000 l, y sat = 8 mg/l, y in (t) = 1 mg/l,. And R(t) has a diurnal variation of WWTP, R SS is steady state of respiration rate and K L a(u(t))=0.0018(1+(r(t)-r SS )/500)U air h -1. This configuration has a same condition of our experimental condition. The sampling time is 30 seconds. And for the similarity with the real process, we added the zero mean white measurement noise with 10% magnitude of process output. Also, we consider the time delay that always exists in the real biological treatment process. The time delay of DO dynamics is two times of the sampling time. In the estimation algorithm, the following setup was used. The RLS initial conditions (λ=0.99, θ ( 0) = θ, P(0)=10 6 ) were used in the 0 simulation runs. The parameters of GDLS are w=1000, λ =0.95 and δ =0.95. Figure 4.9 shows the estimation comparisons of RLS and GDLS method under PID control. The basic RLS algorithm with deadband update within 0.05 is used. At initial of RLS estimation, the estimated parameters of RLS are accurate and bounded more or less. But in the absence of any input excitation using good feedback control, 51

68 the estimated parameters escape from real value and diverge after set point change of DO controller. It is originated that the covariance matrix of RLS grows unbounded, estimation windup, whenever the system excitation is insufficient during the closed-loop control. We can see that continued identification during periods of low excitation leads to parameter drifting and bad estimates in Figure 4.9. On the other hand, estimation result of GDLS method shows good estimation performance in spite of feedback control. Note that these estimation results are operated under feedback control and low exciting signal. Based on GDLS estimation, adaptive model-based DO controller with equation (4.34) is used with the constraint10 u ( t) 10, 000. The tuning parameters of GMC are tuned by Lee s reference trajectory shape (Lee and Sullivan, 1988), which are K 1 =9.50 and K 2 =47.5. In spite of the time-varying R(t) and K L a, the proposed controller shows the good control performance in Figure However, the PID controller shows some offset since DO dynamics has continuously time-varying influent load and respiration rate. On the other hand, influent load and respiration rate is compensated by adaptive and feed-forward action and oxygen transfer rate is compensated by nonlinear gain in an adaptive model-based control. This means that the adaptive model-based DO control can cope with the operation condition changes such as the various load, respiration rate and other process changes. These dynamic variations are frequently occurred in WWTP. And the estimated respiration rate (soft sensor) under closed-loop can give the information about the biological activity and can be used monitoring index. 4.4 Conclusions In this research, we propose a simple and systematic estimation method for the estimation windup problem. On the basis of analysis, we concluded that a GDLS had the same properties as the least squares algorithm and more robust numerical 52

69 properties. Simulation examples show that GDLS method keeps the tracking ability of process parameters under the closed loop control. Based on the robust estimation performance, the model-based DO controller can efficiently cope with the time varying characteristics and operating condition changes that are frequently occurred in WWTP 53

70 D(t)hUair K Estimator (GDLS) L a ( ) ( ) 1 -h θ( N) = wφ ls( N) +Φadd( N) R(t) wy ( N) + Y ( N) ls add Figure 4.1 Robust estimator of K L a(u(t)) and R(t) with a GDLS method 54

71 y s AGMC u(t) DO dynamics y(t) Soft sensor (GDLS) [ Rˆ( t), Kˆ a( t ] T θˆ ( t ) = ) L Figure 4.2 Adaptive model-based DO control strategy with a GDLS algorithm 55

72 Soft sensor and model-based DO control in WWTP Choose AGMC trajectory parameter (K 1, K 2 ) Measure the process input/output values (u(t), y(t)) Soft sensing of K L a(u) and R(t) by GDLS algorithm Calculate the adaptive model-based DO control input Figure 4.3 Flow chart of soft sensor and the model-based DO control algorithm 56

73 Process Output Sampling Intervals Process Input Sampling Intervals Figure 4.4 Process output and input during the simulation (a) process output (b) control input 57

74 1.5 a real a RLS a Sampling Intervals 1.5 b real b RLS 1.0 b Sampling Intervals Figure 4.5 Parameter estimates a ˆ( t) and b ˆ( t ) of RLS with exponential data weighting 58

75 Trace of P Sampling Intervals Figure 4.6 Trace of P with RLS estimation method 59

76 a real a RLS 0.5 a Sampling Intervals b real b RLS b Sampling Intervals Figure 4.7 Parameter estimates a ˆ( t) and b ˆ( t ) of RLS with constant trace 60

77 a real a GDLS 0.5 a Sampling Intervals b real b GDLS b Sampling Intervals Figure 4.8 Parameter estimates a ˆ( t) and b ˆ( t ) with GDLS 61

78 Respiration rate [mg/l/h] Respiration rate [mg/l/h] 50 (a) R(t) 45 R e (t) of GDLS time [h] R(t) 100 (b) R e (t) of RLS time [h] Figure 4.9 Estimation comparisons of RLS and GDLS method under PID control 62

79 4 (a) set point PID proposed 5000 (b) PID Proposed DO [mg/l] 2 U AIR [l/h] time [h] time [h] Figure 4.10 Model-based DO control result with a GDLS method (a) process output, (b) control input 63

80 V. Disturbance Detection and Isolation in WWTP 5.1 Introduction The increase in environmental restrictions in recent times has led to an increase in efforts aimed at attaining higher effluent quality from WWTP. Achieving this goal requires the advanced monitoring of plant performance. Most of the changes in biological WWTP are slow when the process is recovering back from a bad state to a normal state. The early detection and isolation of faults in the biological process are very efficient because they allow corrective action to be taken well before the situation becomes dangerous. Some changes are not very obvious and may gradually grow until they become a serious operational problem. The discrimination between serious and minor anomalies is of primary concern in the monitoring of these processes. To make this distinction, a reliable procedure for the detection and isolation of disturbances is needed. In the case of the activated sludge process (ASP), multivariate statistical process control (MSPC) has been developed to extract useful information from process data and utilize it for monitoring and detection (Krofta et. al, 1995; Rosen and Olsson, 1998; Olsson and Newell, 1999; Teppola, 1999). Krofta et al. (1995) applied MSPC techniques to fault detection in dissolved air flotation. Rosen and Olsson (1998) adapted multivariate statistics based methods to the wastewater treatment monitoring system using simulated and real process data. Teppola (1999) used an approach that combined multivariate techniques, fuzzy clustering and multiresolution analysis for wastewater data monitoring. However, MSPC has fundamental weakness as a method for monitoring the ASP. These problems arise because the biological nutrient removal process changes gradually over time, making the process non-stationary. Thus, ASP hardly ever operates normally for long periods, and the non-stationary nature of the process 64

81 causes the definition of normality to change over time. One shortcoming of MSPC is that it cannot detect the change in correlation among process variables when Hotelling s T 2 and sum of squared prediction error (SPE) are inside the control limit. Hence, MSPC is not suited for monitoring non-stationary processes because it assumes stationary data. This is a problem for developing statistical control charts, as they should be developed from a set of normal operating data. Several methods have been suggested to model these non-stationary natures of the WWTP (Bakshi, 1998; Rosen and Lennox, 2000). One method that has shown potential for treating non-stationary processes is the use of adaptive algorithms for MSPC (Rosen and Lennox, 2000). Another proposed method employs a multiscale model through the use of wavelet transforms (Bakshi, 1998; Teppola, 1999; Rosen and Lennox, 2000). Another important issue in process analysis is the ability to diagnose the source of abnormal behavior. Chemometric methods such as principal component analysis (PCA) have been utilized for merging detection with the diagnosis of the causes of abnormal situations (Ku et al., 1995; Kano et al., 2000a,b). Ku et al. (1995) proposed a diagnostic method in which the out-of-control observation was compared to PCA models for known disturbances. Using refinements of statistical disturbances, discriminant analysis was then used to select the most likely causes of the current out-of-control condition. Kano et al. (2000a, b) proposed a new statistical processmonitoring algorithm. This method is based on the idea that a change of operating condition can be detected by monitoring the distribution of time-series process data, because this distribution reflects the corresponding operating condition. They did not, however, consider the individual contributions of each transformed constituent in the normalization of the dissimilarity index. In this research, we propose a modified dissimilarity measure and disturbance detection method for the successive data sets. Using eigenvalue monitoring, the proposed method can also detect the disturbance and isolate the type of disturbance 65

82 scale. 5.2 Theory Modified Dissimilarity Measure The dissimilarity measure that has been traditional used is based on the Karhunen-Loeve (KL) expansion and is identical to the PCA. This measure compares the covariance structures of two data sets and represents the degree of dissimilarity between them. In the computational procedure, the variance of a transformed data vector is normalized by its corresponding eigenvalue. The dissimilarity measure therefore considers not the absolute magnitude but the relative magnitude of the variance change, and neglects the importance of each transformed variable. We suggest a modified dissimilarity measure that considers the importance of each transformed variable. Using this modified measure we propose the fault detection and isolation (FDI) technique. This technique is divided into two major steps: a training phase using an historical data set representing the process in normal operation, and the on-line monitoring and isolation using the test data set. The modified dissimilarity measure algorithm is as follows. First, the data window size and step size are determined, where the window size is the number of samples in each data set and the step size is the monitoring interval. Second, two successive data sets are selected and normalized with the sample mean and sample variance (X i, i = 1, 2). Figure 5.1 represents the concept of window and step size using a moving window. Third, the sample covariance matrix is found and singular value decomposition (SVD) is applied to it. The algebraic representation of these steps is S = 1 X1 1 X N 2 T X X 1 2 = N1 1 S N 1 1 N S N 1 2 (5.1) 66

83 1 T S i = X i X i i = 1,2 (5.2) N 1 i SP = PΛ (5.3) where P is the loading matrix andλ is the diagonal matrix. In this procedure, input variables are transformed into orthogonal variables (transformation X i to Y i ). Y i N 1 T N 1 i = i (T i = X i P) (5.4) Fourth, the sample covariance matrix (R) of two transformed data sets (Y 1 and Y 2 ) is found and SVD is applied to R. i j i i i R 1 + R 2 = Λ (5.5) Ri q j = λ jq j, i = 1, 2 and j = 1,..., r (5.6) ( q : loading vector, λ i j :eigenvalue, r: dimension of data in PCA). Combining equations (5.5) and (5.6) and using some algebra, we obtain R ( Λ λ ) q j λ jq j and R2q j = jj j q j = (5.7) This result shows that two sample covariance structures share eigenvectors, whose eigenvalues satisfy 1 λ j + λ 2 j = Λ jj (5.8) where λ is the jth eigenvalue in the ith data set and i j Λ is the eigenvalue in the jj total data set. As two of the data sets are more similar than others, their eigenvalues are closer 0.5 Λ jj. In general, the first few principal components, r, explain most of the variation of the data. Next, the modified dissimilarity index D is found, 2 r Λ r jj 2 D = 4 λ j Λ jj (5.9) j= 1 2 j= 1 D has a value between 0 and 1. The more similar two data are, the closer D is to 0, and the more dissimilar two data are, the closer D is to 1. Finally, the (1-α)100% 67

84 confidence interval of each eigenvalue is determined. For many samples, it is reasonable to assume that each eigenvalue is a normal random variable by the central limit theorem. For the samples obtained from a normal operation, the interval containing 99% of eigenvalues calculated above is obtained by where i j i i i i i ( α / 2; N 1) s{ λ } + λ λ t( 1 α / 2; N ) s{ λ } + λ t 1 1 (5.10) i λ is the mean of a sample, { } j j j j s λ is the variance of a sample, and α is j j 99%. That is, (1-α)100% of λ i j are below the limit value and the remainder are above it (Johnson and Wichern, 1992) Fault Detection and Isolation (FDI) For on-line monitoring, the normal operation data is used as a training data set. And confidence limits are calculated from the previous step. In addition, the sample representing a current operating condition is scaled by the sample mean and sample variance obtained in previous steps. The corresponding modified dissimilarity index and eigenvalues are then calculated using the previous step. The modified dissimilarity index, which evaluates the difference between two data sets, can quantitatively detect a change of operating condition and monitor a distribution of time-series data. If the index is outside the control limit or deviates from a value of zero, the operating condition is changed and a disturbance is said to have occurred. In particular, we can focus on the individual variation of several eigenvalue. Only a few eigenvalues are considered as monitoring indexes because most of the variation is captured by the first few eigenvectors. The remaining variation that is not captured by the principal eigenvectors is negligible and it is not critical to identify whether it is caused by changes in the process or noise. If any of the principal eigenvalues exceeds its corresponding confidence limit, the current operation at that eigenvalue is changed indicating that an operating change has 68

85 occurred. In this eigenvalue, a disturbance detected. Monitoring at each eigenvalue allows us to distinguish a process change from an instantaneous fault or sensor noise. Because it represents the corresponding characteristics at each eigenvalue, this technique gives information about the eigenvalue on which a disturbance occurs, and makes it possible to analyze the physical/biological reasons for the disturbance. This method automatically gives us the capability to isolate and interpret the disturbance source. If adaptive scaling is to be used to tackle non-stationary or dynamical problems, the sample mean and variance should be successively updated to detect changes in continuous processes. And a forgetting factor can be introduced to reduce the effect of previous measurements (Li et al., 2000). Another important consideration in monitoring changes in the process or operating condition is the determination of appropriate window and step sizes. These quantities should be carefully selected taking into consideration the process characteristics. We suggest that the window size should be large in comparison to the time constant of the process, and the step size should be small in comparison to the sampling time. 5.3 Simulation Studies Simulation of Benchmark Plant For the monitoring purpose, the proposed method was applied to the detection of various disturbances in the simulated data obtained from a benchmark simulation. Four types of disturbance were tested using the FDI method: External disturbance, internal disturbance, setpoint change, and sensor fault (see Table 5.1). External disturbances are defined as measurable disturbances, which are outside of the process and are detectable from the sensor signal. Examples of such disturbances are changes in the influent flow rate or nitrogen concentration. Internal disturbances are 69

86 caused by changes within the process affecting the process behavior. These disturbances include factors such as decreased nitrification, non-measurable inhibition of influent or gradual reduction of the settling velocity in the secondary clarifier (denoted as bulking phenomena). The two other simulated disturbances were a set point transfer signal with low frequency information and a sensor failure event in the high frequency band. Three events in the influent data developed by the benchmark are associated with the influent flow rate (dry, storm and rain weather). The training model was based on a normal operation period of one week of dry weather. The data used were the influent file and outputs with noises suggested by the benchmark. The variables used to build the X-block in the disturbance detection were the influent ammonia concentration (S NH,in ), influent flow rate (Q in ), total suspended solid in aerator 4 (TSS 4 ), DO concentration in aerators 3 and 4 (S O,3, S O,4 ), and oxygen transfer coefficient in aerator 5 (K L a 5 ). The conditions used for on-line monitoring were a window size of 20 samples (5 hours) and a step size of 5 samples (1.25 hours). The mean and variance were the values calculated from the normal data. External process disturbances We tested a storm event that suddenly occurs two times after a long period of dry weather. This example shows how external disturbances appear within the proposed method. The pattern of measurement variables during the storm weeks was the same as the storm condition in the benchmark. The pattern of measurement variables during the storm weeks is presented in Figure 5.2. And Figure 5.3 shows the monitoring results obtained using the FDI technique during the storm weeks. The dissimilarity index sharply increases at around samples 850 and 1050, which correspond to the first and the second storm. The two storm events are largely detected as changes in the first and second eigenvalues, as shown in Figure 1(b-d). The magnitude of each eigenvalue represents the proportion of the variation 70

87 captured by its corresponding eigenvector. Internal process disturbances The first internal disturbance was imposed by decreasing nitrification rate in the biological reactor through a decrease in the specific growth rate of the autotrophs (µ A ) is decreased. The autotrophic growth rate at sample 300 was linearly decreased from 0.5 to 0.3 day -1. As shown in the left of Figure 5.4(a), the decrease in nitrification is detected for the first time at around sample 330, which is 30 samples after the event occurred. This event is quickly detected by the second eigenvalue, while the first eigenvalue increases continuously after this event. After the deterioration of nitrification ends, the dissimilarity index shows peaks at around samples 500 and 600. These sudden increases in the dissimilarity index are caused by the increase of the first eigenvalue. The gradual increase of first eigenvalue means that the process has undergone this type of internal disturbance such as nitrification or denitrification rate decrease. The second internal disturbance imposed on the system was a linear decrease in the settling velocity in the secondary settler between samples 300 and 500. For the early detection, it was necessary to add another measurement of the effluent total suspended solid (TSS e ) to the general X-blocks. The right side of Figure 5.4(a) shows an increase in the dissimilarity index after sample 330. As in the case of the decrease in nitrification, the dissimilarity index is constant for 30 samples after the onset of the decrease in settling velocity. The increase in the dissimilarity index at around sample 330 is caused by increases in the second and the third eigenvalues. The first eigenvalue jump up about sample 410, contributing greatly to the increase in the dissimilarity index observed shortly afterwards. The jump up of first eigenvalue means that the process has undergone this type of internal disturbance such as bulking or biomass decay events. 71

88 Sensor faults and setpoint change To identify the usefulness of the FDI method for detecting sensor faults with high frequency information, we corrupted the nitrate sensor in the secondary anoxic tank. In the sensor fault case, it was also necessary to add the nitrate concentration (S NO,2 ) to the general X-blocks. The fault was introduced during the sample period Prior to that period, the sensor was operating properly except for the imposition of sensor noise. Monitoring results are presented in the left side of Figure 5.5. The sensor fault is detected by a change in the second eigenvalue, indicating that the sensor fault is caused by the variation change along the second contributing axis. On the other hand, the disturbance caused by the setpoint change with low frequency information is demonstrated in the right side of Figure 5.5, which shows when the DO controller setpoint in the 5 th biological reactor was suddenly changed from 2 to 1 mg/l at sample 300. In contrast to the sensor fault, the setpoint change causes a variation along the third contribution axis Full-scale WWTP We now consider a second example, using real data from the coke WWTP of an iron and steel making plant in Korea. The treatment system is a general activated sludge process that has five aeration basins and a secondary clarifier. We selected 16 general variables to describe the process state of the WWTP; these variables are described in Table 5.2. The data set consisted of daily mean values from 1 January, 1998 to 9 November, 2000, comprising a total of 1034 observations. The first 720 observations were used for the training model of the mean-centered and auto-scaled data. The remaining 314 observations were used as a test data set to test the proposed method. In addition to the proposed algorithm, the PCA method was used to monitor the 72

89 WWTP characteristics. PCA results were then compared with those of the proposed FDI algorithm. We managed to capture only above 55% of the variance by projecting the variables with four latent variables. Figure 5.6 shows the Hotelling s T 2 and the squared prediction error (SPE) chart. The two horizontal lines correspond to the 95% significance levels of the original training data. The data deviated slightly in samples 120 to 125. From Figure 5.6, we can see certain deviations in some of the variables within these intervals. To make the cause of the deviation more obvious, the contributions from every measurement variable were calculated. However, it is difficult to detect this disturbance from the plots. Moreover, it is not possible to diagnose and isolate the disturbance frequencies. The monitoring performances of the proposed FDI method in WWTP are given in Figure 5.7. To monitor changes in the process and operating condition, a window and step sizes of 20 and 5 samples were used, respectively. The dissimilarity indexes of the test data set are shown in Figure 5.7 (a). The dissimilarity index has high values around samples (19 April, May, 2000), leading us to predict that a large process change happens at this time. Five eigenvalues which correspond to a range of disturbance scales are depicted in Figure 5.7 (b-f). The remaining eigenvalues give little information because they provide only high frequency information such as measurement noise. It is evident from Figure 5.7 that the third and fourth eigenvalues, which are representative of middle scale disturbances, contribute greatly to the increase in the dissimilarity index. At this time, the WWTP received a high input of cyanide and chemical oxygen demand (COD) load. This load reduced the activity of the micro-organisms and diminished the settling performance, causing an increase in the solid volume index in the secondary settler. We found that the increase in an influent load started out as an external disturbance but subsequently transformed into an internal disturbance that changed the process operation region in WWTP. These process changes are detected 73

90 by the dissimilarity index, and the disturbance sources were isolated by the proposed FDI method. We can draw the following conclusions from the results of several simulations. First, the modified dissimilarity index unifies all the scales into one monitoring value and provides a compact index. Second, the eigenvalue at each eigenvalue can discern the dominant dynamics and detect the scale on which a disturbance occurs. In this analysis it is presumed that the eigenvalue with a large magnitude represents the effect of low frequency information such as a large change in the process or the occurrence of a large and long disturbance. The eigenvalue of intermediate magnitude represents a small change in operation condition such as a short external disturbance, while the eigenvalue with a small magnitude represents high frequency information such as sensor faults and measurement noises. The fault isolation approach therefore provides intelligence on the scale at which a disturbance occurs, and can be used to analyze and interpret the physical cause and effect of disturbances. Third, because the proposed method is based on evaluation the difference between successive time series data sets with a moving window, as is done in adaptive PCA, the proposed method can tackle the non-stationary problem of WWTP. 5.4 Conclusions The strategy proposed in the present work is able to detect and isolate the effect of various disturbances occurring in the activated sludge process. This strategy uses a modified dissimilarity index and monitoring of individual eigenvalues. One merit of this technique is that it can simultaneously detect the disturbance and isolate its source, in contrast to conventional MSPC. The strength of the isolation technique is that it gives information about the scale on which a disturbance occurs, assisting in the interpretation of the disturbance. Experimental results show that it is an 74

91 appropriate monitoring technique for the activated sludge process, which is characterized by a variety of fault and disturbance sources and non-stationary characteristics. This fault detection and isolation method provides us with a new analysis tool for acquiring a deeper understanding of process monitoring methodology through the detection and isolation of disturbances at different scales. 75

92 Value Step size Window size Time Figure 5.1 Moving windows between successive two datasets. 76

93 TSS x Qin Snh,in So3 So4 KLa sample number Figure 5.2 Measured variables during the storm weeks 77

94 0.20 (a) 60 (b) 50 D st eigen value sample number sample number 3.5 (c) (d) nd eigen value rd eigen value sample number sample number Figure 5.3 Monitoring performances of the external disturbance (a) modified dissimilarity index (b-d) individual eigenvalue plot. 78

95 D D (a) (b) 0.04 (a) 25 (b) st eigen value st eigen value sample number (c) sample number (d) sample number (c) sample number (d) nd eigen value rd eigen value nd eigen value rd eigen value sample number sample number sample number sample number Figure 5.4 Monitoring performances under internal disturbances caused by decreasing nitrification (left plot) and settler bulking (right plot) (a) dissimilarity index, (b-d) individual eigenvalue plots 79

96 D D 0.14 (a) sample number 4.0 (c) 1st eigen value 8 (b) sample number (d) (a) sample number (c) 1st eigen value (b) sample number (d) nd eigen value rd eigen value nd eigen value rd eigen value sample number sample number sample number sample number Figure 5.5 Monitoring performances of sensor faults (left plot) and setpoint change (right plot) (a) dissimilarity index, (b-d) individual eigenvalue plots 80

97 Hotelling T (a) 95% UL SPE (b) sample number % UL sample number Figure 5.6 PCA monitoring performances (a) Hotteling s T 2 chart, (b) SPE plot 81

98 D EV 2 (a) (b) (c) time (days) (d) time (days) (e) time (days) 0.6 (f) time (days) EV EV 1 EV time (days) EV time (days) Figure 5.7 FDI monitoring performances (a) dissimilarity index, (b-f) the 1, 2, 3, 4, 5 th eigenvalues 82

99 Table 5.1 Fault (disturbance) sources in the simulation benchmark Disturbance type External Internal Internal Sensor faults Operation change Disturbances Storm events Decreasing nitrification Decreasing settling velocity Nitrate sensor failure Set point change of DO controller Simulation conditions Abrupt change of influent flow rate at around samples 850 and 1050 Specific growth rate for autotrophs: from 0.5 to 0.3 day -1 in a linear fashion during sample Settling velocity in a secondary settler: from 250 to 150 mday -1 in a linear fashion during sample Nitrate sensor noise in the second anoxic tank: noise mean changed from 0 to 1 mg N/l during sample DO controller set point: from 2 to 1 mg/l at sample

100 Table 5.2 Process variables in full-scale WWTP Variable Unit Mean STD Variable Unit Mean STD Q 2 m 3 /h MLSS AT mg/l Q 3 m 3 /h MLSS R mg/l CN 2 mg/l DO AT mg/l CN 3 mg/l T influent CN PS mg/l T AT C C COD 2 mg/l SVI AT ml/g COD 3 mg/l SVI R ml/g COD PS mg/l PH AT ml/g

101 VI. Modeling and Multiresolution Analysis in WWTP 6.1 Introduction Up to date, monitoring in WWTP has mostly focused on a few key effluent quantities upon which regulations are enforced. However, since the environmental restriction becomes more rigid nowadays, the increasing effort for higher effluent quality is required in the monitoring of WWTP performance. Specially, monitoring of the biological treatment process is very important because the recovery from failures is time-consuming and expensive, where some of changes are not very obvious and may grow gradually until they produce a serious operational problem. Therefore, early fault detection and isolation in the biological process are very efficient to execute corrective action well before a dangerous situation happens. At the same time the discrimination between serious and minor abnormalities is of primary concern. To accomplish this task, a reliable detection procedure is needed. However, few monitoring techniques are available to utilize large on-line data sets despite of the increasing popularity and the decreasing price of on-line measurement systems in the field of WWTP. Multivariate statistical process monitoring (MSPM) or multivariate statistical process control (MSPC) is a possible solution to multivariate, collinear, auto or cross-correlated processes. This comprises chemometrics methods such as principal components analysis (PCA) or partial least squares (PLS) combined with standard sorts of control charts. In order to extract useful information from process data and utilize it for the monitoring of WWTP, several applications of MSPM or MSPC have been developed (Krofta et. al, 1995; Rosen, 1998; Van Dongen and Geuens, 1998; Teppola, 1999). However, the biological treatment process has several peculiar features unlike chemical or industrial engineering. Above all, it is nonstationary, which means 85

102 that the process itself changes gradually over time. For example, systematic seasonable variations show a dynamic pattern, for example, the process normal condition evolves according to the seasonal variations. In addition, many underlying phenomena of WWTP takes place simultaneously and it may be difficult to separate specific phenomenon among them. Namely, it has multiscale characteristics that have multiple simultaneous phenomena affecting the data at different time or frequency scales. If these synchronous characteristics interfere or mask other time or frequency variations, called the disturbing effects, the situation turns troublesome because the multiscale variations are enlarging up the confidence limits. This is unfavorable because the actual events can stay undetected by the monitoring algorithm while the plant is being under way of the events. To solve these problems, several methods have been suggested recently using adaptive PCA, multiscale analysis with dynamic PCA and multiresolution analysis with wavelet (Bakshi, 1998; Kano et al., 2000a,b; Rosen and Lennox, 2000; Teppola and Minkkinen, 2000, 2001; Ying and Joseph, 2000; Choi et al., 2001; Yoo et al., 2001). Bakshi (1998) used a multiscale model through the use of wavelet transforms. Kano et al. (2000) proposed a dissimilarity index based on the distribution of timeseries process data. Rosen and Lennox (2000) applied and developed adaptive PCA and multiresolution analysis of wavelet. Ying and Joseph (2000) evaluated the feasibility of sensor fault detection using multi-frequency signal analysis of noise. Teppola and Minkkinen (2000, 2001) suggested several multiresolution analyses using wavelet-pls regression model for interpreting and scrutinizing a multivariate model. Choi et al. (2001) suggested a generic monitoring algorithm utilizing a modified dissimilarity index in the benchmark simulation and Yoo et al. (2001) confirmed these results using a PCA-type monitoring algorithm in a real WWTP. Shortly, this research applies two methods, one is for prediction and the other is for multiresolution monitoring technique. In this way, it is possible to take into 86

103 account the multivariate, nonstationary and multiscale natures of WWTP. These approaches are organized by putting PLS model and multiresolution analysis together. In the first approach, PLS model is used for the prediction and data analysis. In the second approach, multiresolution analysis using a generic dissimilarity measure and singular value decomposition to PLS score matrix is proposed. The statistical confidence limit of detection and isolation is suggested and its ability is verified by using real plant data. 6.2 Theory Partial Least Squares (PLS) Very often in industrial applications, the data are severely corrupted by noise and collinearities among a high number of variables. To treat these problems, it is convent to apply latent variables models, particularly PLS modeling. PLS maximizes the covariance between process variables and responses. In PLS, the matrix X (process variables) is decomposed and modeled in such a way that the information in Y (responses) can be predicted as well as possible. In addition, PLS uses only the variation in X matrix that is significant in the prediction of the variation in Y matrix. Moreover, one does not assume that the X variables are free of noise as in multiple linear regression (MLR). The noise and insignificant variations are not used in modeling. In PLS, the standardized sample matrices Z X of X and Z Y of Y are decomposed as follows. nx m nx m = T T T T X TP = ti pi = ti pi + ti pi = ti i= 1 i= 1 i= m+ 1 i= 1 Z p + E = Zˆ + E ny m ny m = T T T T Y UQ = uiqi = uiqi + uiqi = i= 1 i= 1 i= m+ 1 i= 1 i T i T i X Y X Y X Y (6.1) Z u q + E = Zˆ + E (6.2) 87

104 In the above representations, t i and u i are score vectors, p i and q i are loading vectors, Ẑ X and Ẑ Y are unbiased estimates of Z X and Z Y respectively, and E X and E Y are residual matrices. PLS is composed of outer and inner relationship. In construction of outer model, score vectors t i and u i are obtained by the projection of Z X and Z Y onto loading vectors p i and q i, respectively. While in construction of inner model, t i is linearly regressed on u i yielding u ˆ = t b, where b i is a regression i i i coefficient. Then Z Y can be expressed as T m Z = TBQ + E = b t q + E (6.3) Y Y where B is a diagonal matrix of the regression coefficient b i. In this respect, PLS can be considered an useful tool which divides multivariate linear regression into simple linear regression. The first several latent variables (LVs) are extracted from the matrix X and Y and they contain most of variance of matrix X and Y, respectively. On the other hand, the last LVs mostly consist of noise and variations that are not related to X and Y. Importantly, the LVs are orthogonal to each other. These features together make it possible to compress information in the presence of collinearity and redundancy. Although PLS is a regression technique, it is a more important technique that visualizing ability enables us to probe search and data sets more minutely (Geladi and Kowalski, 1986; Höskuldsson, 1996; Rosen, 1998). PLS projects X and Y variables simultaneously onto the same subspace, T, in such a manner that there is a good relation between the position of one observation on the X-plane and its corresponding position on the Y-plane. Once a PLS model has been derived, it is important to grasp its meaning. For this, the scores t and u are considered. They contain information about the observations and their similarities/dissimilarities in X and Y space with respect to the given problem and model. X and Y weights provide the way how the variables combine to form t and u, which in turn express the quantitative relation between X and Y. Hence, these i= 1 i i T i Y 88

105 weights are essential for the understanding which X variables are important for modeling Y, which X variables provide common information, and also for the interpretation of the scores t. In order to detect the occurrence of process faults and disturbances, PCA-type monitoring is based on the statistical analytical approach of the score values and the residuals. The scores are monitored by using Hotelling s T 2 statistics or viewing the corresponding score plots directly. The residuals are monitored by Q statistics, that is, sum of squared prediction error of X variables (SPE X ). T 2 statistics is a measure of the distance from the multivariate mean to the projection of the operating point on the principal component (PC) plane. Q statistics is the Euclidean distance of the operating point from the plane formed by the retained PCs. T 2 monitors systematic variations in the latent variable space while SPE X represents variations, not explained by the retained PCs (Kourti and MacGregor, 1995; Wise et al., 1990;Wise and Gallagher, 1996; Teppola, 1999). However, the conventional MSPM method, such as T 2 and Q statistics, does not always function well, because it cannot detect the changes of correlation among process variables if T 2 and Q statistics are inside the confidence limits Generic Dissimilarity Measure (GDM) Recently, several dissimilarity indices with the distribution between two datasets have emerged (Kano et al., 2000; Choi et al., 2001; Yoo et al., 2001). They are based on the idea of that a change of process operation can be detected by comparing the distribution of successive datasets because the data distribution reflects the corresponding process operating condition. In previous section (5.2.1), we introduced and developed a generic dissimilarity measure (GDM) algorithm. It compares covariance structures of two datasets and represents the degree of dissimilarity between them by considering the importance of each transformed 89

106 variable (see the section 5.2.1) Multiresolution Analysis (MRA) PLS monitoring is different from the PCA-type monitoring algorithm. In PLS, principal component decomposition of X blocks should be rotated (by introducing the loading weights) in order to maximize the covariance between X and Y blocks. Therefore, these multivariate control charts are only approximations. A comparison of X block loadings and loading weights is one way to check at least a partial validity. In this case, there were no significant differences between the loading and the loading weights (Teppola, 1999). Therefore, it is required a new monitoring method for PLS monitoring that can effectively treat the peculiar characteristics of the biological treatment process and isolate and diagnose their fault sources with a multiscale approach. Figure 6.1 shows the scheme of multiresolution analysis (MRA) for PLS monitoring. In the first place, a PLS model is constructed with normal historical data in order to solve the multivariate and collinear problems in a biological WWTP. It is used to represent the process behavior and the common-cause variations of WWTP and excludes noise, measurement errors, and those variations that are uncorrelated to Y variables. Then, MRA for score values is executed by GDM and principal eigenvalues contribution to detect the process change and to diagnose or isolate different kinds of faults and disturbances. The motivation of this work is to identify the type of event which has occurred. It is believed that different events can result in different process measurement values, which could be projected into the change of data distribution and be manifested into different areas of the PC space. Here, each successive dataset in GDM consists of PLS score values with a moving window because PLS score values are normally distributed than the original variables themselves. This is a consequence of the central limit theorem, which can 90

107 be stated as follows: If the sample size is large, the theoretical sampling distribution of the mean can be approximated closely with a normal distribution. Thus, we would expect the scores, which are a weighed sum like a mean, to be distributed approximately normally (Neter et al., 1996; Wise and Gallagher, 1996). Figure 6.2 demonstrates the normality comparison between the original value and PLS score value. Therefore, as the abnormality will manifest itself as a shift or time series distribution change in the score value than the original variables. As the abnormality will manifest itself as a shift in the score plane like T 2 statistics of PCA and PLS monitoring, it will be shown in this case as a dissimilarity value between successive two datasets, that is, GDM. Exactly, a moving window concept of PLS score values for GDM is a remedy of nonstationary problem of the PLS monitoring algorithm. On the other hand, if the relationships between the process variables are rapidly changed and the correlation structure has a breakdown, SPE X of PLS residual error value should be included in two datasets of the proposed MRA algorithm. In this case, moving window matrices combined with score values and residual error values of PLS model are processed in the GDM and MRA method. Since the process inner relationship in WWTP, however, is slowly changed, only score matrices are sufficient for monitoring. Additionally, the confidence limit of individual eigenvalue is a proposal for multiscale fault detection and isolation. If each of eigenvalues exceeds to its corresponding confidence limit, the current process at that scale is changing and a certain event is occurring. By monitoring at each scale, we can diagnose diverse process variations and events, i.e., diagnosis of slow variations (seasonal fluctuations or other long-term dynamics), middle scale variation (internal disturbance, process operation change), and instantaneous variations (input disturbances, faults or sensor noises). Because it represents the corresponding characteristics at each scale, multiscale technique can discover information on the 91

108 scale where process changes, faults and events occur and analyze the physical/biological reasons. The proposed MRA automatically gives us the diagnosis and interpretation capability of events and fault sources. Note that it can get rid of nonstationary problem systematically by comparing successive datasets with a moving window concept. Moreover, it does not bring about the zero padding problems unlike other MRA, such as wavelet. 6.3 Result and Discussion PLS modeling The process data were collected from a biological WWTP that treated the coke wastewater of the iron and steel making plant in Korea (Figure 2.3). Eleven process and manipulated variables, X blocks, were used to model three process output variables, Y blocks. Y blocks consist of the solid volume index (SVI), the reduction of cyanide ( CN), and the reduction of COD ( COD). Table 6.1 describes the process variables and presents the mean and standard deviation (SD) values of X and Y blocks. The process data consisted of daily mean values from 1 January, 1998 to 9 November, 2000 with a total number of 1034 observations. The first 720 observations were used for the training of PLS model of mean-centered and autoscaled data. And the remaining 314 observations were used as a test data set in order to verify the proposed method. For the determination of the latent variable number of PLS model, a cross-validation method was used and four LVs were selected in PLS model. It managed to capture about 54% of the X block variance and 61% of the Y block variance by projecting the variables from dimension 14 to dimension 4, which is originated from the troublesome and difficult treatment of coke wastewater. The results of PLS model are represented in Table 6.2. An appealing feature of PLS method is the modeling ability, that is, predictive 92

109 capability. Figure 6.3 shows the real and predicted value from PLS model and displays the residual of Y blocks. The prediction values of the reduction of COD and the reduction of CN are explained very well in the test periods and manifest the prediction power of PLS model for the response Y variables. However, the prediction of SVI of secondary settler is not satisfied unlike the process quality variables. That may result from measurement inaccuracy and the operator s carelessness. It needs a precise measurement skill to the operator. The residual value of Y blocks shows the sum of differences between the real and predicted values for three response variables, which is mostly caused by the residual error of SVI prediction. Interpretation of PLS model For the interpretation of WWTP, we consider the PLS loading weights to see how X and Y variables are interrelated. Figure 6.3 represents that specific X and Y variables load strongly in the first two latent variables dimension, where COD 3, COD 2, and T aerator for COD reduction are closely correlated as seen in the left middle side of Figure 6.4. The first Y variable, COD reduction of WWTP is influenced by the COD load from BET2 and BET3 and the temperature in aerators which is certified by the heterotrophic biomass activity effects of the temperature in a biological treatment for the carbonaceous nutrients. The second group is formed by CN 2, CN 3, T influent, Q 2 and Q 3, and DO of aerator for CN reduction in the low section of Figure 6.4. It also presents that the reduction of cyanide is affected by the cyanide loads, influent flow rate, influent temperature, and dissolved oxygen level. Specially, microorganisms related with cyanide are counter-connected with the heterotrophic organisms and cyanide compounds are toxic and inhibitory to the growth of heterotrophs, which is shown in the opposite direction of each other in the loading plots. So, shock loading of cyanides in the wastewater influent causes a deterioration of the biological WWTP. Those facts are well reported with experimental results in 93

110 a technical paper (Lee, 2000). The third group is made up of MLSS_R and MLSS_%E for the SVI of secondary settler in the right upper side region, which exemplifies that the settleability of biomass is related to the microorganism amount (MLSS_R) and activity (MLSS_%E) included in the aerator and settler. Those are excellent results taken into account biological similarity and the fact that process layout represents. Sometimes it may be quite useful to overview the PLS weights with large number of latent variables, especially over 3 LVs. A real wastewater plant has generally more than 3 LVs, in our case, four latent variables. The variable influence on projection (VIP) informs us of the relevance of each X block pooled over all dimensions and Y blocks (Eriksson et al., 1995). Thus, VIP in square is a weighted sum of squares of the PLS weights, w, considering also the amount of Y variance explanted by each latent variable. VIP plot is shown in Figure 6.5 and this reveals that COD 3 is the most important variable, followed the T aerator, MLSS_R, MLSS_%E, and so on. This can be interpreted that COD influent from BET3 is most important to the plant treatment efficiency of the aeration basins and the settling ability of the secondary clarifier. In Figure 6.6, the monitoring results of T 2 and SPE X statistics are shown. The horizontal line corresponds to the 95% significance level of the training data. From this figure, we can see two deviations in the monitoring chart of T 2 and SPE X statistics. During samples 75 to 80 in the T 2 chart, statistics have been deviated slightly, which indicates that the deviations are large within the internal model. However, SPE X does not increase and it is an indication that the internal mutual relations are not altered according to PLS model. Since 15 days of solid retention time (SRT) from the T 2 deviation at sample 75, the SPE X chart begins to being changed from samples 90 to 110. In this event, the SPE X value represents a similar shape of the T 2 change, except that it occurs after a SRT period. And around sample 94

111 250, the T 2 statistics has a peak value, while SPE X is maintained in the vicinity of 95% confidence limit for a long time. We infer that the process has experienced the large transition in the operation at this time, but does not know its cause correctly. In order to identify more obvious cause for the deviation, the contributions from every measurement variable are calculated. Also, it cannot diagnose and isolate their fault (or disturbance) scale from the viewpoint of the process dynamics. This result arises from a weak point of T 2 and Q statistics. This is, although the events were within the confidence limits, the changes or upsets in the operating condition sometimes occurred in practice. This result illustrates that the principal components might undergo a change though the correlation structure is unchanged, when the variances of scores and residual error are similar to each other. Multiresolution analysis After the construction of PLS model, MRA was processed to the score matrix (T) of X blocks of PLS model. To monitor the process change or fault and event change, window and step sizes are 15 samples considering the SRT and 3 samples considering the hydraulic retention time (HRT), respectively. MRA to the PLS score values of the test dataset are shown in Figure 6.6. As shown in Figure 6.7(a), GDM started to change at sample 65 and deviated during around samples (March 3, 2000 April 27, 2000), where a large process change happened at this time. It shows more rapid and critical detection ability than the conventional MSPM method. Three eigenvalues which indicate their own specific scale disturbance are depicted in Figure 6.7(b-d). The remaining eigenvalues have little information and gives only high frequency information such as measurement noises. From Figure 6.7, we can know that the first and second eigenvalues largely contribute to the increase of GDM and are representative of middle scale disturbances. In detail, the process change is first detected in GDM, which is caused by the peaks of the second eigenvalue and then has experienced the 95

112 systematic variations of the first eigenvalue. It is easily identified and visualized by monitoring each eigenvalue pattern at two scales. At this time, WWTP received high input cyanide and COD load, while a small influent flow rate, that is, a highly concentrated load. It reduced the activity of the microorganisms and diminished the settling performance, then turned up the SVI increase in the secondary settler. From this result, it has been seen that sludge and floc formation changes due to high load and influent quality. Figure 6.8 shows the contribution plot at this time. From this result, we found that a large influent load broke out the external disturbance and were transformed into an internal disturbance, and then it changed the process operation region in the activated sludge process. Meanwhile, GDM deviated again from sample 230 to the last of test dataset (August 16, 2000 November 9, 2000). During the summer, WWTP was modified and a number of treatment equipments and facilities were appended. This made it feasible for operators to change the operation strategy which increased the MLSS concentration and maintained the high DO concentration. It invokes the large process changes, which is shown as a gradual increase of the first eigenvalue in Figure 6.8(b). This result confirms that it is distinctly better than other conventional methods for a multiscale process change in a nonstationary signal of unknown characteristics. They indicated that the proposed MRA could be effectively used to extract information resulting form the change in process operation and as a result could be contributed the localization of different process faults and events. 6.4 Conclusions In this research, a new approach of a multiresolution monitoring algorithm for the PLS model is presented in order to solve the distinctive problems in WWTP, such as collinear, multivariate, noisy, nonstationary, and multiscale. It is achieved by combining the PLS technique for the modeling and multiresolution analysis for the 96

113 monitoring. PLS model is used for the prediction and data analysis that take full advantage of the multivariate nature of the data and MRA of the PLS score and residual error value is utilized to detect and diagnose the fault and disturbance with a multiscale concept. It would give us the prediction, detection, and diagnosis power at a time and make the investigation about nonstationary and multiscale phenomena practicable. Experimental results from the industrial coke WWTP demonstrated that it had the prediction and analysis ability of a complex plant and simultaneously the suitable power of detection and isolation about various faults and events occurring in the biological treatment. Moreover, it can distinguish small failures from process upsets. 97

114 X Y PLS regrssion T GDM of moving score matrix Fault Detection U EV 1 EV... 2 EV n Confidence Limits Prediction Diagnosis Biplots & Contribution plot Figure 6.1 Multiresolution analysis for PLS monitoring 98

115 4 (a) 4 (b) expected value expected value original data score value (t 1 ) 100 (c) 100 (d) Figure 6.2 Normal probability plot and histogram of original data and PCA score values (a) normal probability plot of original data (b) probability plot of score values (c) histogram of original data (d) histogram of score values 99

116 140 (a) 40 (b) SVI CN reduction time (days) time (days) 1000 (c) 2.0 (d) COD reduction SPE Y time (days) time (days) Figure 6.3 Prediction results of the PLS model with real Y value (solid line with squares) and predicted value (dotted line) (a) SVI (b) reduction of CN (c) reduction of COD (d) residual error of Y variables (SPE Y ) 100

117 MLSS_R LV COD3 SVI_R MLSS_#E 0.3 COD_red COD2 0.2 DO_Aerator T_Aerator 0.1 Q2 Q3 0.0 CN2-0.1 CN_red CN3 T_Influent LV 1 Figure 6.4 The second PLS weight vector plotted against the first for the PLS model 101

118 COD3 T_aerator MLSS_R MLSS_#E COD2 DO_aerator CN3 T_Influent CN2 Q3 Q2 VIP Figure 6.5 Variable influence on projection (VIP) for the predictor variables 102

119 25 20 (a) T (b) time (days) 3 SPE X time (days) Figure 6.6 Monitoring performances based on T 2 and SPE X statistics with 95% confidence limits 103

120 0.10 (a) 6 (b) GDM 0.04 EV time (days) (c) time (days) (d) EV EV time (days) time (days) Figure 6.7 Monitoring performance of MRA for the PLS score values with 95% confidence limits 104

121 Figure 6.8 Contribution plot of PLS score value for the first event values 105

122 Table 6.1 Process Input/Output Variables in WWTP No Variable Description Unit Mean SD X 1 Q 2 Flow rate from BET2 m 3 /h X 2 Q 3 Flow rate from BET2 m 3 /h X 3 CN 2 Cyanide from BET2 mg/l X 4 CN 3 Cyanide from BET3 mg/l X 5 COD 2 COD from BET2 mg/l X 6 COD 3 COD from BET3 mg/l X 7 MLVSS at final aeration MLSS_%E mg/l basin X 8 MLSS _ R MLSS in recycle mg/l X 9 DO aerator DO at final aeration basin mg/l X 10 T influent Influent temperature C X 11 T aerator Y 1 SVI settler Temperature at final aerator C Solid volume index at settler mg/l Y 2 CN red Cyanide reduction mg/l Y 3 COD red COD reduction mg/l

123 Table 6.2 Variations explained by the PLS model of four latent variables X blocks (cumulative) Y blocks (cumulative) LV LV LV LV

124 VII. Process Monitoring for Continuous Process with Cyclic Operation 7.1 Introduction Recently, due to the increasingly stringent environmental regulations, advanced monitoring and control strategies for WWTP are attracting a lot of interests. However, some specific features about this process are yet to be addressed fully. First, most changes in this biological process are slow and recovery from failures can be very time-consuming and expensive. Sometimes it takes several months for the process to recover from an abnormal operation. Therefore, early detection of developing abnormalities is especially important for this process. Secondly, most WWTPs are subject to large diurnal fluctuations in the flow rate and compositions of the feed stream. So, these biological processes exhibit periodic characteristics, where strong diurnal fluctuations are observed in the flow rate and compositions of the feed waste stream. Since the variables of such processes tend to fluctuate widely over a cycle, their mean and variance do not remain constant with time. Because of this, conventional statistical process monitoring (SPM) methods like principal component analysis (PCA), which implicitly assume a stationary underlying process, may lead to many false alarms and missed faults. Better treatment performances can be expected by accounting for this periodic pattern when applying advanced monitoring and control strategies to this process. Recently it has been made several attempts to treat the characteristic features of a particular WWTP. First approach is solving a nonstationary problem, which are an adaptive PCA, dynamic PCA, PLS, a monitoring based on state space method, and so on. Second approach is solving a multiscale problem, which are a wavelet, multiscale PCA, multiresolution analysis, and so on. Rosen (1998) used conventional methods of PCA and PLS in the simulation 108

125 benchmark. For the monitoring purpose, a PCA model was built from a period of 14 days of dry weather condition with diurnal and weekly variations. Based on PCA model, it tested the storm, rain weather and decreasing nitrification rate. For the prediction model, they compared the linear PLS, dynamic PLS and nonlinear PLS method in predicting the effluent nitrogen concentration. Because it did not deliberate the non-stationary and periodic features of benchmark influent, T 2 and SPE chart also showed the periodicity and frequently revealed the false alarm and bad prediction performance. Teppola (1997) applied the dynamic PLS method for the purpose of process modeling and analysis in real wastewater plant. Twenty process and control variables were used to model four purification efficiency-related variables, i.e. the diluted sludge volume index (DSVI) and the reductions of chemical oxygen demand (COD), nitrogen and phosphorus. The data set consisted of daily values for each variable during two years. Dynamic PLS model was used for extracting relevant information form X to predict Y. The prediction performances were varied typically from 60 to 90%. DSVI has been explained very well in cross-validation compared to other effluent quality related variables. Some results of the prediction model were poor. Especially, some of high peaks of nutrient were difficult to predict because there was not enough information in those X-variables. Moreover, it did not include the seasonal variations of process. Rosen and Lennox (2000) pointed out the limitations of conventional PCA method, that is, stationary and one time-scale. The data used for the examples are industrial WWTP which are fourteen available variables form the on-line measurement system and sampling time is five minutes. For these solutions, they applied and compared the adaptive PCA and multiscale PCA. First approach, adaptive PCA, in terms of updating mean, variance and covariance structure overcomes the problems of non-stationary process data. The monitoring model is 109

126 continuously updated using an exponential memory function. These adaptations made most of the variation the score plane (T 2 ) from the residual plane (SPE). Second approach, multiscale PCA used a wavelet transform to decomposed measurement data into different time-scales and separate PCA models were used to monitor each scale. And for the interpretation of a disturbance, they recombined the scales to more physically interpretable scales, fast scale (hydraulic dynamics), medium scale (concentration dynamics) and slow scale (population dynamics). multiscale PCA increases the sensitivity of the monitoring and makes it easy to interpret the disturbance. But adaptive PCA has the limitations, such as adaptation into abnormal changes, test of information content (information windup) and the interpretation difficulty of updating the covariance matrix. MSPC makes the interpretation cumbersome by using a PCA model on each scale and its covariance structure is static which may introduce errors. But two approaches are too complex and simplest model should be used for monitoring. A trade-off between complexity and information should be considered. Teppola (2000) suggested the combined approach of PLS and multiresolution analysis (MRA). In this work, a PLS model was built for removing the collinearity problem and parsimonious modeling. Then the score values of the PLS model were processed by using wavelet and MRA to extract the process trends and to detect different kinds of fault and disturbance. It is shown how seasonal trends, faults, and disturbances can be separated and discriminated by observing at different scales and also diagnosed by studying biplots at multi-scale and computing variable contribution. In the next paper, Teppola (2001) presented the monitoring algorithm to remove the periodic seasonal fluctuation and long-term drifting problems, which these low-frequency variations mask and interfere with detection of small and moderate-level transient phenomena. By trending, this relatively common problem of autocorrelated measurements can be avoided. It first applies the wavelet filter to 110

127 the original data and detrends the low-frequency components and then constructs the PLS model. These particular data, the PLS monitoring results are shown to be superior compared to conventional PLS model. This is because it removes lowfrequency fluctuations and results in a more stationary filtered data set that is more suitable for monitoring. Although these processes are non-stationary, their dynamic behavior tend to repeat from cycle to cycle and hence their cycle-to-cycle behavior may be assumed stationary. Hence, it is plausible to calculate and use different means and covariances for different time points within each cycle. One can also establish correlations among the samples at different time points of a cycle, much like in Multiway-PCA (M-PCA) used for batch process monitoring (Nomikos and MacGregor, 1994, 1995). Beyond that lies the possibility to capture correlation in the variations from cycle to cycle for quicker detection of small mean shifts and slow drifts. An efficient way of doing this is to describe the variations (from their mean behavior) by a periodically time-varying (PTV) state-space model. However, PTV system models are difficult to build and we need a systematic framework for it. In order to provide this monitoring ability, we propose the monitoring method based on state space model to capture and use the period-to-period correlation structure. 7.2 Theory Model development To make the modeling task manageable, we adopt the technique of lifting. In the lifted form of a PTV model, all samples within one cycle are collected as a single vector (Dorsey and Lee, 2001). Let y k (t) represent the vector of the mean centered and scaled process measurements available at sample time t during cycle k. y k (t) R ny where n y is the number of variables selected for monitoring purposes 111

128 (including the controller outputs and process outputs). Assume that there are N time samples in each cycle. Then, the lifted vector looks like Y T T T [ y ( 1), y (2), L, y ( N ] = (7.1) k k k k ) First, a cycle-to-cycle invariant model is constructed using the subspace identification technique (Van Overschee and De Moor, 1993, 1994, 1996; Dorsey and Lee, 2001). x k +1 Y k = = Ax + K e k Cx + e k k k (7.2) where A, K, C are the system matrices, Y k is the lifted vectors of all data made for the k th cycle, x k is the state sequence that is extracted the process data based on the relevancy of the previous cycle measurements for predicting the future cycle measurement, that is, the state is defined to be a holder of information from previous cycles that is relevant for predicting future cycles, e k is the innovation vector that is the residual between the process data and its estimate, and Ke k. is a stochastic input vector as a state disturbance. The dimension is usually very high for Y k (N*n y ) and there exist strong correlations among its elements. To reduce the dimension and facilitate the identification step, PCA can be applied to Y k to obtain score vector r Y k. Now the stochastic state-space model can be constructed with the reduced output vector. x k + 1 Y r k = = Ax + Ke k r C x + e k k k (7.3) Y = ΘY + E (7.4) k Based on the above cycle-to-cycle system model (7.3), a time evolution PTV model can be constructed for the on-line monitoring purpose. r k k 112

129 113 ) ( ) ( ) ( ) ( ) ( 1) ( t t x t H t y t x t x k ε k + = = + k k k (7.5) [ ] r C t I t H Θ = L L ),0, (, 0, ) ( (7.6) where ε k (t) includes both the appropriate elements of e k and the residual E k from PCA. The cycle-to-cycle transition is then described as Ke k N Ax x + = + -1) ( (0) 1 k k (7.7) Hence, the terminal state of one period becomes the initial state of the next period. This means the states of successive periods are naturally connected through the dynamics of the process. This is the main difference from the case of batch systems where the sate is reset at the start to each run. However, this model formulation may not be valid for Kalman filter implementation, since the residual could become correlated series, instead of white noise, that is, the state noise Ke k and the output noise ε k are correlated. To deal with this, one can use the following augmented form of the model: k r k r r k e I K Y x C A Y x + = k k [ ] = r k r r k Y x C Y 1 0 k (7.8) where [ ] Γ = = = = + + I K, C H, C A, Y x z r r r k Φ 1 1 k k Then, within a cycle, the time model becomes = + r k r k Y t x Y t x 1 ) ( 1) ( k k [ ] [ ] ) ( ) ( 0 ),0, (, 0, ) ( t ν Y t x I t I t y PCA r k k k + = L Θ L (7.9) where

130 Finally, cycle transition model is, xk Y r k [ 0, L, I( t),0, ] [ I ] H ( t) = L Θ 0 (0) (0) = A C r 0 x 0 Y + 1 k r k 1 ( N-1) K + e N I ( -1) k Y r k = x ( N-1) r k [ C 0] e k ( N-1) (7.10) Now, a periodic Kalman filter can be applied to the model (7.9) and (7.10). The PTV Kalman filter can be designed using the standard equations to update the state and the output score vectors recursively on the basis of incoming measurements. Assume Φ, Η, Γ, Q, R represent the state transition, output, state noise coefficient, state noise covariance, and measurement noise covariance matrices respectively. The Kalman filter equation is as follow, [ y( t+ 1) - Hxˆ ( t 1 )] xˆ ( t+ 1 t+ 1) = Φ xˆ ( t+ 1 t) + K( t+ 1) + t T [ HP( t+ 1 t) H + R( + 1) ] 1 T K( t+ 1) = P( t+ 1 t) H t P( t+ 1 t) = Φ P( t t) Φ T + ΓQ Γ T [ I- K( t+ 1) H ] P( t 1 ) P ( t+ 1 t+ 1) = + t (7.11) When t = 0,, N-2, Φ=I, Q=0, H= H (t), R can be estimated from the PCA residual. When t = N-1, Φ = Φ, Q is the error covariance matrix obtained from subspace identification, H = H, R = 0. Process Monitoring Measure The Hotelling s T 2 monitoring statistics based on state space has been proposed by Negiz and Cinar (1997). The Hotelling s T 2 statistics is a metric that includes information for both mean and covariance structure of the state variables. 114

131 2 T 1 ( N 1) n Tk = xk Σ xk ~ Fα ( n,n n) (7.12) ( N n) where subscript k indicates time, N is the number of samples and n is the dimension of the state vector x. The T 2 is obtained by assuming that the state variables follow a Gaussian distribution, that is, zero mean and multinormal distribution with estimated covariance matrix Σ while they are orthogonal at zero lag. F α (n, N-n) is the upper 100α% critical point of the F-distribution with n and N-n degree of freedom, which can be used to establish control limit with significance level α. The updated state vector x and the output score vector Y r can be monitored separately using T 2 statistics. Note that T 2 monitoring of Y r amounts to the periodby-period M-PCA monitoring implemented in a real-time manner. The monitoring of state vector x(t) could give extra information about small mean shifts or slowly developing abnormalities that are hard to detect with PCA. That is, the state has an information form previous cycles that is relevant for predicting future cycles. Therefore, a monitoring algorithm built around the developed state space model gives a efficient ability to detect abnormal deviation in the process dynamics from cycle-to-cycle dynamics point of view as well as deviation in process measurements (Dorsey and Lee, 2001). On the other hand, in many chemical and biological processes, some quality variables cannot be measured on-line and their lab measurements are available only after long delays. And advanced nutrient sensors are expensive in the aspect of cost and maintenance. In such cases, inferential sensing can be very useful for on-line monitoring and control. An added advantage of the proposed framework that it is very easy to implement inferential sensing. For the purpose of inferential sensing, the quality variables q y k can be augmented with all other measured output before doing PCA, such as, 115

132 T q T T q T q T [ y ( 1), y (1), y (2), y (2), L, y ( N ] Y = ) (7.13) k k k k Note that quality measurements do not have to be available at a same rate as process measurements. All the presented model formulations remain the same. Once the model is built, the Kalman filter can be designed to use whatever measurements available at each each time. Inferential predictions on the quality variables can be obtained just by picking out the proper element after the Kalman filter estimate and transforming it back from the score space to the original full space. x ( t) r k [ 0, L, I ( t),0, ] Θ[ C I ] q L k ŷ q ( t) = (7.14) ek ( t) k 7.3 Simulation Study The proposed a process monitoring method for cyclic process is tested on data generated form a simulation of benchmark plant. The exiting dry weather data set in benchmark may not be proper for constructing cycle-to-cycle model. There are few variations from day to day influent data. While, constructing dynamic day to day model needs data which contains enough variations and correlations form day to day. And benchmark data is not long enough. If considering to construct daily model, only 14 days of data are available, which is far from enough, especially for later subspace identification. To generate data which shows day to day correlation as well as some in-cycle correlation a variation (or disturbance) model is used of both S NH,in and Q in. The used functions are: For time model variation For cycle-to-cycle model, E ( t+ 1) = ae ( t) e( t) (7.15) k k + 116

133 θ 1 = b + E (7.16) k + θ k where θ k is a vector which stores all the variations of each sample. Through the simulation, data set in-cycle correlation would have only 4 times a day which means every 6 hours, a variation is assumed to occur. It comes from that too much in-cycle correlation would require more PCA scores to capture the whole cycle dynamics which will complicate the subsequent state space modeling and makes the PTV Kalman filter hard to follow the trend. We generated 300 days normal data set with influent data file, which used last 200 days data as modeling data set for PCA and N4SID and used any part of first 100 days as prediction test. Figure 7.1 show the measured variables of the first 10 days of normal data set, which has a cyclic and diurnal variation. Two disturbance scenarios were simulated. The first deals with slowly linear decrease of nitrification rate. The second disturbance is a small mean (step) decrease of nitrification rate. Table 7.1 represents the simulation conditions of two disturbance cases. For the comparisons, we applied the general PCA and PLS method for both original measured data set and data set with subtracting mean variation. For a data set with subtracting a mean variation, PCA technique is applied to the remaining dataset which averaging trajectory within a day is subtracted. So, periodicity is removed within one day, which normal trajectory within a day is subtracted averaging within one day (96 sample time). And then PCA model for the auto-scaled data set with mean zero and unit variance is treated. Monitoring and quality prediction performances are compared with those of the static PCA and PLS methods. The variables used to build the X-block in the disturbance detection were the influent ammonia concentration (S NH,in ), influent flow rate (Q in ), nitrate concentration in the second aerator (S NO,2 ), total suspended solid in aerator 4 (TSS 4 ), DO concentration in aerators 3 and 4 (S O,3, S O,4 ), oxygen transfer coefficient in aerator 5 (K L a 5 ), and internal recirculation rate (Q int ). A PCA k 117

134 model of 95% and 99% confidence limit is built from training periods of 200 days of dry weather dataset (normal operation). Three PCs are selected for PCA model and captured variability is shown in Table 7.2. Figure 7.2 shows the T 2 and SPE plot of conventional PCA method during first 10 days of normal data set. During normal operation with diurnal influent, the PCA monitoring result shows apparent cyclic and non-stationary characteristics. This non-stationary (periodic) behavior of T 2 score values is the cause of false alarm and missed fault by widen confidence limit. Monitoring of nitrification linear decrease case Figure 7.3 shows the general PCA monitoring results with stationary statistics assumption for nitrification linear decrease case. T 2 and SPE plot which focuses on during samples show the bad monitoring results for this linear shift type disturbance, where T 2 plot cannot detect this type of event and SPE plot has a delay about 200 samples. It originated from the widened confidence limit of their mean behavior. For these non-stationary (periodic) behaviors, it tends to give false alarm at peak value, but not sensitive at lower value period. Figure 7.4 shows the general PCA monitoring results with subtracting mean variation for nitrification linear decrease case during samples. The control limits of PCA method for cyclic removal data set have higher control limits than limits of the conventional dataset. T 2 and SPE plot for nitrification linear decrease case show better monitoring performances than the previous data set, which T 2 plot shows the removal of the periodic variations and shows some other variations. From the Figure 7.4, we know that T 2 measure shows a delayed detection performance and SPE plot has a delayed detection about 100 samples. However, T 2 and SPE plot still have some false alarms. The monitoring result of the proposed method is shown in Figure 7.5. The 95% confidence limits is calculated and is shown by the horizontal dotted line on the plots. The result represents a high sensitivity of the proposed state space monitoring 118

135 method, where T 2 around the state vector can immediately detect the linear shift event of the nitrification rate. The earlier detection capability of the proposed method could allow for a corrective action and operation change before any serious situations such as the biomass decay and sludge bulking would be occurred. Monitoring of nitrification step decrease Figure 7.6 shows the conventional PCA monitoring results for small step decrease of nitrification rate. PCA model shows the bad monitoring result for this step type disturbance, which both T 2 and SPE plot cannot detect the disturbance of small step decrease. It originated from the widened confidence limit of their mean behavior. For non-stationary behavior, it tends to give false alarm at peak value, but not sensitive at lower value period. Because small decrease of nitrification rate is introduced at lower value time, PCA monitoring method which is based on the average of the whole trajectory of each variable cannot detect this type of small mean shift. Figure 7.7 shows the general PCA monitoring results with subtracting mean variation for nitrification step decrease case during samples. Here, the normal trajectory within a day is subtracted. As the previous case, T 2 and SPE plot of small step decrease show slightly better monitoring performances than the original data set. But both T 2 and SPE plot have a delay about samples and also have some false alarm. It comes from that PCA method with subtracting variant mean only considers the time variant mean and assume the constant variance which can be time variant. The monitoring result of the proposed method is shown in Figure 7.8. The result represents a high sensitivity of the proposed state space monitoring method, where T 2 around the state vector can immediately detect the small step decrease of the nitrification rate. The earlier detection capability of the proposed method could allow for a corrective action and operation change before some undesirable 119

136 situations such as the reduced plant efficiency and the decreased settling ability would be occurred. Inferential sensing Quality prediction performances of inferential sensing are compared with those of PLS method for both original data set and data set with subtracting mean variation. We made 300 days normal data set, where used last 200 days data as modeling data set for PLS and used any part of first 100 days as validation test. Because we are interested in the prediction ability based on integrated monitoring model in this research, we constructed the inferential model which was based on 200 days of normal operation and did not consider any time lags between the input and output. Quality variables are the effluent ammonia and nitrate, S NH,e and S NO,e. Linear PLS model of original data set is built for the prediction of quality variables, where four latent variables (LVs) are selected. The prediction results of validation data set are shown in Figure 7.9. PLS model shows bad prediction results of both S NH,e and S NO,e which has nonlinear and periodic dynamics. Linear PLS model cannot model these nonlinear process behaviors. And then, PLS model for the periodic removed data set is built, where four LVs are also selected. Compared to the previous PLS model, Figure 7.10 shows highly good prediction performance. It means that subtracting the average trajectory from the periodic process removes the major nonlinear behavior of S NH,e. Finally, Figure 7.11 shows the prediction performance of the proposed inferential sensing. Table 7.3 compares the mean square of prediction error (MSE) of three prediction methods. As PLS prediction for periodic removed data set, it removes the periodic dynamics and shows the good prediction performance. This is not a surprising result as the proposed method captures the most variability of the normal data set and the main dynamics of the system. So, the proposed method offers an advantage for the process monitoring as well as a chance to predict the quality variable in integrated model. 120

137 7.4 Conclusions In this research, we propose a monitoring method based on state space models for diurnal cyclic characteristics in domestic WWTP. A state-space model is identified to extract within cycle and between-cycle correlation information from historical data using subspace identification method. First, a cycle-to-cycle invariant model is constructed using the subspace identification method. Second, time-varying Kalman filter model is constructed for on-line monitoring. For the purpose of inferential sensing, integrated framework augmenting the quality variables is also suggested. Simulation results show that the proposed method is an appropriate monitoring technique for WWTP with cyclic operation. Specially, it can detect more rapidly the changes of cycle-to-cycle behavior, linear decrease of nitrification, and more accurately detect mean shift, step decrease of nitrification, than conventional monitoring methods. 121

138 Figure 7.1 Measured variables of the first 10 days of normal data set 122

139 14 Value of T 2 with 95 and Process Residual Q with Value of T Sample Number Residual Sample Number Figure 7.2 Conventional PCA monitoring result during the first 10 days of normal data set 123

140 Figure 7.3 Conventional PCA monitoring result for nitrification linear decrease: T 2 and SPE plot 124

141 Figure 7.4 PCA monitoring result with periodic removal for nitrification linear decrease: T 2 and SPE plot 125

142 Figure 7.5 Monitoring result of the proposed method for nitrification linear decrease 126

143 Figure 7.6 Conventional PCA monitoring result for nitrification step decrease: T 2 and SPE plot 127

144 Figure 7.7 PCA monitoring result with periodic removal for nitrification step decrease: T 2 and SPE plot 128

145 Figure 7.8 Monitoring result of the proposed method for nitrification step decrease 129

146 Figure 7.9 Prediction results of S NH,e and S NO,e for validation data set with static PLS method 130

147 Figure 7.10 Prediction results of S NH,e and S NO,e for validation data set with static PLS method (after periodic removal) 131

148 Figure 7.11 Prediction results of S NH,e and S NO,e for validation data set with the proposed method 132

149 Table 7.1 Two disturbances in the benchmark Disturbances Linear/step Simulation conditions Decreasing nitrification rate Decreasing nitrification rate Linear Step Specific growth rate for autotrophs: from 0.5 to 0.4 day -1 in a linear fashion during samples 288 (3 day) 480 (5day) Specific growth rate for autotrophs: from 0.5 to 0.47 day -1 in a step fashion during sample 288 (3 day) 133

150 Table 7.2 Percent variance captured by PCA model Original Periodic removal PCs % Variance captured this PC % Variance captured total % Variance captured this PC % Variance captured total

151 Table 7.3 MSE of two PLS methods and the proposed method Model type MSE Conventional PLS Periodic removal PLS The proposed method

152 VIII. Simultaneous Prediction and Classification in the Secondary Settling Tank 8.1 Introduction Since the environmental restriction becomes harder and harder nowadays, the increasing effort for higher effluent quality from wastewater treatment plant is required in the advanced monitoring of plant performance. To reduce the effluent pollutants from the wastewater treatment plant (WWTP), it should precede any other procedure to analyze the current state of each unit plant of WWTP. Most of the changes in WWTP are slow when the process is recovering back from a bad state to a normal state. Early fault detection and current status classification in the biological process are very efficient to execute corrective action well before a dangerous situation starts. So, the monitoring and multivariate analysis of activated sludge process or the secondary settler have long been noted (Olsson et. al., 1988; Hasselblad et. al., 1996; Stefano, 1998; Teppola et. al, 1997, 1999; Van Dongen et. al., 1998; Rosen et. al., 1998). Secondary sedimentation in the WWTP separates the biomass from the treated wastewater and its performance is crucial to the operation of an activated sludge system. Its operation depends on the status of sludge that relies on many other parameters such as temperature, organic loading, influent flow rate, and flock properties. The solid volume index (SVI) primarily describes the settling properties of the sludge. High SVI values are indicators of a bulking state and overgrowth of filamentous microbes, which is one of the major upsets of the activated sludge process leading to the deterioration of the purification efficiency. Therefore, the prediction of SVI value is very important from the viewpoint of the strategy of the settler operation. In this research, we forecasted the solid volume index (SVI) of the secondary 136

153 settler using an adaptive RLS method. ARX model parameters are proposed and verified as a good feature in a secondary clarifier monitoring by observing the evolution of ARX model parameters through power spectrum. The capability of monitoring the secondary clarifier is illustrated with the application of a neural network classifier which, combined the adaptive processing scheme, proved to be suitable for the monitoring and classification application in the real wastewater treatment plant. 8.2 Theory Monitoring system in the secondary settler has been developed based on a neural network of parallel distributed processing, powerful learning and generalization capability of the pattern information. This system is composed of three fundamental parts. First, auto-regressive with exogenous input (ARX) models of time series data of SVI value in the secondary settler have been used to predict the SVI value. Its parameters are adaptively estimated by RLS method and provide feature vectors. Its ability is showed by the power spectrum analysis of ARX parameters. Second, in the classification process, we design neural network classifier to identify the current state of the secondary classifier. After training the neural networks, we can recognize the state class of the settler from the values of output nodes is chosen according to the maximum selection rule. Third, in the structure of the classifier, we decide the optimal number of hidden nodes using GA. SVI prediction with RLS method To model SVI of the secondary settler, we introduce the identification methods for the ARX model using recursive least square (RLS) method (Ljung, 1987; Ko and Cho, 1996). A general form of the discrete ARX model is as follows. y t) + a y( t 1) + L + a y( t n ) = b u( t 1) + b u( t 2) + L+ b u( t n ) + e( ) (8.1) ( 1 n a 1 2 n b t a b 137

154 The objective of the ARX is to estimate the adjustable parameters of a i and b i to minimize the difference between the predicted process output and the measured process output. Because the secondary settler is time varying process and has inherently dynamic characteristics, it is required to use the adaptive ability. For this purpose, we use the RLS algorithm that makes the modeling technique well suited for time varying environment. The RLS algorithm is as follows. ˆ( θ t) = ˆ( θ t 1) + K( t)( y( t) yˆ( t)) T yˆ( t) = ϕ ( t) ˆ( θ t 1) K( t) = p( t 1) ϕ( t) T λ + ϕ ( t) P( t 1) ϕ( t) P( t 1) ϕ( t) ϕ ( t) P( t 1) P( t 1) T λ + ϕ ( t) P( t 1) ϕ( t) P( t) = λ T (8.2) where K(t) is an adaptation gain, θ(t) parameter estimation at time, ŷ(t) is the prediction value based on observations at time t-1, ϕ(t) regression vector, λ forgetting vector, and P(t) covariance matrix of estimates. This recursive form is very convenient for updating the model at each time, so that the model follows the gradual change in the characteristic of the settler process. If the parameters of ARX model are well tuned, a change in dynamic characteristics of the settler process will cause gradual change in the parameter vector and prediction error. Therefore, the status of the secondary settler can be observed by a gradual change in ARX model parameters. Power Spectrum In order to see the sensitivity of ARX coefficients of SVI at each state and verify its discriminant ability, the comparison of the power spectrum at each state is required. The power spectrum of a stationary process is defined as the Fourier transform of its covariance function (Ljung, 1987). While a deterministic signal can be expressed as a mixture of sine and cosine functions at different frequencies, a 138

155 time series response or stochastic system response of a function of time doesn t belong to the class of functions dealt with in the usual Fourier transform theory. The frequency decomposition of these random functions can be obtained by taking the Fourier Transform of the auto-covariance function for which the usual Fourier transform can be used. For stochastic process, y(t) can be given by y ( t) = G( q) u( t) + H ( q) e( t) (8.3) where u(t) is a quasi-stationary, deterministic signal with a spectrum, and e(t) is white noise with a variance. Let G(q) and H(q) be stable filters. Then y(t) is quasistationary and 2 iw iw Φ ( ω ) = G( e ) Φ ( ω) + σ H ( e ) (8.4) y Φ yu u 2 a iw ( ω) = G ( e ) Φ ( ω) (8.5) where Φ y (ω) is a power spectrum of y(t) and Φ yu (ω) is a cross spectrum of y(t) and u(t). It should be noted that this type of spectrum estimates is inherently smooth because they are obtained based on a parameter representation of the system. The result has a physical interpretation, where G(e iw ) 2 is the steady-state amplitude of the response of the system to sine wave with a frequency. The value of the spectral density of the output is then the product of the power G(e iw ) 2 and the spectral density of the input Φ u (ω). If the power spectrum was separated and had a dissimilarity value at a different state, analyzing the power spectrum of ARX parameters can make the decision on the state of the secondary settler. Pattern classification (neural network) While different states are not completely separable in the original input and output dimensional space under a wide range of conditions, the classes become separable in the dimensional feature of ARX parameters space using neural network u 2 139

156 classifier that has the ability of nonlinear mapping. So, ARX parameters were used as input features for the neural network classifier. The explanation of neural network classifier is as follows. Pattern recognition methods such as neural networks are important for the classification problems because they do not require accurate process models, which are often difficult to obtain for many biological and chemical processes. And neural network computing ability outperforms the conventional statistical approach in many engineering application because of its non-linear transformation (Bishop, 1995; Lin and Lee, 1996; Haykin, 1999). Neural network maps a set of input patterns (e.g., process operating conditions) to respective output classes (e.g., categorical groups). We use an input vector and an output vector to represent the input pattern and output class, respectively. The output vector, y, from the neural network is bipolar, with -1 indicating that the input pattern is not within the specific, and 1 indicating that it is within a specific class (e.g., -1 = not in class I; 1 = in class I). The actual output from the neural network is a numerical value between -1 and 1, and can be viewed as the probability that the input pattern corresponds to a specific class. The output vector (y) contains three possible classes, that is, y={class I, class II, class Ⅲ}. Note that for every point within the input space, there must be only one class specified. In this paper, we have only three possible output vectors for training the network, for example, y = {[1,-1,-1], [-1,1,-1], [-1,-1,1]}. After calculation of the neural network classifier output, the values of output nodes are passed to the maximum selector. The output node selected by the maximum selector gives information on the class that includes a current input. In theory, for an M-class classification problem in which the union of the M distinct classes forms the entire input space, we need a total of M outputs to represent all possible classification decisions. It can be expressed as follows. If y i (x j ) > y k (x j ) for all k (k=1,2,,m : k i), then x j s i (8.6) 140

157 where x j is jth input vector, y i is the ith output node value of the neural network classifier for input x j, s i is the ith state of secondary clarifier, and M is the number of output nodes. A unique largest output value exists with probability 1 when the underlying posterior class distributions are distinct. Genetic algorithm (GA) The GA is a derivative-free stochastic optimization technique in which the stochastic search algorithm is based on the idea of the principle of natures such as natural selection, crossover, and mutation (Marsili-Libelli, 1996; Wang, et al., 1998). One of the GA s characteristics is the multiple points search, which discriminate the GA from other random search methods. In this paper, the string, which is a model of chromosome, represents the number of hidden layer of the neural network. The GA typically starts by randomly generating initial population of strings. Each string is transformed into the fitness value to obtain a quantitative measure. On the basis of the fitness value, the strings undergo genetic operations. The goal of genetic operations is to find a set of parameters that search the optimal solution to the problem or to reach the limited generation. Since the ultimate objective of a pattern classifier is to achieve an acceptable rate of correct classification, this criterion is used to judge when the variable parameters of the neural network are optimal. In addition, GA is the useful tool to select features for neural network classifiers. For example, GA can be used to learn or train neural network structure or to initialize the reasonable weight that is generally assigned randomly. This paper uses the hybrid algorithm for the optimization of the neural network structure using GA in order to improve the behavior and the design of neural networks. GA was used to find the optimal number of hidden nodes. Hierarchy structure 141

158 Monitoring system in a secondary settler has been developed based on a neural network of parallel distributed processing, powerful learning and generalization capability of the pattern information. This system is composed of three fundamental parts. These are an adaptive ARX estimation processing, neural network classification and maximum decision rule making part. Figure 8.1 represents the schematic diagram of the proposed hierarchy structure. First, adaptive estimation processing is performed to predict SVI value and provide feature vectors. That is, ARX model has been used to predict SVI value in a secondary settler. Its parameters are adaptively estimated by RLS method and input vector of the neural network classifier. Second, in the neural network classification, the feature vectors are associated with the desired output decision. In the structure of the neural network classifier, we decide the optimal number of hidden node using GA. After training neural networks, the classifier output is calculated by the trained weight. Third, in the maximum decision rule, only one of the values of the classifier output is chosen according to the rule "the minority is subordinated to the majority. 8.3 Simulation Study In this research, we used the industrial wastewater treatment facility data of the iron and steel making plant in Korea. It is a general activated sludge process that has five aeration basins and a secondary clarifier. Figure 2.3 shows the layout of the WWTP. The data set consisted of daily mean values from January 1, 1997 to December 22, The data are divided into two parts. A training set consisted of the values during first two years and a test data set during the remaining one year are used to see how the monitoring proceeds with the proposed algorithm. First, the ARX model structure is as follows. Its inputs are four which are the 142

159 influent flow rate, influent COD, dissolved oxygen (DO) of the final aeration basin and mixed liquor suspended solid (MLSS) in the final aeration basin. Output variable is SVI of the settler. The state of a secondary settler is divided three classes that were judged by the experienced operator. The choice of the order of the model is a non-trivial problem that requires trade-off between precise description of data and model complexity. We determined the model order using cross-validation and numerous simulation. The prediction model uses ARX structure whose parameter is adapted by RLS with the forgetting factor, where order of AR part is 3 and the order of each exogenous input is 2. The applied ARX model has a following form. b 1,1 + b y( t) + a u ( t 1) + b 3,1 1 u ( t 1) + b 3 1 y( t 1) + a 1,2 u ( t 2) + b 3,2 1 y( t 2) + a u ( t 2) + b 3 2 2,1 y( t 3) = u ( t 1) + b 4,1 2 u ( t 1) + b 4 3 2,2 u ( t 2) 4,2 2 u ( t 2) 4 (8.7) where y(t) is SVI, u 1 (t) is influent flow rate, u 2 (t) is influent COD, u 3 (t) is DO and u 4 (t) is MLSS. To remove data redundancy, we normalize the raw training data. The RLS method uses the dead-zone method to remedy the estimation windup. Figure 8.2 shows the result of the one-step ahead prediction value of SVI that forecasts reasonably. The dot point is real value and solid line is the prediction value. In order to see the sensitivity of the ARX coefficients at each state, the parameter values of each state were shown in Figure 8.3. In this Figure, ARX parameters have different values according to each state, which means that the decision on the state of the secondary clarifier can be achieved by quantitatively analyzing the ARX parameters. To conform the difference between parameters in each class theoretically, we display the power spectrum analysis of the parameters in the Figure 8.4. Second, neural network classifier has a MLP structure with two hidden layer, which its nodes are decided by GA. To speed up training and stabilize the learning algorithm, we use the momentum term, adaptive learning rate, normalized weight 143

160 updating and batch learning techniques. The neural network is trained using three patterns according to the state of a secondary settler. The number of ARX parameters, which is used as the input variables of neural network, is eleven. And other operating conditions can be taken as additional features to compensate for sensitivity of the ARX parameters to the variation of operation conditions, such as toxic occurrence, aeration basin status. The simulation results showed little improvement of classification ability. In this paper, we did not use this additional information for the clarity. The input features were normalized in [-1, 1] ranges in order to prevent saturation of an activation function. The corresponding target values of output nodes were set to normal state (0.9, -0.9, -0.9), bad state (-0.9, 0.9, -0.9), bulking state (-0.9, -0.9, 0.9) for each state of three classes. In the application of GA for the structure of a neural network, the initial population size of parents was 30 and generation number was 100. Ranked-base selection as a selection operator, and mutation and uniform crossover as a search operator were used. We have set the mutation rate for 0.01 and crossover rate for 0.6. GA can find the optimal number of each hidden node quickly, because the search space is small. The number of first and second hidden layer is 7 and 4, respectively. In this experiment, the neural network with two hidden layers have a better result than with only one hidden layer. In addition, three or more hidden layers have no improvement of performance. In testing mode, the maximum value of neural network classifier outputs was chosen in determining the present states. It indicates what state is the current state. The test data has not a bulking state but only the normal and bad state. Table 8.1 shows the confusion matrix from the result of the test set using the neural network classifier. This is a matrix A whose (i, j) element is the number of vectors that originate from the ith distribution and are assigned to the jth cluster. Though output values don t completely agree with the corresponding desired outputs, they are reasonable to recognize the present state. From the trained neural network, the 144

161 classification rate was about over 80.9% on an average, even though the system was tested under a wide range of operating condition. Because the process has an abrupt load variation during the latter part of test set, the misclassification rate was higher in this period. 8.4 Conclusions The recognition of the process state of a secondary settler is very important in the operation decision. We can monitor the current state through the mixed structure of ARX model and neural network classifier. We found the optimal structure with second order of ARX model and neural network with three layers. From the experiment, a strong correlation between the settler states and the values of the ARX parameters could be used as effective features for secondary clarifier monitoring. The training and decision making for pattern recognition were successfully performed through neural network classifier. The proposed method is useful to predict the SVI value of the secondary settler and to classify the current state of a secondary settler simultaneously. And the suggested method can also be used as the classifier of the other process in the wastewater treatment plant. 145

162 SVI Prediction Number of Hidden Nodes Evolution Fitness y 1 Inpu ARX MODEL Featur e y 2 y 3 MAXIMUM DECISION MAKING Class s i Reproduction of Figure 8.1 Schematic diagram of the proposed hierarchy structure 146

163 SVI, ml/g data Figure 8.2 One-step ahead prediction value of SVI using RLS method 147

164 2 Parameter Value Normal Bad Bulking Model Parameter Figure 8.3 Sensitivity of the ARX model parameters of each state 148

165 5.5 (a) 15 (b) 8 (c) Power frequency(ω) frequency(ω) frequency(ω) Figure 8.4 Power spectrum in each state (a) normal (b) bad (c) bulking state 149

166 Table 8.1 Confusion matrix of the test data Predicted True Normal Bad Bulking Normal Bad Bulking 0 150

167 IX. Nonlinear Fuzzy PLS Modeling 9.1 Introduction Statistical data analysis has been widely used in establishing models from experimental or historical data. Typical problems in multivariate statistical analysis are high dimensionality and collinearity in a sparse sample data set. The partial least squares (PLS) modeling method is one of the most useful measures for overcoming these problems. PLS is a multivariate statistical data analysis and regression method which uses projection into latent variables to reduce high dimensional and strongly correlated data to a much smaller data set that can then be interpreted. The PLS method is used in a variety of areas where multivariate data emerge, both in the laboratory and in the real world. Typical lab-scale examples are multivariate calibration, and quantitative structure-property and compositionproperty relationships. Real world examples include the monitoring of industrial and environmental processes, geochemistry, and clinical, atmospheric, and marine chemistry. As PLS uses a statistical data reduction and regression algorithm, it is employed primarily in data analysis (Teppola et al., 1997, 1998; Rosen and Olsson, 1998; Wikström et al., 1998). Although the original linear PLS (LPLS) regression method provides good remedial measures to the problems of correlated inputs and limited observations, it has the major limitation that only linear information can be extracted from data. Since many practical data are inherently nonlinear, it is desirable to have a robust method that can model any nonlinear relation. A successful step towards nonlinear PLS modeling was the quadratic PLS (QPLS) method proposed by Wold et al. (1989). In QPLS quadratic functions are used for the inner regression in PLS. However, the nonlinearity of the QPLS method is very limited. To create a PLS method of greater nonlinearity, several more generic approaches have been 151

168 developed such as spline PLS (SPLS), neural networks PLS (NNPLS), and locally weighted regression PLS (LWR-PLS) (Wold, 1992; Qin and McAvoy, 1992; Centner and Massart, 1998; Baffi et al., 1999). As their names suggest, SPLS uses spline inner models and NNPLS uses neural networks inner models. LWR-PLS uses LPLS as a regression method to build a locally weighted model for every sample. In general, NLPLS algorithms use the criterion of minimum regression error to select inner model parameters. However, the resulting models suffer from over-fitting or local minima. In many cases modeling experts can easily detect these kinds of poor modeling results by inspecting the PLS score plots. However, correcting NLPLS models by changing model parameters is not an easy task because the relationship between model parameters and model shape is not clear, and the models were not developed taking into consideration the need for this kind of measure. The proposed FPLS model remedies the shortcomings of NLPLS outlined above. 9.2 Theory PLS modeling method Basically, the PLS method is a multivariable linear regression algorithm that can handle correlated inputs and limited data. The algorithm reduces the dimension of the predictor variables (input matrix, X) and response variables (output matrix, Y) by projecting them to the directions (input weight w and output weight c) that maximize the covariance between input and output variables. Through this projection decomposes variables of high collinearity into one-dimensional variables (input score vector t and output score vector u). The decomposition of X and Y by score vectors is formulated as follows: X = m t h h= 1 p T h + E (9.1) 152

169 Y = m h= 1 T u q + F (9.2) where p and q are loading vectors, and E and F are residuals. This relation is known as the PLS outer relation. The relation between score vectors t h and u h is known as the inner relation. The original PLS algorithm was developed as a linear regression method that uses a linear inner relation on the latent space. This LPLS algorithm has many beneficial properties for use as a data analysis tool. For example, w and c can be used to find the contributions of different variables to each score, and t and u can be used to detect outliers. Moreover, the method has a well-developed statistical foundation and results can be illustrated using biplots that enhance intuition into the underlying system. However, LPLS is limited to modeling linear relationships, and the real world is not limited to linear systems. Various nonlinear PLS algorithms have been proposed to cope with the problems introduced by nonlinearity. However, each of these approaches has shortcomings such as simplicity, lack of analytical interpretability of regression coefficients, and so on. The FPLS algorithm proposed here applies the TSK fuzzy model to the PLS inner regression. This method was developed because the interpretability of the TSK fuzzy model overcomes some handicaps of extant nonlinear PLS algorithms TSK Fuzzy Modeling A fuzzy inference system is an effective means of creating models based on human expertise in a specific application by a selection of fuzzy IF-THEN rules, which form the key components of the system. Having selected the IF-THEN rules, fuzzy set theory provides a systematic calculus to deal with information linguistically, and it performs numerical computation by using linguistic labels stipulated by membership functions. The fuzzy inference system therefore has the properties of a structured knowledge representation in the form of fuzzy IF-THEN h h 153

170 rules. This system therefore provides a good framework for applying human expertise in the construction of inference models. The fuzzy inference system proposed by Takagi, Sugeno and Kang, known as the TSK model, provides a powerful tool for modeling complex nonlinear systems (Yen et al., 1998). Typically, a TSK model consists of IF-THEN rules of the form R i : if x 1 is A i1 and and x r is A ir then y i = b i0 + b i1 x b ir x r for i = 1, 2,, L (9.3) where L is the number of rules, x i = [x 1 x 2 x r ] T are input variables, y i are local output variables, A ij are fuzzy sets that are characterized by the membership function A ij (x j ), and b i = [b i0 b i1 b ir ] T are real-valued parameters. The overall output of the model is computed by y L i= τ y + b 1 i i i= 1 i i0 i1 1 = = L L τ i= 1 i L τ ( b x + L+ b x ) τ i= 1 i ir r (9.4) where τ i is the firing strength of rule R i, which is defined as τ = A x ) A ( x ) L A ( x ) (9.5) i i1( 1 i2 2 ir r Figure 9.1 shows a schematic block diagram of the TSK fuzzy model. In general, Gaussian-type membership functions are used to build the model. They are defined by A ( x ) ir ( x r c ) exp 2σ i 2 ir r = 2, i = 1, 2,, L (9.6) where c ir is the center of the ith Gaussian membership function of the rth input variable x r and σ i is the width of the membership function. The TSK model presented above is sometimes called a first-order TSK model, because it formulates its rules using a first-order polynomial. In general, any function can be used for the fuzzy rules as long as it can appropriately describe the output of the model within the fuzzy region specified by the antecedent of the rule. 154

171 For example, when the function is a constant it is called a zero-order TSK model. Moreover, the zero-order TSK model is functionally equivalent to a radial basis function network. In the present research, we refer only to the first-order TSK model as the TSK model to avoid complication. The great advantage of the TSK fuzzy model is its representative power, which stems from its ability to describe complex nonlinear systems using a small number of rules. Moreover, the output of the model has an explicit functional form (equation 9.4), and the individual rules give insights into the local behavior of the model. The good interpretability of the fuzzy system may match the utility of the PLS method in intuitive data analysis Nonlinear FPLS Modeling Since many practical data are inherently nonlinear, there is a need for a nonlinear PLS modeling approach which can not only represent any nonlinear relationship but also attain the robust regression property of the LPLS method. We propose the FPLS method as such a nonlinear modeling method. The FPLS method is basically a combination of the PLS method and the TSK fuzzy model. The PLS outer projection is used as a dimension reduction tool to remove collinearity, and the TSK fuzzy inner model is used to capture the nonlinearity in the projected latent space. An advantage of using the TSK fuzzy model as the inner regressor is its interpretability, which facilitates in the design of the FPLS model structure by allowing human experts to participate in the design process. The FPLS method differs from the direct TSK fuzzy modeling approach in that the data are not used directly to train the TSK model, but are preprocessed by the PLS outer transform. This transformation decomposes the multivariate regression problem into a few univariate regression problems and simplifies the TSK model. The TSK method is a type of kernel regression method, where the input variables are transformed nonlinearly to feature space variables and the transformed data set is 155

172 regressed linearly. Well-designed nonlinear transformation procedures usually reduce the collinearity problem. In the kernel regression method, the method of nonlinear transform is related directly to the regression performance. However, designing an optimal nonlinear transformation for high dimensional and collinear data set is very difficult, and the resulting models often suffer from over-fitting or local minima. However, the robust data reduction characteristic of the PLS method can compensate for this problem in the TSK fuzzy modeling method. In the following subsections we propose the FPLS and IFPLS algorithms. The basic FPLS algorithm keeps the weight vectors the same as for LPLS, whereas IFPLS is an extended version of the FPLS algorithm that iteratively updates its weight vectors according to inner relation functions. The weight updating algorithm used in the IFPLS algorithm was developed because a complete PLS algorithm should have an algorithm that updates the weights according to the inner relation. However, the automatic updating of the weights to meet a certain object function diminishes the knowledge-based modeling aspect of FPLS because it automatically changes the score plots. When such a feature is used, the model that experts judged to have an appropriate structure for the previous score plot may not be good for the present plot. Therefore, we do not include the weight update scheme in the modeling procedure of FPLS. However, as IFPLS is an advanced PLS method that follows the main stream of NLPLS convention, we present it as an extended FPLS algorithm. FPLS algorithm Figure 9.2 shows a schematic of the basic FPLS method, which uses the PLS outer transform to generate score variables from the data. Score vectors (t h and u h ) of the same factor h are used to train the inner TSK fuzzy model f h ( ), which obeys the following relation u = f ( t ) + e (9.7) h h h h 156

173 where e h represents the regression error. The parameters of f h ( ) should be selected to minimize e h without over-fitting. To summarize, by not updating the outer relation FPLS keeps the LPLS property that variables are projected into the directions maximizing the covariance, and it captures nonlinearity through the large modeling capacity of the TSK model. The proposed FPLS algorithm can be formulated as follows. 1. Scale X and Y to have zero-mean and unit-variance. Let E 0 = X, F 0 = Y and h = For each factor h, take u h from one of the columns of F h PLS outer transform: T T T wh = uh E h 1/ ( uh uh ) (9.8) T w = w / w (9.9) h h th h 1 h = E w (9.10) T h T ch = th F h 1/ ( th th ) (9.11) c = c / c (9.12) h h uh h 1 h = F c (9.13) Iterate this step until it converges. This step is called the nonlinear iterative partial least squares (NIPALS) algorithm. Although there exists a faster and more stable algorithm using eigen vectors (Höskuldsson, 1988), we use NIPALS to give readers a clearer picture of PLS outer projection. 4. Find the TSK fuzzy-type inner relation function, f h ( ), which predicts the output score u h with the input score t h. f h ( ) has the functional form where h L i= 1 i h f ( t) = G ( b + b t (9.14) i0 i1 ) 157

174 G i τ i = L τ i = 1 i (9.15) ( t c ) 2 i τ i ( t) = exp, i = 1, 2,, L (9.16) 2 2σ i G i is the normalized firing strength and τ i is a Gaussian-type firing strength for the ith rule. First, the number of fuzzy rules, L, should be estimated by the model designer at an integer value that minimizes the regression error of f h ( ) without creating an over-fitted model. The designer may use intuition gained from the score plot or some numerical criteria such as the sum of squared errors (SSE) for cross validation. The designer can then decide the other parameters, such as c i, σ i and b i, using a numerical curve fitting function to minimize the SSE. 5. Calculate the X and Y loadings T T T ph = th E h 1/ ( th th ) (9.17) T T T q ˆ ( ˆ ˆ h = uh F h 1/ uh uh ) (9.18) where uˆ f ( ) = [ f ( t (1) ), f ( t (2)), L, f ( t ( N) )] T h h t h = for N samples. 6. Calculate the residuals for factor h. h h h = Eh 1 h h h T h h E t p (9.19) h h h T h F = F û 1 q (9.20) 7. Let h = h + 1, then return to step 2 until all m principal factors are calculated. The number of factors m is decided by the designer. The designer may use intuition gained from the score plot or some numerical criteria such as SSE for cross validation. The parameters of f h ( ) can be decided by various heuristics. In this research, the initial values of c i, σ i and b i are decided using the fuzzy c-means (FCM) h 158

175 algorithm (Jang et al., 1997), Moody and Darken s (M&D) rule (Moody and Darken, 1989) and the global learning procedure (Yen et al., 1998) (see the Appendix for the mathematical formulations of these methods). Then a numerical nonlinear least squares curve fitting function is applied for the optimization of the parameters with the object function of minimizing the SSE. However, if the optimized model shows signs of over-fitting such as very steep changes in its trend, the designer can change and fix some parameters and then optimize the other parameters to make a smoother and more reliable model within the criteria of his or her expertise. As is shown in the algorithm, the designer s decisions are emphasized in the calibration of a FPLS model. This aspect of FPLS represents an improvement over other PLS algorithms. Generally, structural parameters such as L and m are selected using cross validation method to avoid the problem of over-fitting. Cross validation is mandatory for high dimensional models, because the model shape cannot be well presented in visible form. Although the fuzzy modeling process gives particular weight to the application of the expert s knowledge in the modeling process, it is also hindered by the problem of high dimensionality. Regardless of the type of modeling, designers should check the validity of their model. The FPLS method aids designers in model validation by providing a simple modeling interface for visual checking, in addition to the typical cross validation method. The visual check comprises checks of the error correlation, high leverage data treatment, local minimum, over-fitting and lower fitting. Checking using visualization is possible because of the robust data reduction and the two-dimensional presentation properties of PLS. Other PLS methods such as LPLS and NNPLS also have these properties, but they lack the interpretability and high nonlinear regression capacity of the TSK inner relation function. The fuzzy rules of the TSK function provide insights into the model that allow us to make a simple linear expectation of its behavior even in the extrapolation range and to interactively change its parameters. These capabilities 159

176 make FPLS a promising modeling and monitoring method. Iterative FPLS (IFPLS) algorithm NLPLS algorithms include weight update methods. The weight update methods can be classified as follows. The first approach is not to update any weight. In this scheme, no iterative weight update procedure is used on either the input or the output weight. So, inner relation functions are calibrated only one time for each latent factor. FPLS uses this method. As a result the first score and weight vectors remain the same as that of LPLS; however, the later vectors change because the reduction of X and Y changes depending on the nonlinear regression performance. The advantages of this method are that the direction of the weights remains in the direction of maximizing covariance and that the calculation time is short. The second method is to use fixed input weights, and update output weights and inner relation functions iteratively, where a weight update method similar to that of the NIPAL algorithm is used (Qin and McAvoy, 1992). As a result only the first input score and the first input weight vectors remain the same as those of LPLS. This method goes half way toward fitting the weights to the inner relation function. On the other hand, it avoids the problem of updating the input weights, which can be controversial. The third approach is to update both the input and output weights iteratively (Wold et al., 1989; Baffi et al., 1999). While the output weights are updated using the same method as that of the second method described above, the input weights are updated iteratively to the vectors that minimize the regression SSE of the each inner relation function that is decided at the previous iteration, where numerical techniques are usually used to find the optimum input weights. However, this method shows an obvious problem when applied to rank deficient data sets. When the input dimension is larger than the number of samples, attempts to find the w which minimizes the SSE of u = f(xw) can yield an uncountable number of w s 160

177 which give the same minimum SSE. Numerical techniques give one of these solutions. This kind of weight update method does not have the robust dimension reduction properties of PLS. One feature of this kind of model is that they capture very large y-variance, but very small x-variance. The fourth approach involves the simultaneous reduction of x- and y-variances along with the updating of both the input and output weights. Wold et al. (1992) proposed a method of this kind that uses the same output weight update method as the second and third methods outlined above, but which updates the input weights using a correlation related algorithm. This method has the basic PLS principle in mind, which places equal emphasis on the approximation of X and on the correlation between X and Y. However, the algorithm of Wold et al. (1989) is not as balanced as NIPALS. In this research we propose a new weight update method. The idea behind this method is to apply the same update scheme to the input and output weights. Our approach uses the weight update method that has been used only on output weights in the past. This is achieved by defining a backward inner relation function, g( ), where a TSK model is used for the functional form of g( ). The core of the algorithm is as follows. 1. Initialize PLS parameters using LPLS. 2. Update the parameters. uˆ = f ( t) (9.21) c T T = uˆ Y/ ( uˆ T uˆ) (9.22) c = c/ c (9.23) u = Yc (9.24) w T tˆ = g( u) ˆT T = t X/ (ˆ t tˆ) (9.25) (9.26) 161

178 w = w/ w (9.27) t = Xw (9.28) where f( ) minimizes SSE between u and û and g( ) minimizes SSE between t and tˆ. Iterate this step until convergence is reached. This is a nonlinear extension of the NIPALS algorithm that reduces to LPLS when the inner relation functions are first order polynomials with no constant term. The function g( ) is used only in the training process not in the prediction procedure. The use of g( ) achieves balanced reductions of X and Y. This algorithm can be applied to other PLS algorithms with minor changes. IFPLS is an extended version of FPLS that uses the weight update algorithm given in equations In the IFPLS modeling, the number of fuzzy rules can be decided using the method employed for FPLS. However, decisions based on intuition lose meaning because the score plots change throughout the iteration process. Therefore, the use of the cross validation method is recommended for IFPLS. Prediction method with FPLS model FPLS and IFPLS models trained on a calibration data set are both identified by scaling information, outer projection vectors and inner relation parameters, i.e., the means and variances of the calibration data sets X 0 and Y 0, loading vectors p and q, input weight vector w, the number of fuzzy rules L, the center of the membership function c = {c 1, c 2,, c L }, the width of the membership function σ = {σ 1, σ 2,, σ L } and the linear regression coefficient b = {b 1, b 2,, b L } of fuzzy rules for all the factors m under consideration. Let us denote the outer projection vectors of the m factors by matrix form, i.e., P, Q and W. Then, for a new input data set X the output data set Y can be predicted using the following steps. 162

179 1. Scale X by the mean and variance of X Calculate the input score matrix T T = XW( P W) 1 (9.29) where T = [t 1, t 2,, t m ] 3. Predict output score vectors using the TSK inner model defined in equation (9.14), with c h, σ h and b h for each factor h. uˆ h = f ( t ) (9.30) h h 4. Predict the scaled Y ˆ ˆ T = (9.31) Y UQ where U ˆ = [ uˆ, uˆ, L, uˆ ] for i = 1, 2,, m. 1 2 m 5. Rescale Ŷ by the mean and variance of Y 0 Using the PLS outer relation and the TSK fuzzy-type inner model, the FPLS method is capable of robustly describing any complex nonlinear system and provides informative biplots. Because FPLS uses the outer relation of PLS, the analytical meaning of the outer projection vectors remains valid. Hence, various PLS monitoring methods are still applicable to FPLS. Moreover, the interpretation based on fuzzy rules gives a new way of monitoring nonlinear systems. For an example, each sample of a system modeled by FPLS can be classified according to the fuzzy rule that has the largest firing strength value on it. 9.3 Results and Discussion The proposed FPLS algorithm is applied to two data sets. First, a simulation data set of benchmark plant is considered, followed by real data of BET plant. TSK fuzzy model were built using the nonlinear least squares optimization function (Bang et al., 2001), which initial point is determined by FCM clustering algorithm 163

180 for identifying the center locations, P-nearest neighborhood method for deciding the width and the global learning procedure for determine the parameters of fuzzy rule (See the appendix). For the comparison, prediction performances of FPLS are compared with LPLS and QPLS. Simulation benchmark Eight variables used to build the X-block in the simulation benchmark were the influent ammonia concentration (S NH,in ), influent flow rate (Q in ), nitrate concentration in the second aerator (S NO,2 ), total suspended solid in aerator 4 (TSS 4 ), DO concentration in aerators 3 and 4 (S O,3, S O,4 ), oxygen transfer coefficient in aerator 5 (K L a 5 ), and internal recirculation rate (Q int ). Quality variables are the effluent ammonia and nitrate, S NH,e and S NO,e. We used 14 days as a normal data set developed by the benchmark, where the training model was based on a normal operation period for one week of dry weather and validation data was used on data set for last 7 days. Because we are interested in the normal operation condition in this research, we constructed the training model which was based on normal operation and did not consider any time lags between the input and output to avoid the complication. The results of three PLS models are represented in Table 9.1, where four LVs are selected in PLS model. Figure 9.3 shows the scatter plot and firing strength of FPLS model. In the score plot, the small circle represents the center c i of a firing strength function shown in the lower plot and the dashed line crossing the circle is its fuzzy rule. In the lower plot, the solid lines represent the firing strength τ i and the dashed lines represent the normalized firing strength G i. These plots clearly show the nonlinear natures of the benchmark plant. LPLS gives no direct way to cope with this nonlinearity; however, FPLS can give a direct and interactive way of treating such nonlinearities. To decide the number of fuzzy rules, we applied various 164

181 numbers of fuzzy rules and heuristic rules to each LV. Then, we found that fuzzy rules for each LV and fixing the center of fuzzy rule of first LV by FCM gave the best regression performances on training and validation data sets. The score plots of the third and fourth LVs showed almost no nonlinearity; hence, we used only one fuzzy rule for each of these LVs. Compared with other NLPLSs, FPLS model gives a visual and interactive design capability which can treat such nonlinearities and avoid overfitting problem. Percent variances captured of training data (%) and mean squared error (MSE) of test data set in benchmark with three PLS models are listed in Table 9.1, which shows the regression performance of all PLS models. Explained variances of X- block using LPLS, QPLS and FPLS model do not show any particular difference and the value of Y-variance captured by the FPLS model is larger than two methods. And the mean squared error (MSE) in the validation data set shows that best prediction performance is achieved by the FPLS method. Figures 9.4 and 9.5 show the prediction results of S NH,e and S NO,e in the validation data set for LPLS and FPLS method. Time series plots and scatter plots illustrate the prediction improvements that are achievable through the fuzzy regression approach. Scatter plots certify the modeling capability of FPLS. These results are not surprising because FPLS model is designed to capture the main variability of the training data set and validation data set is generated with the similar statistical properties to the training data. However, the above results are valid on only the normal data set. In other situations, such as other disturbances cases, other models may be better than FPLS model. The situation and the aim of the models can determine their best model structure. Full-scale WWTP The process data were collected from a biological WWTP (BET) that treated the coke wastewater of the iron and steel making plant in Korea (Figure 2.3). 165

182 Twelve process and manipulated variables, X blocks, were used to model three process output variables, Y blocks. Y blocks consist of the solid volume index (SVI), the reduction of cyanide ( CN), and the reduction of COD ( COD). Table 2.1 describes the process variables and presents the mean and standard deviation (SD) values of X and Y blocks. The process data consisted of daily mean values from 1 January, 1998 to 9 November, 2000 with a total number of 1034 observations. The 720 observations were used as the calibration of PLS models, where the samples of odd number were used as a training set and those of even number were used as a validation set. And the remaining 314 observations were used as a test data set. The results of three PLS models are represented in Table 9.2, where six LVs are selected for each PLS model. Figure 9.6 shows the scatter plot and firing strength of FPLS model with six LVs (the fifth and sixth LV are not shown). Unlike the expectation, the data from BET showed no obvious nonlinearity. However, we did find some nonlinear characteristics at the second LV, which leads us to use three fuzzy rules for this factor. The first and later factors showed almost no nonlinearity; hence, one fuzzy rule was used for each of these LVs. To avoid complication, we did not consider the nonlinearity of these factors further. The value of X and Y-variance captured by the FPLS model is larger than those of LPLS and QPLS methods and the mean squared error (MSE) of validation shows even best result in FPLS. Contrary to our expectation, MSE in the test data set represents that LPLS and QPLS have better prediction performance than FPLS. During the test data set, WWTP had received a large influent load and experienced the large change of operating condition. These process transitions altered a sort of microorganism and sludge, which changed the process dynamics in BET. Because FPLS model is designed to capture the nonlinear behavior and statistical properties of the training data set, FPLS model showed inferior prediction result in these disturbances cases. Figure 9.7 and 9.8 shows the time series and scatter plot of real 166

183 and predicted value with LPLS and FPLS model during the validation periods. The prediction performances of COD and CN reduction are satisfactory. But, the prediction of SVI of secondary settler is not so good as those of the other process quality variables. LPLS and FPLS show a similar prediction performance. After we performed several experiments comparing FPLS with the other NLPLS models. We concluded that FPLS shows similar regression performance to the other NLPLS models; however, it is difficult to make a fair comparison between models, because each algorithm has its own characteristics. For this reason we will not present a detailed comparison between models, but below we will outline the difference between FPLS and the other NLPLS in two aspects. First, inner relation models of FPLS usually take on gentler curvature than those of other NLPLS, as they are locally weighted averages of linear fuzzy rules and model designers would not favor highly nonlinear shapes of inner relation models whose variables are the results of linear computations. In contrast, other NLPLS models can take on any nonlinear shape to minimize the SSE, providing this shape is permitted by cross validation. If a FPLS model were built referring only to the cross-validation result, with no input from the experts, it could have greater curvature. Hence, it ultimately depends on the experts decision whether to use a conservative model or an SSE-minimizing model. At seconds, the number of regression parameters estimated for each NLPLS inner model depends on a few structural parameters, such as the order of a polynomial for QPLS, the order of polynomials and the number of knots for SPLS, the number of neurons for NNPLS and the number of rules for FPLS. They also vary depending on the nonlinearity of the modeled system. If the value of the structural parameters is increased the regression SSE of the model will decrease and the model will take on a more nonlinear shape. Because these structural parameters have different physical meanings, their values cannot be compared with those of another 167

184 NLPLS. However, if the values are the same, FPLS generally uses more parameters than other NLPLS methods. For example, if the values of the structural parameter are L for both NNPLS and FPLS, an inner model of NNPLS needs 2L + 1 regression parameters for the input and output weights of the neurons plus a bias term, whereas that of FPLS needs 4L parameters for c, σ and b. However, this does not mean FPLS is a more complex model to interpret. Because FPLS analyzes the system using submodels represented by fuzzy rules, the 2L parameters used for b help in the preparation of submodels and the 2L parameters used for c and σ help to interpret the relationship between the input data and the submodels. Therefore, although FPLS uses more regression parameters than other NLPLS methods for a same structural parameter, its superiority as an informative model will rate it highly among the elemental NLPLS methods 9.4 Conclusions In this research we proposed a fuzzy PLS model and presented experimental results showing the application of this algorithm. The proposed model uses a PLS framework, which gives the model robust regression performance when used on high dimensional and collinear data. Moreover, as the model uses TSK fuzzy models, it can represent highly nonlinear systems. Most importantly, the proposed model has higher interpretability than any other NLPLS modeling method, creating a modeling environment that is favorable to the use of experts knowledge. The interpretability of the FPLS model is embodied in the following elements. First, the model can be presented in intuitively simple biplots; second, the fuzzy rules of the TSK function provide insight into the model; and finally, the effects of the fuzzy rules can be estimated using plots of the firing strength. Another property that distinguishes the FPLS model from other NLPLS models is that the TSK fuzzy model is a combination of linear submodels. This feature causes the FPLS model to provide 168

185 more stable estimations of output on extrapolation. Using these properties, a model designer can interactively revise FPLS model and construct a more robust nonlinear model with fewer instances of local minima and over-fitting. 9.5 Appendix A1. Fuzzy c-means (FCM) algorithm The center of a Gaussian-type membership function, c i, can be decided by using the FCM algorithm, that is where is a membership grade. A2. Moody & Darken s rule c N j = 1 i = N µ j= 1 2 ij µ t 2 ij j, i = 1, 2,, L (9.32) 1 µ ij = (9.33) 2 = 1 L t j ci k t j ck The width of a Gaussian-type membership function, σ i, can be decided by using the P-nearest neighborhood heuristic suggested by Moody and Darken (1989), that is p 1 σ i = p l= 1 ( c c ) i 1/ 2 2 l, i = 1, 2,, L (9.34) where c l (l = 1, 2,, p) are the p (typically p = 2) nearest neighborhoods of the center c i. A3. Global learning algorithm 169

186 The parameters, b i, of a fuzzy rule can be decided by using a global learning algorithm. Global learning chooses the parameters of fuzzy rules that minimize the objective function J G. J G = N [ u k) uˆ( k) ] k= 1 2 ( (9.35) Equation (9.35) can be rearranged into a simple matrix form. J G where [ ] T G T ( u T bg ) ( u T b ) = (9.36) u = u = u ( 1) u(2) L u( N) R N 1 G G G G G T G G1 (1) = G1 (2) G1 ( N) G (1) t(1) G (2) t(2) G ( N) t( N) G G (1) (2) G ( N) G (1) t(1) G (2) t(2) G ( N) t( N) L L M L G G (1) (2) G ( N) L L L GL (1) t(1) G L (2) t(2) GL ( N) t( N) G R N 2 (9.37) [ b b b b b ] T b = R 2L 1 (9.38) b L L0 L1 Appling singular value decomposition (SVD) to T G yields ~ ~ ~ T TG = UΣ V (9.39) where ~ U = R N N (9.40) ~ V = [ u ~ u ~ ~ ] T 1 2 L u N [ v ~ v ~ L v ~ ] T 1 2 2L ~ Σ = diag( ~ σ, ~ σ, L, ~ σ 1 2 2L R 2L 2L (9.41) ) R N 2 (9.42) where ~ σ ~ σ L ~ σ 0. Then the minimum Euclidean norm solution of the 1 2 2L fuzzy rule parameters, b G, is computed as b G = s i= 1 T u ~ i u ~ σ i G v ~ i (9.43) where s is the number of nonzero singular values in Σ ~. 170

187 171

188 Rule 1 x is A 1 y 1 = x T b 1 τ 1 Rule 2 x x is A 2 y 2 = x T b 2 τ 2 L i= 1 L i= τ 1 i yi τ i y τ L Rule L x is A L y L = x T b L Figure 9.1 Block diagram of the TSK fuzzy model 172

189 X E E E E m E w 1 t 1 p 1 T w 2 t 2 p 2 T w m t m p m T... u 1 f û 1 1 ( ) u 2 f û 2 2 ( ) u m f m ( ) û m c 1 q 1 T c 2 q 2 T c m q m T Y F F F F m F First factor Second factor Last factor Figure 9.2 Block diagram of the FPLS method 173

190 (a) (b) (c) (d) Figure 9.3 Scatter plots and firing strength plots of FPLS model in benchmark (a) first LV (b) second LV (c) third LV (d) fourth LV 174

191 (a) (b) Figure 9.4 Comparisons of LPLS and FPLS for the predicted and actual S NHe in benchmark (a) Time series plot (b) Scatter plot 175

192 (a) (b) Figure 9.5 Comparisons of LPLS and FPLS for the predicted and actual S NOe in benchmark (a) Time series plot (b) Scatter plot 176

193 (a) (b) (c) (d) Figure 9.6 Scatter plots and firing strength plots of FPLS model in BET (a) first LV (b) second LV (c) third LV (d) fourth LV 177

194 (a) (b) (c) Figure 9.7 Time series plots of predicted and actual output in BET (a) SVI with LPLS and FPLS (b) CN with LPLS and FPLS (c) COD with LPLS and FPLS 178