# Interaction Analysis and Control of Bioreactors for Nitrogen Removal ¨ B

IT Licentiate theses 2007-006 Interaction Analysis and Control of Bioreactors for Nitrogen Removal B J ÖRN H ALVARSSON UPPSALA UNIVERSITY Department of Information Technology Interaction Analysis and Control of Bioreactors for Nitrogen Removal BY B J ÖRN H ALVARSSON December 2007 D IVISION S YSTEMS AND C ONTROL D EPARTMENT OF I NFORMATION T ECHNOLOGY U PPSALA U NIVERSITY U PPSALA S WEDEN OF Dissertation for the degree of Licentiate of Philosophy in Electrical Engineering with Specialization in Automatic Control at Uppsala University 2007 Interaction Analysis and Control of Bioreactors for Nitrogen Removal Björn Halvarsson [email protected] Division of Systems and Control Department of Information Technology Uppsala University Box 337 SE-751 05 Uppsala Sweden http://www.it.uu.se/ c Björn Halvarsson 2007 ISSN 1404-5117 Printed by the Department of Information Technology, Uppsala University, Sweden Abstract Efficient control of wastewater treatment processes are of great importance. The requirements on the treated water (effluent standards) have to be met at a feasible cost. This motivates the use of advanced control strategies. In this thesis the activated sludge process, commonly found in the biological wastewater treatment step for nitrogen removal, was considered. Multivariable interactions present in this process were analysed. Furthermore, control strategies were suggested and tested in simulation studies. The relative gain array (RGA), Gramian based interaction measures and an interaction measure based on the H2 norm were considered and compared. Properties of the H2 norm based measure were derived. It was found that the Gramian based measures, and particularly the H2 norm based measure, in most of the considered cases were able to properly indicate the interactions. The information was used in the design of multivariable controllers. These were found to be less sensitive to disturbances compared to controllers designed on the basis of information from the RGA. The conditions for cost-efficient operation of the activated sludge process were investigated. Different fee functions for the effluent discharges were considered. It was found that the economic difference between operation in optimal and non-optimal set points may be significant even though the treatment performance was the same. This was illustrated graphically in operational maps. Strategies for efficient control were also discussed. Finally, the importance of proper aeration in the activated sludge process was illustrated. Strategies for control of a variable aeration volume were compared. These performed overall well in terms of treatment efficiency, disturbance rejection and process economy. Keywords: activated sludge process; biological nitrogen removal; bioreactor models; cost-efficient operation; interaction measures; multivariable control; wastewater treatment. Acknowledgements First of all, I would like to express my sincere gratitude to my supervisor, Professor Bengt Carlsson, for all his help and encouragement during my research so far. Special thanks also go to my co-author and “mentor” Dr. Pär Samuelsson at Dalarna University (formerly with us here at the Division of Systems and Control) for all fruitful discussions and good advice. He read previous versions of this thesis and his suggestions certainly improved the quality. Furthermore, I wish to thank all colleagues at the Division of Systems and Control and the Division of Scientific Computing for providing such a pleasant working atmosphere. Part of this work has been financially supported by the EC 6th Framework programme as a Specific Targeted Research or Innovation Project (HIPCON, Contract number NMP2-CT-2003-505467). Furthermore, I would like to thank Stiftelsen J. Gust. Richerts Minne for financial support. I am also grateful to Dr. Ulf Jeppsson for letting me use his Simulink implementation of BSM1 and to Assistant Professor Torsten Wik, Chalmers University of Technology, Göteborg, for taking the time of being my licentiate opponent. Finally, very special thanks go to my friends and to my family. Contents 1 Introduction 1.1 Interaction measures . . . . . . . . . . . . . 1.1.1 Motivational example . . . . . . . . 1.2 Wastewater treatment systems . . . . . . . 1.2.1 The activated sludge process (ASP) 1.2.2 The benchmark model BSM1 . . . . 1.2.3 Control of WWTPs . . . . . . . . . 1.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 7 9 10 12 13 14 15 2 Controllability and Interaction Measures 19 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Systems description . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.1 State controllability for continuous-time systems . . . 21 2.3.2 State controllability for discrete-time systems . . . . . 22 2.3.3 Output controllability . . . . . . . . . . . . . . . . . . 23 2.4 The Relative Gain Array (RGA) . . . . . . . . . . . . . . . . 24 2.5 Gramian based interaction measures . . . . . . . . . . . . . . 25 2.5.1 The Hankel norm . . . . . . . . . . . . . . . . . . . . . 25 2.5.2 Energy interpretations of the controllability and observability Gramians for discrete-time systems . . . . 28 2.5.3 The Hankel Interaction Index Array (HIIA) . . . . . . 28 2.5.4 The Participation Matrix (PM) . . . . . . . . . . . . . 29 2.5.5 The selection of proper scaling . . . . . . . . . . . . . 30 2.6 An interaction measure based on the H2 norm . . . . . . . . 30 2.6.1 The Σ2 interaction measure . . . . . . . . . . . . . . . 30 2.6.2 The H2 norm . . . . . . . . . . . . . . . . . . . . . . . 31 2.6.3 Calculation of the H2 norm . . . . . . . . . . . . . . . 32 2.6.4 Energy interpretation for discrete-time systems . . . . 33 2.6.5 Properties of the H2 norm based interaction measure Σ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.6.6 The H2 norm and induced norms . . . . . . . . . . . . 36 2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3 2.8 2.7.1 Example 2.7.2 Example 2.7.3 Example 2.7.4 Example 2.7.5 Example Conclusions . . 1. 2. 3. 4. 5. . . . . . . . . . . . . . . 3 Interaction Analysis in a 3.1 Introduction . . . . . . 3.2 The bioreactor model 3.3 RGA analysis . . . . . 3.4 HIIA analysis . . . . 3.5 Discussion . . . . . . . 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bioreactor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 39 42 46 46 47 . . . . . . 49 49 50 53 54 55 56 4 Interaction Analysis and Control of the Denitrification Process 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The bioreactor model . . . . . . . . . . . . . . . . . . . . . . 4.3 Analysis of the model . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Linearization and scaling of the model . . . . . . . . . 4.3.2 RGA analysis of the model . . . . . . . . . . . . . . . 4.3.3 HIIA analysis of the model . . . . . . . . . . . . . . . 4.4 Control simulations . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Decentralized control . . . . . . . . . . . . . . . . . . . 4.4.2 Multivariable control . . . . . . . . . . . . . . . . . . . 4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Economic Efficient Operation of a Pre-denitrifying Activated Sludge Process 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The model and the operational cost functions . . . . . . . . . 5.2.1 The nitrate fee . . . . . . . . . . . . . . . . . . . . . . 5.2.2 The ammonium fee . . . . . . . . . . . . . . . . . . . . 5.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Simulation results for the denitrification process . . . 5.3.2 Simulation results for the combined denitrification and nitrification process . . . . . . . . . . . . . . . . . . . 5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 57 57 58 60 61 63 64 65 66 67 69 73 75 75 76 78 79 80 80 85 86 93 6 Aeration Volume Control in an Activated Sludge – Discussion of Some Strategies Involving On-Line nium Measurements 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 6.2 The simulation setup . . . . . . . . . . . . . . . . . . 6.3 Description of the proposed control strategies . . . . 6.3.1 The reference aeration control strategies . . . 6.3.2 Feedforward aeration volume control I and II 6.3.3 Supervisory feedback aeration volume control 6.4 Simulation results . . . . . . . . . . . . . . . . . . . 6.4.1 The reference aeration control strategies . . . 6.4.2 Feedforward aeration volume control I . . . . 6.4.3 Feedforward aeration volume control II . . . 6.4.4 Supervisory feedback aeration volume control 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . A The minimized condition number 5 Process Ammo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 95 97 97 97 98 99 100 100 100 102 102 108 109 6 Chapter 1 Introduction This thesis concerns the interaction analysis and the control of bioreactors for nitrogen removal. More precisely, models of the activated sludge process commonly found in the biological treatment of wastewater are considered. The interactions present in these processes will be analysed and different controller structures will be compared in simulation studies. The influence of various input signals on the treatment efficiency, both in terms of the treatment performance and in terms of the process economy, will also be investigated. 1.1 Interaction measures Many control systems of today are multivariable. This means that they have multiple inputs and multiple outputs. Such systems are called multiple-input multiple-output (MIMO) systems. Compared to single-input single-output (SISO) systems, the control design for MIMO systems is more elaborate. One reason for this is that different parts of a multivariable system may interact and cause couplings in the system. As an example, consider a shower with separate flow control for hot and cold water. This is a MIMO system since the two inputs, the flow of hot water and the flow of cold water, are utilized to control the two outputs which are the flow from the tap and the temperature of the effluent water. Evidently, when changing one of the inputs, both of the outputs will be affected. This means that there are significant couplings in the system. In other words, interaction occurs if a change in one input affects several outputs. Often, an easy way to control a fairly decoupled MIMO system is to use a multi-loop strategy, i.e. to separate the control problem into several single-loop SISO systems and then use conventional SISO control on each of the loops, see Kinnaert (1995) and Wittenmark et al. (1995). This gives rise to the pairing problem: Which input signal should be selected to control which output signal to get 7 the most efficient control with a low degree of interaction? In real-life applications the considered MIMO system could be rather complex: In the chemical process industry a complexity of several hundred control loops is not unusual, see Wittenmark et al. (1995). The proper pairing selection is thus often not at all obvious. Also, the choice of pairing is crucial since a bad choice may give unstable systems even though each loop separately is stable. This problem could arise due to interaction between the different loops. Generally, the stronger the interactions, the harder it is to obtain satisfactory control performance using a multi-loop strategy. Evidently, there is a need for a measure that can both give some advise when solving the pairing problem and that also quantifies the level of interaction occurring in the system. One such measure is the Relative Gain Array (RGA) developed by Bristol (1966). The RGA considers steady-state properties of the plant and gives a suggestion on how to solve the pairing problem in the case of a decoupled (decentralized) control structure. Such structure will be diagonal. It also indicates which pairings that should be avoided due to possible stability and performance problems. Later, a dynamic extension of the RGA was proposed in the literature, see e.g. Kinnaert (1995) for a survey. With the extension, the RGA could be used to analyse the considered plant at any frequency but still only at one single frequency at a time. A recent approach to define a dynamic relative gain array was made by Mc Avoy et al. (2003). Moreover, the RGA can be generalized for non-square plants and be employed as a screening tool to get a suggestion on what inputs or outputs that should be removed in the case of excess signals, see Skogestad and Postlethwaite (1996). Over the years, several resembling tools have been developed. One such example is the Partial Relative Gain (PRG) suggested by Häggblom (1997) that is intended to handle the pairing problem for larger systems in a more reliable way than the conventional RGA. Other examples are the µ interaction index (Grosdidier and Morari, 1987) and the Performance Relative Gain Array (PRGA) (Hovd and Skogestad, 1992). An interesting novel approach is found in (He and Cai, 2004) where pairings are found by minimizing the loop interaction energy characterized by the General Interaction (GI) measure. This measure is used in combination with the pairing rules of the RGA and of the Niederlinski Index (NI) (Niederlinski, 1971). The NI can be used as an indicator of possible instability issues when solving the pairing problem. In the Effective RGA (ERGA) proposed by Xiong et al. (2005) the steady state gain and the bandwidth of the process are utilized to form a dynamic interaction measure. He et al. (2006) suggest an algorithm for control structure selection where the ideas by He and Cai (2004) are further developed. Other examples are given by Kinnaert (1995) where a survey of interaction measures for MIMO systems can be found. 8 The RGA provides only limited knowledge about when to use multivariable controllers and gives no indication of how to choose multivariable controller structures. A somewhat different approach for investigating channel interaction was therefore employed by Conley and Salgado (2000) and Salgado and Conley (2004) when considering observability and controllability Gramians in so called Participation Matrices (PM). In a similar approach Wittenmark and Salgado (2002) introduced the Hankel Interaction Index Array (HIIA). These Gramian based interaction measures seem to overcome most of the disadvantages of the RGA. One key property of these is that the whole frequency range is taken into account in one single measure. Furthermore, these measures seem to give appropriate suggestions for controller structures both when a decentralized structure is desired as well as when a full multivariable structure is needed. The use of the system H2 norm as a base for an interaction measure has been proposed by Birk and Medvedev (2003) as an alternative to the HIIA. 1.1.1 Motivational example As a first motivational example, consider a system previously analysed by Goodwin et al. (2005). The system has a transfer function G(s) = −10(s+0.4) (s+4)(s+1) 2 s+2 −2.1 s+3 0.5 s+1 20(s−0.4) (s+4)(s+2) 3 s+3 −1 s+1 1 s+2 30(s+0.4) (s+4)(s+3) and a steady-state gain of −1.0000 0.5000 −1.0000 G(0) = 1.0000 −1.0000 0.5000 . −0.7000 1.0000 1.0000 The interaction measures are: Λ(G(0)) = ΣH = Φ = Σ2 = 2.8571 −1.2857 −0.5714 −2.8571 3.2381 0.6190 , 1.0000 −0.9524 0.9524 0.1330 0.0324 0.0648 0.0648 0.2827 0.0324 , 0.0454 0.0648 0.2798 0.0768 0.0036 0.0144 0.0144 0.4377 0.0036 , 0.0071 0.0144 0.4279 0.0915 0.0011 0.0044 0.0088 0.2992 0.0022 0.0065 0.0132 0.5732 9 where Λ is the RGA, ΣH is the HIIA, Φ is the PM and finally, Σ2 is an H2 norm based interaction measure. All of these will be defined in Chapter 2. The aim in this example is to find the decentralized pairing recommendation so that each input signal is paired uniquely with one output signal. In the case of the RGA input-output pairings corresponding to elements close to one should be selected and negative elements should be avoided. The other of the considered interaction measures recommend the input-output pairings that result in the largest sum when adding the corresponding elements in the measure. Evidently, all interaction measures suggest the diagonal pairing: input 1 – output 1, input 2 – output 2 and input 3 – output 3. However, no useful pairing information can be found by inspecting G(0) or G(0). This demonstrates the need of dedicated interaction measures even for pairing suggestions relevant for operation in steady state. Even though, the considered interaction measures are rather similar in this particular example, this is not generally the case. Further examples and theoretical differences between these will be examined in Chapter 2–4. 1.2 Wastewater treatment systems Until some time during the 19:th century, the activity of man had not affected the environment to any appreciable extent. When the industrial revolution came, a rapidly increased standard of living as well as a substantially population growth followed. The society became more and more urbanized and the problem of taking care of the human waste products and waste disposal became a serious (hygienic) problem. The introduction of the water closet solved the problem locally, but only locally, since the problem was instead moved to the surrounding environment with an increased load on the recipients (e.g. lakes and rivers). This could not be handled by the recipients without heavily disturbed local ecosystems. The degradation of organic material present in the wastewater, consumes oxygen and the recipient will thus suffer from lack of oxygen after some while. Even if most of the organic matter is removed before the wastewater reaches the recipient, chemical compounds such as phosphorous and nitrogen are still present, and may cause eutrophication (i.e. over-fertilization). Eventually, this will also result in a lack of oxygen. Therefore, the aim of wastewater treatment should be to remove both the content of organic matter and suspended solids as well as the content of nitrogen and phosphorous to a reasonable extent. In the beginning of the 20:th century, the first wastewater treatment plants were introduced in Sweden. They were simple plants using only a mechanical treatment step. This step could consist of a grid and a sand filter to remove larger objects and particles. In the late 1950’s the biological treatment step, was introduced. Hereby, microorganisms (e.g. bacteria) are used to remove organic matter present in the incoming wastewater. Later, in 10 the 1970’s, the chemical treatment step, was employed to reduce the content of phosphorous. Nowadays, the biological step is also utilized to reduce the content of nitrogen and phosphorous. A general wastewater treatment plant (WWTP), consisting of the above mentioned steps, is given in Figure 1.1. The sludge also needs to be treated. The main procedures are depicted in Figure 1.1. In the thickening procedure, the sludge is concentrated. Then, the sludge is stabilized in order to reduce odor and pathogenic content. Finally, the moisture content of sludge can be reduced by the use of dewatering. For a description of how to practically realize these steps, see e.g. Hammer and Hammer Jr. (2008). 1 Mechanical treatment 2 Biological treatment Activated sludge Grid Sand filter Primary Sedimentation 3 Chemical treatment Chemicals Effluent water Preciptation 4 Sludge treatment Dewatering Dewatered sludge Sludge thickening Stabilization Supernatants + Backwashing Figure 1.1: A general WWTP (Kommunförbundet, 1988). In the complex process of wastewater treatment, many different causeeffect relationships exist, and therefore, there are many possible choices of input and output signals, see Olsson and Jeppsson (1994). This makes the WWTP models particularly interesting to study with respect to the interactions present and the selection of proper control structures. When treating wastewater, the aim is to reduce as much as possible of the undesired constituents such as organic matter, nitrogen and phosphorous. This is commonly done using wastewater treatment plants. In a WWTP several biological processes occur simultaneously. These processes need to be properly controlled in order to maintain the concentrations of undesired constituents in the outlet water within the legislated limits. As the public awareness of environmental issues increases, the environmental legislation becomes stricter, and thus, the requirements on WWTPs become even harder to fulfill. The used control strategies need then to be as efficient as possible, see e.g. Olsson and Newell (1999). Therefore, models of the WWTP processes are interesting to study with respect to the choice of e.g. control structure. An example of such models are the bioreactor models. From a theoretical point of view, the bioreactor models are non-linear 11 multivariable systems that may contain a significant degree of coupling. Hence, this also gives an interesting opportunity to test the performance of the methods for input-output pairing selection mentioned in the previous section. The aim of Section 1.2 is to give a brief description of the bioreactor models that will be analysed in the forthcoming chapters. 1.2.1 The activated sludge process (ASP) The biological treatment step can be realized in several different ways. One of the most common is the activated sludge process where activated sludge, i.e. microorganisms (mainly bacteria), is employed to degrade (i.e. oxidize) organic material. The basic set-up consists of an aerated basin where oxygen is added by blowing air into the water, and a settler tank, see Figure 1.2. In the aerated basin, the bacteria degrade the incoming organic material while consuming oxygen. In this way the microorganisms fulfill their need of energy and as a result bacterial growth will occur. Together with decayed microorganisms and other particulate material, the living microorganisms form sludge. To separate the sludge from the purified water a settler, where the sludge settles, can be used directly after the aerated tank. Since the amount of microorganisms needs to be kept at a high level, some sludge is recirculated as shown in Figure 1.2, while the rest is removed as excess sludge. With the excess sludge, some nitrogen (and phosphorus) is removed, but still far too much remains. Influent Effluent Aerobic Settler Sludge recirculation Excess sludge Figure 1.2: A basic activated sludge process with an aerated basin and a settler. However, if the activated sludge process is extended to consist of both aerated and non-aerated (anoxic) basins, then bacteria may be employed for efficient nitrogen removal. In the aerated basins, bacteria oxidize ammonium to nitrate in a two-step process called nitrification: − + NH+ 4 + 1.5O2 → NO2 + H2 O + 2H , − NO− 2 + 0.5O2 → NO3 . For these processes to occur, the concentration of dissolved oxygen (DO) 12 must be sufficiently high and a long sludge age (the average time each particle stays in the system) is required due to slow bacteria growth. In the anoxic tanks, another type of bacteria is employed in the denitrification process, described by + 2NO− 3 + 2H → N2 (g) + H2 O + 2.5O2 i.e., the bacteria convert nitrate into nitrogen gas using the oxygen in the nitrate ions. However, no dissolved oxygen should be present for this process to take place, instead, a sufficient amount of readily biodegradable substrate is needed. Hence, together, nitrification and denitrification convert ammonium into nitrogen gas which is harmless to the environment. For further descriptions of these processes, see Henze et al. (1995). Nitrogen removal can be performed in several different types of WWTPs. One of the most popular is the pre-denitrification system (ibid.). In this design, the anoxic tanks are placed before the aerated basins, and thus, denitrification is performed before the nitrification process, see Figure 1.3. Influent Effluent Anoxic Anoxic Aerobic Aerobic Aerobic Internal recirculation Settler Sludge recirculation Excess sludge Figure 1.3: An activated sludge process configured for nitrogen removal (predenitrification). To supply the denitrification process with nitrate, there is a feedback flow from the last tank as shown in Figure 1.3. In some cases, when the influent water has a low content of carbon, the bacteria in the anoxic tank need to be fed with an external carbon source. For this purpose, methanol or ethanol is often used. For a further discussion about the ASP, see e.g. Olsson and Newell (1999) and Hammer and Hammer Jr. (2008). 1.2.2 The benchmark model BSM1 The comparison between different control strategies for a WWTP is often difficult due to the variable influent conditions and the high complexity of a WWTP. Therefore, to enable objective comparisons between different control strategies, a simulation benchmark activated sludge process, Benchmark Simulation Model No.1 (BSM1), has been developed by the COST 682 Working Group No.2, see Copp (2002) and IWA (November 19, 2007). 13 In the BSM1 a typical activated sludge process with pre-denitrification is implemented. It consists of five biological reactor tanks configured in-series. The first two tanks have a volume of 1000 m3 each, and are anoxic and assumed to be fully mixed. The remaining three tanks are aerated and have a volume of 1333 m3 each. All biological reactors are modelled according to the ASM1 model. Finally, there is a secondary settler modelled using the double-exponential settling velocity function of Takács et al. (1991). To get an objective view of the performance of the applied control strategy, it is important to run the BSM1 simulation with different influent disturbances. Therefore, influent input files for three different weather conditions – dry, stormy and rainy weather – are available together with the benchmark implementation. A number of different performance criteria are defined, such as various quality indices and formulas for calculating different operational costs. 1.2.3 Control of WWTPs As previously stated, WWTPs may be seen as complex multivariable systems. Therefore, to obtain satisfactory control performance, it is often necessary to use more advanced control strategies. However, since wastewater treatment traditionally has been seen as non-productive compared to the industry, the extra investments needed to employ such advanced control strategies have been hard to justify economically. Nowadays, as the effluent demands get tighter, the interest for more advanced control strategies is again awakening, see Olsson and Newell (1999). The plant has to be run economically and at the same time the discharges to the recipient should be kept at a low level. The control problem is hence twofold. The economical aspect involves minimizing operational costs such as pumping energy, aeration energy and dosage of different chemicals. Consequently, the main problem is how to keep the effluent discharges below a certain pre-specified limit to the lowest possible cost (ibid.). One way of solving this conflict of interest is to design the control algorithms in such a way that the overall operational costs are minimized. To make sure that also the wastewater treatment performance demands are fulfilled, the effluent discharges can be economically penalized. The corresponding cost can then be included together with the actual costs (energy and chemicals) in the calculation of the overall cost. Control handles for nitrogen removal In the nitrogen removal process, there are several variables that can be used as actuators, or control handles, to control the outputs. In a predenitrification system, there are five main control handles, as stated by Ingildsen (2002): 14 1. The airflow rate (in the aerated compartments); 2. The internal recirculation flow rate; 3. The external carbon dosage; 4. The sludge outtake flow rate (excess sludge); 5. The sludge recirculation flow rate. In this thesis, only the three first of these are considered. The last two control handles are described by for example Yuan et al. (2001) and Yuan et al. (2002). The first control handle, the airflow rate, is employed to affect the DO concentration in the aerated compartments. Hereby, the performance of the autotrophic nitrification bacteria will be influenced. Most common today is to control the airflow rate to maintain a specific DO level. Another way is to make use of online-measurements of the ammonium concentration in the last aerated compartment, and let these control the time-varying DO set point, see e.g. Lindberg (1997). The internal recirculation flow rate affects the supply of nitrate for the denitrification process but also the DO concentration in the anoxic compartments since some DO may be transported from the last aerated compartment. The DO transportation between the processes, can however, be reduced by introducing an anoxic tank after the last aerated basin. External carbon dosage can be applied when the influent water does not have enough readily biodegradable substrate to feed the denitrification bacteria. Controlled output signals for nitrogen removal The primary outputs from a WWTP are the effluent ammonium concentration, the organic matter, the nitrate concentration and the suspended solids, see Ingildsen (2002). For a more thorough discussion on cause-effect relationships in activated sludge plants, see Olsson and Jeppsson (1994). 1.3 Thesis outline Chapter 2 In Chapter 2 different interaction measures are reviewed and compared for some MIMO plants. In addition a simulation study is performed where the influence of a time delay on the coupling is examined. State controllability and output controllability are also discussed and further motivations for incorporating the concept of output controllability in an interaction measure are given. A H2 norm based interaction measure is investigated. 15 Chapter 3 In Chapter 3 the interactions in a multivariable ASP model configured for nitrogen removal are studied. The RGA and the HIIA are utilized to quantify the degree of coupling present in the system. Both the nitrification and the denitrification process are studied since the output signals (the controlled signals) are the effluent concentration of ammonium and the effluent concentration of nitrate. The input signals (control handles) are the dissolved oxygen concentration set point in the aerobic compartment and the internal recirculation flow rate. The material is based on: Halvarsson, B., P. Samuelsson and B. Carlsson (2005). Applications of Coupling Analysis on Bioreactor Models. In: Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, July 4-8. Chapter 4 Chapter 4 once again considers the interactions present in an ASP. Here, the focus is on controlling the denitrification process when an external carbon source is added. Thus, one of the two considered control handles (input signals) is the readily biodegradable organic substrate in the influent water (which has the same influence as an external carbon source would have). The other input signal is the internal recirculation flow rate. The output signals (controlled signals) are the nitrate concentration in the anoxic compartment and the nitrate concentration in the effluent. The model is analysed using the RGA and the HIIA. The results are discussed from a process knowledge point of view, and are also illustrated with some control experiments. The chapter is based on: Samuelsson, P., B. Halvarsson and B. Carlsson (2005). Interaction Analysis and Control Structure Selection in a Wastewater Treatment Plant Model. IEEE Transactions on Control Systems Technology 13(6). Samuelsson, P., B. Halvarsson and B. Carlsson (2004). Analysis of the Input-Output Couplings in a Wastewater Treatment Plant Model. Technical Report 2004-014. Division of Systems and Control, Department of Information Technology, Uppsala University, Uppsala, Sweden. Chapter 5 In this chapter, the focus is on finding optimal set-points and cost minimizing control strategies for the activated sludge process. Both the denitrification and the nitrification process are considered. In order to compare different 16 criterion functions, simulations utilizing the COST/IWA simulation benchmark (BSM1) are considered. By means of operational maps the results are visualized. It is also discussed how efficient control strategies may be accomplished. The material is based on: Halvarsson, B. and B. Carlsson (2006). Economic Efficient Operation of a Predenitrifying Activated Sludge Process. HIPCON Report number HIP06-86-v1-R Deliverable D6.5. Uppsala University, Uppsala, Sweden.1 which is an extended version of: Samuelsson, P., B. Halvarsson and B. Carlsson (2007). Cost-Efficient Operation of a Denitrifying Activated Sludge Process. Water Research 41(2007) 2325-2332. Samuelsson, P., B. Halvarsson and B. Carlsson (2005). Cost Efficient Operation of a Denitrifying Activated Sludge Plant – An Initial Study. Technical report 2005-010. Division of Systems and Control, Department of Information Technology, Uppsala University, Uppsala, Sweden. In these two references only the denitrification process is studied. Chapter 6 In the concluding chapter the influence of the aeration on the efficiency of the nitrogen removal in an ASP is studied. Different strategies for controlling the DO set point as well as the aerated volume are compared in terms of efficiency in a simulation study. Chapter 6 is based on: Halvarsson, B. and B. Carlsson (2006). Aeration Volume control in an activated sludge process – Discussion of some strategies involving on-line ammonia measurements. HIPCON Report number HIP06-86v1-R Deliverable D6.5. Uppsala University, Uppsala, Sweden.1 1 This paper is an internal EU project report which is available from the author. 17 18 Chapter 2 Controllability and Interaction Measures In this chapter the concept of controllability is discussed and different interaction measures are reviewed and compared. In particular, the two Gramian based interaction measures the Hankel Interaction Index Array (HIIA) and the Participation Mtarix (PM) are considered. Moreover, motivations for incorporating the concept of output controllability in an interaction measure are given and a H2 norm based interaction measure is investigated. 2.1 Introduction There are today several different measures for quantifying the level of inputoutput interactions in multivariable systems. The perhaps most commonly used is the Relative Gain Array (RGA) introduced by Bristol (1966). The RGA is a measure that can be employed in order to decide a suitable inputoutput pairing when applying a decentralized control structure. It can also be used to decide whether a certain pairing should be avoided. This measure, however, suffers from some major disadvantages. For instance it only considers the plant in one frequency at the time and it often provides limited knowledge about when to use multivariable controllers. Neither is the RGA able to give advice on how to select an appropriate multivariable controller structure. The RGA is also unable to suggest a proper pairing in the case of plants with triangular structure or large off-diagonal elements (this particular situation is further investigated in Chapter 3). A different approach for investigating channel interaction was employed by Conley and Salgado (2000) when considering observability and controllability Gramians in so called Participation Matrices (PM). In a similar approach Wittenmark and Salgado (2002) introduced the Hankel Interaction Index Array (HIIA). These Gramian based interaction measures seem to overcome most of the disadvantages of the RGA. One key property of 19 these is that the whole frequency range is taken into account in one single measure. Furthermore, these measures seem to give appropriate suggestions for controller structures both when a decentralized structure is desired and when a full multivariable structure is needed. For applications and comparisons between the RGA and various types of Gramian based interaction measures, see for instance Salgado and Conley (2004), Birk and Medvedev (2003), Samuelsson et al. (2005c) and Halvarsson et al. (2005). The use of the system H2 norm as a base for an interaction measure has been proposed by Birk and Medvedev (2003) as an alternative to the HIIA. In HIIA the Hankel norm forms the basis. In this chapter this H2 norm based interaction measure is investigated. Further motivations for incorporating the concept of output controllability in an interaction measure are also given. This chapter is organized in the following way: Section 2.2 gives a brief description of the systems that will be analysed and some general assumptions. In Section 2.3 state controllability and output controllability are defined. Section 2.4 introduces the reader to the RGA. Section 2.5 presents the Gramian based interaction measures the Hankel Interaction Index Array and the Participation Matrix and their theoretical foundations. Section 2.6 defines the H2 norm based interaction measure and investigates its relation to the concept of output controllability. In Section 2.7 different interaction measures are compared in the analysis of the interactions present in some MIMO systems. A simulation study is also performed. Finally, the conclusions are drawn in Section 2.8. 2.2 Systems description Consider a continuous-time linear time-invariant system, with inputs at time t given by the N × 1 vector u(t) and outputs at time t given by the p × 1 vector y(t). The system can be described as a state-space realization ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t) (2.1) where A, B, C and D are matrices of dimension N × N , N × q, p × N and p × q, respectively. x(t) is the state vector. Furthermore, a discrete-time system x(t + 1) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t) (2.2) will be considered as well. Note that (A, B, C, D) both denote the continuoustime system matrices and the discrete-time system matrices. These do not generally coincide; what quadruple of matrices that are referred to will be clear from the context. 20 2.3 2.3.1 Controllability State controllability for continuous-time systems Most often, the term controllability refers to the property of a system as being state controllable. The concepts of state controllability and state observability were introduced by Kalman, see for example Kalman et al. (1963), Kreindler and Sarachik (1964) and the references therein. With an initial state x(t0 ) and an input u(t), the solution of (2.1) for t ≥ t0 is given by Z t x(t) = eA(t−t0 ) x(t0 ) + eA(t−τ ) Bu(τ )dτ. (2.3) t0 This is a standard result found in many text books such as (Skogestad and Postlethwaite, 1996; Zhou et al., 1996). Since the system is time-invariant t0 can be set to 0. A system with an arbitrary initial state x(0) = x0 is said to be state controllable if there exists a piecewise continuous input u(t) such that x(t1 ) = x1 for any final state x1 and t1 > 0. Equivalently, a state controllable system can be transferred from any initial state x(t0 ) to any final state x(t1 ) in finite time. It can be verified using (2.3) that one input that satisfies this criterion is given by (ibid.) u(t) = −B T eA T (t −t) 1 Wc (t1 )−1 (eAt1 x0 − x1 ) where Wc (t) is a Gramian matrix defined as Z t T Wc (t) = eAτ BB T eA τ dτ. (2.4) (2.5) 0 Clearly, for the solution in (2.4) to exist, the inverse of Wc (t) needs to exist, i.e. Wc (t) must have full rank for every t > 0. For a stable timeinvariant system it is enough to require Wc (∞) to have full rank. Hence, state controllability can be investigated by considering the controllability Gramian, P , defined for stable time-invariant systems as Z ∞ T P , eAτ BB T eA τ dτ. (2.6) 0 If P has full rank the system is state controllable. Similarly, a stable system will be state observable if the observability Gramian, Q, defined as Z ∞ T Q, eA τ C T CeAτ dτ (2.7) 0 has full rank. These Gramians can be obtained by solving the following continuous-time Lyapunov equations (Skogestad and Postlethwaite, 1996): AP + P AT + BB T = 0, 21 (2.8a) AT Q + QA + C T C = 0. (2.8b) The rank of P is the dimension of the controllable subspace corresponding to the given system, and correspondingly, the rank of Q is the dimension of the observable subspace of the same system. State controllability and state observability can also be examined by considering the matrices1 Wc , [B AB ... AN −1 B], (2.9a) C CA .. . (2.9b) Wo , . N −1 CA The system (A, B) is then state controllable if Wc has full rank N where N is the number of states. Similarly, the system (A, C) is state observable if Wo has full rank N . Even though a system is state controllable, it should be noted that there is no guarantee that the system can remain in its final state x1 as t → ∞. Furthermore, nothing is said about the behaviour of the required inputs. These can both be very large and change suddenly. Therefore, state controllability is rather a result of theoretical interest than a result of practical importance. 2.3.2 State controllability for discrete-time systems The discrete-time case can be treated similarly. The discrete controllability Gramian is given by (Weber, 1994) P = Wc WcT (2.10) and the discrete observability Gramian by Q = WoT Wo . (2.11) Similarly to the continuous-time case, these Gramians can also be obtained as the solutions to the (discrete-time) Lyapunov equations AP AT − P + BB T = 0, 1 (2.12a) It can be verified that the controllable states can be expressed as linear combinations of the matrices B, AB, . . . , An−1 B by considering the solution to (2.1) given in (2.3) (let x0 = 0) and expressing eAt as a power series and using the Cayley-Hamilton theorem. This motivates the introduction of Wc . The Cayley-Hamilton theorem says that every quadratic matrix satisfies its own characteristic equation. For details, see for instance Glad and Ljung (1989) where also further motivations for the definition of Wo are given. 22 AT QA − Q + C T C = 0. (2.12b) Once again, the same symbols are used for both the continuous-time quantities and the discrete-time counterparts. Note also that in the continuoustime case, P and Q cannot be obtained from Wc and Wo as in (2.10) and (2.11) for discrete-time systems. 2.3.3 Output controllability Whereas state controllability considers the ability of affecting the states of a given system by manipulating the inputs, output controllability rather considers the situation of affecting the outputs by means of manipulating the inputs. In practical control problems it is often more relevant to be able to control the outputs rather than the states (see e.g. Kreindler and Sarachik (1964)). State controllability is “neither necessary nor sufficient” to be able to control the outputs as pointed out by Kreindler and Sarachik (1964). According to Skogestad and Postlethwaite (1996), state controllability is rather a “system theoretical concept” and it “does not imply that the system is controllable from a practical point of view.” For this reason the concept of output controllability was introduced. Kreindler and Sarachik (1964) discuss time-varying plants of the form given in (2.1) and defines a plant as being “completely output-controllable on [to , tf ] if for given t0 and tf any final output y(tf ) can be attained starting with arbitrary initial conditions in the plant at t = t0 .” For a plant without a direct term, i.e. D(t) = 0, this holds if and only if the Gramian Poc (t0 , tf ) , Z tf t0 Hy (tf , τ )HyT (tf , τ )dτ (2.13) is non-singular (Kreindler and Sarachik, 1964) where Hy (t, τ ) is the impulse response matrix (Skogestad and Postlethwaite, 1996), For linear timeinvariant stable plants with t0 set to 0 the Gramian in (2.13) transforms to the output controllability Gramian given by Z ∞ T Poc = CeAτ BB T eA τ C T dτ 0 Z ∞ T = C eAτ BB T eA τ dτ C T 0 = CP C T . (2.14) For plants including direct transmission (i.e. D 6= 0) the matrix DDT has to be added to the Gramian in (2.14) for the output controllability criteria to be valid. In contrast to the state controllability Gramian, P , Poc is independent of the selected state-space realization. To see this, change the state coordinates 23 by multiplying the state vector x(t) with a linear non-singular transformation matrix T . This is a similar transformation that transforms the state vector x(t) to z(t) = T x(t). The plant can now be described by ż(t) = T AT −1 z(t) + T Bu(t), y(t) = CT −1 z(t) + Du(t). (2.15) For the new realization, the output controllability Gramian becomes Z ∞ −1 −1 ∗ ∗ ∗ ′ Poc = CT −1 eT AT t T BB ∗ T ∗ e(T ) A T t (T −1 )∗ C ∗ dt Z0 ∞ ∗ = CT −1 T eAt T −1 T BB ∗ T ∗ (T −1 )∗ eA t T ∗ (T −1 )∗ C ∗ dt 0 Z ∞ ∗ = C eAt BB ∗ eA t dt C ∗ 0 = CP C ∗ = Poc (2.16) where it has been utilized that n o −1 A2 t2 eT AT t = eAt = I + At + + . . . = . . . = T eAt T −1 2! and that the plant is assumed to be time-invariant so that C is independent ′ = P of time. Clearly, Poc oc and thus Poc is independent of the selected state-space realization. 2.4 The Relative Gain Array (RGA) The static RGA for a quadratic plant is given by RGA(G) = G(0). ∗ (G(0)−1 )T (2.17) where G(0) is the steady-state transfer function matrix and “.∗” denotes the Hadamard or Schur product (i.e. elementwise multiplication). Each element in the RGA can be regarded as the quotient between the open-loop gain and the closed-loop gain. Hence, the RGA element (i, j) is the quotient between the gain in the loop between input j and output i when all other loops are open and the gain in the same loop when all other loops are closed. For a full derivation of the RGA, see e.g. Bristol (1966), Kinnaert (1995) or Skogestad and Postlethwaite (1996). In the case of a 2×2 system, the following RGA matrix is obtained: λ 1−λ RGA(G) = . (2.18) 1−λ λ Depending on the value of λ, five different cases occur (Kinnaert, 1995): 24 λ = 1: This is the ideal case when no interaction between the loops is present. The pairing should be along the diagonal, i.e. u1 − y1 and u2 − y2 ; λ = 0: This is the same situation as above, except that now the suggested pairing is along the anti-diagonal, i.e. u1 − y2 , u2 − y1 0 < λ < 1: This case is not desirable since the gain increases (i.e. ĝij increases) when the loops are closed, hence, there is interaction; λ > 1: Now, the gain decreases when the loops are closed. This situation is therefore also undesirable. λ < 0: This situation corresponds to the worst case scenario since now, even the sign changes when the loops are closed and this is highly undesirable. The conclusion is that u1 should only be paired with y1 when λ > 0.5, otherwise it should be paired with y2 . For the higher-dimensional case, the rule should be to choose pairings that have an RGA-element close to one. Negative pairings should definitely be avoided. 2.5 2.5.1 Gramian based interaction measures The Hankel norm The controllability and observability Gramians as defined in (2.6) and (2.7) can be seen as measures of how hard it is to control and to observe the states of the given system. Unfortunately, both of these Gramians depend on the chosen state-space realization. However, as can be verified, the eigenvalues of the product of these will not. The Hankel norm for a system with transfer function G (continuous-time or discrete-time) can be calculated as p kGkH = λmax (P Q) = σ1H (2.19) where σ1H is the maximum Hankel singular value (HSV). Clearly, this measure is invariant with respect to the state-space realization and it is therefore well suited as a combined measure for controllability and observability. In fact, the Hankel singular values can be interpreted as a measure of the joint controllability and observability of the states of the considered system, see for instance Farsangi et al. (2004), Skogestad and Postlethwaite (1996) and Lu and Balas (1998). Furthermore, the HSV:s of G can be regarded as measures of the gain between past inputs and future outputs since these are the singular values of the Hankel matrix (defined below) for discretetime systems, or equivalently, for the Hankel operator (defined below) for 25 continuous-time systems (Zhou et al., 1996; Skogestad and Postlethwaite, 1996; Wilson, 1989; Antoulas, 2001; Weber, 1994; Glover, 1984; Wittenmark and Salgado, 2002). To see this, consider the discrete-time time-invariant system given in (2.2) and let D = 0. Assume starting from zero initial state, x(−L) = 0, the influence of the L past inputs on the state x(0) is given by (Weber, 1994; Glover, 1984) x(0) = [B AB u(−1) u(−2) .. . ... AL−1 B] u(−L) = Wc u(−1) u(−2) .. . u(−L) (2.20) and the influence of the initial state x(0) on the L future outputs is given by (Weber, 1994; Glover, 1984) y(0) y(1) .. . y(L − 1) = C CA .. . CAL−1 x(0) = Wo x(0) (2.21) where it is assumed that u(t) = 0 for t ≥ 0. When L = N , Wc and Wo are the controllability and observability matrices, respectively. For L > N these are the extended controllability matrix and the extended observability matrix. To be able to reconstruct all of the states in the state vector at time 0, i.e. x(0), from the past inputs according to (2.20), Wc must have full rank N . Similarly, Wo must have full rank N so that the outputs can be found from (2.21). For a derivation of (2.20) and (2.21), see Weber (1994). Combining (2.20) and (2.21) the result is the following expression that links the past inputs to the future outputs via the state x(0) at time zero (Weber, 1994; Antoulas, 2001) y(0) y(1) .. . y(L − 1) = W x(0) = W W o o c u(−1) u(−2) .. . u(−L) = Γ u(−1) u(−2) .. . u(−L) . (2.22) Γ is the Hankel matrix which in the considered time-invariant case is defined as (Antoulas, 2001; Weber, 1994) Γ= S1 S2 .. . S2 S3 .. . ... ... SL SL+1 .. . SL SL+1 . . . S2L−1 26 (2.23) where {Sk } are the Markov2 parameters of the system. For multiple-input multiple-output (MIMO) systems the Markov parameters are matrices and consequently, the Hankel matrix is a block matrix. The impulse response in the discrete-time case for the system given in (2.2) with D = 0 is given by (Antoulas, 2001) CAt−1 B t > 0 h(t) = (2.24) 0 t < 0. For this reason the Hankel matrix can be expressed as CB CAB . . . CAL−1 B CAB CA2 B . . . CAL B Γ= .. .. .. . . . CAL−1 B CAL B . . . CA2L−2 B . (2.25) Clearly, it follows that Γ = Wo Wc as stated in (2.22). The Hankel singular values equal the non-zero singular values of the Hankel matrix. In the continuous-time case, the counterpart to the Hankel matrix is the Hankel operator Γ given by (see e.g. Antoulas (2001)) Z ∞ y(t) = (Γu)(t) = g(t + τ )u(−τ )dτ t≥0 (2.26) 0 where g(t) is the continuous-time impulse response matrix given by 0 t<0 g(t) = (2.27) At Ce B t≥0 when D in (2.2) is assumed to be 0. Similarly to the discrete-time case, the Hankel operator relates the past inputs to the future outputs. The Hankel singular values are the same as the singular values of the Hankel operator. However, note that these do not coincide with the singular values of the corresponding Markov parameters as in the discrete-time case. The Hankel norm can also be regarded as an induced norm3 . In fact, it is the induced operator norm of the Hankel operator. For a stable system 2 A strictly proper continuous transfer function G(s) can be expressed as a power series P∞ −k in the Laplace variable s as G(s) = where {gk } = {Sk } are the Markov k=1 gk s parameters. In the discrete-time case, the Markov parameters are the impulse response. (Weber, 1994) 3 Let || · || be some vector norm. Then the norm ||A||i = max x6=0 ||Ax|| ||x|| is said to be an induced norm of the current vector norm. ||A||i can be interpreted as the maximum gain for all possible input directions of a system with amplification A and input x. See Skogestad and Postlethwaite (1996), Zhou et al. (1996) and Horn and Johnson (1985) for a more detailed description of induced norms. 27 G(s), the Hankel norm is given by (Skogestad and Postlethwaite, 1996) qR ∞ 2 0 ||y(τ )||2 dτ ||G(s)||H = max qR . (2.28) 0 u(t) 2 dτ ||u(τ )|| 2 −∞ This expression can be interpreted as follows (Skogestad and Postlethwaite, 1996, p.155): Apply an input u(t) up to time t = 0 and then measure the resulting output y(t) for t > 0 and maximize the 2-norm ratio between these signals. For a more thorough description of the continuous-time Hankel operator, see for example Glover (1984), Zhou et al. (1996), Antoulas (2001), Wilson (1989), Birk and Medvedev (2003) and Weber (1994). 2.5.2 Energy interpretations of the controllability and observability Gramians for discrete-time systems The controllability and observability Gramians can also be interpreted in terms of energy (Weber, 1994). One way of expressing the energy in a signal is to calculate its square-sum. From Equation (2.21) the energy released from a given state x(0) is L−1 X |y(n)|2 = xT (0)WoT Wo x(0) = xT (0)Qx(0) (2.29) n=0 where Q is the discrete-time observability Gramian. Hence, a small Q (i.e. the eigenvalues of Q are small) corresponding to low observability implies that the state variables release a small amount of energy in the outputs. Similarly, the controllability Gramian may be seen as a measure of the amount of energy that is needed in the inputs to obtain a given state x(0). This energy can be expressed as L−1 X |u(−n)|2 = . . . = xT (0)P −1 x(0). (2.30) n=0 If the plant is hard to control, P will have small eigenvalues and the eigenvalues of P −1 will be large. Therefore a large amount of energy is needed in the inputs to reach the desired state x(0). Similar interpretations can be made in the continuous-time case but are omitted here. For further details see Weber (1994) and Glover (1984). 2.5.3 The Hankel Interaction Index Array (HIIA) A stable MIMO system represented by (A, B, C, 0) can be split into fundamental SISO subsystems (A, Bj , Ci , 0) with one input uj and one output yi 28 each, where Bj is the j:th column in B, Ci is the i:th row in C (Conley and Salgado, 2000; Salgado and Conley, 2004). For each of these, the controllability and the observability Gramians can be calculated. Furthermore, the controllability and observability Gramians for the full system will be the sum of the Gramians for the subsystems. If the Hankel norm is calculated for each fundamental subsystem and arranged in a matrix Σ̃H given by [Σ̃H ]ij = kGij kH (2.31) this matrix can be used as an interaction measure. A normalized version is the Hankel Interaction Index Array (HIIA) proposed by Wittenmark and Salgado (2002): kGij kH [ΣH ]ij = P . (2.32) kl kGkl kH With the normalization, the sum of the elements in ΣH is one. The larger the element, the larger the impact of the corresponding input signal on the specific output signal. Hence, expected performance for different controller structures can be compared by summing the corresponding elements in ΣH . Clearly, due to the normalization, the aim is to find the simplest controller structure that corresponds to a sum as near one as possible. Of course, a big difficulty could be to decide whether an entry in the HIIA matrix is large enough to be relevant or not, and there are currently no clear rules for this. If the intention is to find a decentralized controller, the HIIA can be used and interpreted in a similar way to the RGA. When Gij = 0 the Gramian product, P (j) Q(i) , will be zero and so will the corresponding element in the matrix ΣH . This implies that the structure of ΣH will be the same as the structure of G and thus, non-diagonal elements will not be hidden as in the case of the RGA (see for instance Halvarsson et al. (2005) or Chapter 3). Hence, the HIIA can also be used to evaluate other controller structures than just the diagonal, decentralized, ones. 2.5.4 The Participation Matrix (PM) The Hankel norm is given by the largest HSV (see Section 2.5.1). For elementary (SISO) subsystems with only one HSV this is no issue. However, for subsystems with several HSV:s it can be argued that a more relevant way of quantifying the interactions is to take into account all of the HSV:s, at least if there are several HSV:s that are of magnitudes close to the maximum HSV. One way of doing this is to calculate the trace of the Hankel matrix (for discrete-time systems) or Hankel operator (for continuous-time systems) – or equivalently of the Gramian product P Q. This is what is done in the participation matrix (PM) approach, proposed by Conley and Salgado (2000). Each element in the PM is defined as φij = tr(Pj Qi ) tr(P Q) 29 (2.33) where tr denotes the trace. tr(Pj Qi ) is then the sum of the squared HSV:s of the subsystem with input uj and output yi . The measure tr(P Q) is, however, in general not an induced norm such as the Hankel norm. Only when the system has rank one (so that only one eigenvalue exists) tr(P Q) is an induced norm (it then coincides with the Hankel norm). Note that tr(P Q) equals the sum of all tr(Pj Qi ), i.e. X tr(P Q) = tr(Pj Qi ). (2.34) i,j See Salgado and Conley (2004) and Salgado and Oyarzún (2005) for a further discussion of PM theory and properties. 2.5.5 The selection of proper scaling All of the considered Gramian based interaction measures depend on the selected scaling of the system. This means that some effort must be spent on finding proper scaling matrices. Salgado and Conley (2004) deal with this issue by normalizing the ranges for the considered signals. However, what seems to matter is that the scaled system has a fairly low condition number. As a guidance what fairly low means, the minimized condition number (see Appendix A) can be of interest. For instance, in the interaction studies of bioreactor models performed by Halvarsson et al. (2005) and Samuelsson et al. (2005c) the scaling matrices were selected so that the maximum deviations from the average point of the considered signals lie in the interval [-1,1]. This scaling procedure significantly reduced the steady state condition number for the plants (i.e. the condition number for G(0)): from between 5046 and 2.4·106 for the different operating points to between 7 and 95, and from 2145 to 6.0, respectively. The minimized condition numbers were 1 and 2.4. 2.6 An interaction measure based on the H2 norm Birk and Medvedev (2003) suggest the use of the H2 norm and the H∞ norm as bases for new interaction measures. The proposed interaction quantifiers share the same form as the HIIA given in (2.32) but with the use of the H2 norm and the H∞ norm instead of the Hankel norm. In this section the H2 norm based interaction measure proposed by Birk and Medvedev (2003) will be defined, properties of the H2 norm will be reviewed and some interpretations of the H2 norm will be given. Finally some properties of the H2 norm based interaction measure will be derived. 2.6.1 The Σ2 interaction measure Birk and Medvedev (2003) suggest a new interaction measure, here denoted 30 Σ2 , similar to the HIIA but with the Hankel norm interchanged by the H2 norm, i.e. kGij k2 [Σ2 ]ij = P . (2.35) kl kGkl k2 This measure is normalized in the same way as the HIIA and the PM and should be used in the same manner as these to analyse the interactions present in MIMO systems. 2.6.2 The H2 norm The system H2 norm for a stable and strictly proper (i.e. D = 0) system with transfer function G(s) is given by (Skogestad and Postlethwaite, 1996) s Z ∞ 1 ||G(s)||2 = tr G∗ (jω)G(jω) dω. (2.36) 2π −∞ By the use of Parseval’s relation, the above equation can be expressed as (ibid.) sZ ∞ tr gT (τ )g(τ ) dτ ||G(s)||2 = ||g(t)||2 = 0 v uX Z u = t i,j ∞ |gij (τ )|2 dτ (2.37) 0 where g is the impulse response matrix. Hence, the H2 norm can be interpreted as the energy of the impulse response, see for example Zuo and Nayfeh (2003) and Zhou et al. (1996). Furthermore, the H2 norm of a given stable system can be seen as the sum of the H2 norm of the outputs that are produced if a unit impulse is applied to each input, one after another. This interpretation follows from (2.37) (Skogestad and Postlethwaite, 1996). For a SISO system (2.36) becomes s Z ∞ 1 |G(jω)|2 dω (2.38) ||G(s)||2 = 2π −∞ and hence, the H2 norm is proportional to the integral of the magnitudes in the Bode diagram. Clearly, the H2 norm can be regarded as a measure of energy. In the case of (continuous) unit variance white noise input the H2 norm is the power, or root-mean-square (RMS), of the output signal y(t). To see this, consider the power semi-norm of y(t) given by (Zhou et al., 1996) Z T 1 ||y||2RM S = lim ||y(τ )||2 dτ = tr Ryy (0) (2.39) T →∞ 2T −T 31 where Ryy is the autocorrelation matrix of y(t) defined as Z T 1 Ryy (τ ) = lim y(t + τ )y ∗ (t)dt T →∞ 2T −T (2.40) where y ∗ (t) is the transposed complex conjugate of y(t). The Fourier transform of Ryy (τ ) is the spectral density Syy (jω). If this function is known, Ryy can be obtained as Z ∞ 1 Ryy (τ ) = Syy (jω)ejωτ dω. (2.41) 2π −∞ If the stable and strictly proper transfer function G relates the input u(t) with the output y(t) then Syy (jω) = G(jω)Suu (jω)G∗ (jω). (2.42) When the input is unit variance white noise, i.e. the spectral density Suu (jω) = I, where I is the identity matrix, the RMS norm of the output is given by ||y||2RM S = tr Ryy (0) Z ∞ 1 = tr Syy (jω) dω 2π −∞ Z ∞ 1 = tr G(jω)G∗ (jω) dω 2π −∞ Z ∞ 1 = tr G∗ (jω)G(jω) dω 2π −∞ = ||G||22 (2.43) where Equations (2.39, 2.41, 2.42, 2.36) and the fact that tr GG∗ = tr G∗ G have been used. This shows that ||G||2 = ||y||RM S in the case of unit variance white noise input. 2.6.3 Calculation of the H2 norm For a continuous-time strictly proper system the H2 norm can be calculated as Z ∞ 2 2 ||G(s)||2 = ||g(t)||2 = tr gT (τ )g(τ ) dτ Z0 ∞ T T = tr (CeA τ B)T CeA τ B dτ 0 Z ∞ T T = tr B eA τ C T CeAτ dτ B 0 T = tr B QB (2.44) 32 where the impulse response matrix for a continuous-time stable system given in (2.27) has been utilized with D =0 and the definition of the observability T T Gramian, Q, in (2.7). Since tr xx = tr x x holds, where x is a vector, the derivation in (2.44) can also be written as Z ∞ 2 2 T ||G(s)||2 = ||g(t)||2 = tr g (τ )g(τ ) dτ Z0 ∞ = tr g(τ )gT (τ ) dτ Z0 ∞ = tr CeAτ B(CeAτ B)T dτ 0 Z ∞ T = tr C eAτ BB T eA τ dτ C T 0 = tr CP C T . (2.45) For a stable strictly proper system G given by the state space description (A, B, C, 0) the H2 norm may therefore be calculated as r r T (2.46) ||G||2 = tr B QB = tr CP C T . Hence, [Σ2 ]ij in (2.35) can be calculated as q tr Ci Pj CiT [Σ2 ]ij = P q . T tr C P C k l k kl (2.47) For SISO systems CP C T reduces to a scalar. Therefore, for the considered fundamental subsystems in Σ2 , tr CP C T = CP C T and consequently, in this particular case, CP C T = B T QB. 2.6.4 Energy interpretation for discrete-time systems Since tr CP C T = tr B T QB it is of interest to interpret the quantity B T QB in terms of energy. Consider the discrete-time system given in (2.2) and let D = 0 and recall that the observability Gramian is given by Q = WoT Wo (Equation (2.11)). Thus B T QB can be expressed as (use (2.9b) and (2.25)) CB CAB T T T T T B QB = B Wo Wo B = B Wo (2.48) = ΓT (:, 1)Γ(:, 1) .. . | 33 CAL−1 B {z } Γ(:,1) where Γ(:, 1) is the first column of the Hankel matrix given in (2.25). In the SISO case, when B and C have dimensions N × 1 and 1 × N , respectively, the first column of the Hankel matrix equals the transpose of the first row, i.e. Γ(:, 1) = ΓT (1, :). Hence, B T QB = ΓT (:, 1)Γ(:, 1) = Γ(1, :)ΓT (1, :). (2.49) In the MIMO case the Hankel matrix is a block matrix and for this reason the property used in (2.49) does not hold. As can be seen from the first row of Equations (2.22) and (2.25) u(−1) u(−2) y(0) = Γ(1, :) (2.50) . .. . u(−L) | {z } U Therefore the power of y(0) may be calculated as |y(0)|2 = y T (0)y(0) = U T ΓT (1, :)Γ(1, :) U = U T Γ̃U. (2.51) Equation (2.51) indicates that the matrix Γ̃ may be interpreted as the matrix that transfer energy from the past inputs to the current output y(0). A way of quantifying this energy transmission is to calculate the trace of this matrix. In the SISO case (use (2.49)) tr Γ̃in = tr ΓT (1, :)Γ(1, :) = tr Γ(1, :)ΓT (1, :) = tr B T QB = B T QB. (2.52) Since B T QB = CP C T in the SISO case – as for the fundamental subsystems considered in Σ2 – this implies that CP C T , or tr(CP C T ) the H2 norm, of each fundamental subsystem can be seen as the coupling in terms of the energy transmission rate (power) between the past inputs U and the current output y(0). This view supports the previously discussed energy interpretations of the H2 norm in Section 2.6.2. One key difference in the interpretations of the HIIA and the PM compared to the H2 norm based interaction measure Σ2 is that the former quantify the interactions between (past) inputs and (future) outputs directly (see Equation (2.22)) whereas the latter relates (past) inputs with (present) outputs in terms of energy (see e.g. Equation (2.51)), i.e. it relates squared inputs and squared outputs. Obviously, there are several interesting interpretations of the H2 norm based interaction measure Σ2 . The definition of output controllability given by Kreindler and Sarachik (1964) and Equations (2.13–2.14) indicate that 34 the considered measure in fact is a measure of the output controllability of the plant. Recall that the HIIA and the PM rather are quantifiers of the combined state controllability and state observability. Furthermore, the measure Σ2 can be interpreted both in terms of energy and as the RMSvalue, depending on the type of inputs, as previously discussed. 2.6.5 Properties of the H2 norm based interaction measure Σ2 Here, some of the basic properties of the considered interaction measure Σ2 will be derived. Independence of realization The Σ2 is independent of the selected realization. This follows from the definition in (2.35). It is also a consequence of CP C T being independent of realization as shown in Section 2.3.3. Preservation of structure The structure of the plant, G, is preserved in Σ2 . To see this, assume Gij = 0 for some i, j 6= 0. Then tr(Ci Pj CiT ) = ||Gij ||2 = 0 and the stated property follows. Frequency scaling One advantage of the HIIA and the PM is that they are insensitive to frequency scaling. It can be verified (Salgado and Conley, 2004) that for a frequency scaled system with transfer function Gij (s/ξ) the corresponding controllability and observability Gramians transform to P̂j = Pj /ξ and Q̂i = ξQi and clearly, P̂ Q̂ = P Q which explains why the PM and the HIIA are preserved. However, the quantity Ci Pj CiT will be affected by the frequency scaling since Ĉi P̂j ĈiT = Ci P̂j CiT = Ci Pj T C . ξ i (2.53) Fortunately, the influence of the frequency scaling on the interaction measure Σ2 given in Equation (2.35) will be canceled in the normalization since it 1 P 1 P corresponds to a division by kl tr(Ck P̂l CkT ) 2 = kl tr(Ck (Pl /ξ)CkT ) 2 . 35 Time delays Since time delays may alter the process dynamics significantly, knowledge of these are important when selecting the controller structure. Since only discrete time systems are able to model time delays in finite-dimensional state-space models the following discussion will address this type of systems. Salgado and Conley (2004) show that the PM is able to indicate the presence of a time delay. In fact, they prove that a pure time delay of 1/z ℓ , ℓ ∈ N, applied at the output of the non-delayed system, gives an extra contribution to the expression tr(P Q) for the delayed system: tr(Pℓ Qℓ ) = tr(P0 Q0 ) + ℓ · C0 P0 C0T = tr(P0 Q0 ) + ℓ||G0 ||22 (2.54) where the index 0 is used for the non-delayed system and the index ℓ for the delayed system. Following the proof therein, it can be verified that the quantity CP C T will not be affected by the time delay, i.e. Cℓ Pℓ CℓT = C0 P0 C0T ℓ ∈ N. (2.55) Hence Σ2 is unaffected by time delays. However, as will be illustrated later in Example 4 in Section 2.7, the presence of a time delay is by itself not a sufficient reason to include – nor exclude — that particular input-output pair in the controller. 2.6.6 The H2 norm and induced norms The Hankel norm is the induced operator norm of the Hankel operator in the continuous-time case. The same interpretation holds in the discrete-time case if the Hankel matrix is seen as an operator. It should be noted that the H2 norm is not an induced norm. For instance it does not satisfy the multiplicative property (Skogestad and Postlethwaite, 1996) ||AB|| ≤ ||A|| ||B||. (2.56) In practice, this means that the behaviour of a connected system cannot be judged from the H2 norm of the components of the system (ibid.). However, as previously discussed, for SISO systems CP C T is a scalar. This implies that tr CP C T = λ(CP C T ) = λmax (CP C T ) (2.57) where λ denotes the eigenvalues and λmax is the largest one. As pointed out by Wilson (1989), Antoulas (2001), Chellaboina et al. (1999) and Dharmasanam et al. (1997) λmax (CP C T ) is in fact the 2-∞ induced operator norm of the convolution operator: q ||y||∞ = λmax (CP C T ) where ||y||∞ = sup max yi (t). ||G||2,∞ = sup i t u6=0 ||u||2 (2.58) 36 The convolution operator G is defined as (Dharmasanam et al., 1997) y(t) = (Gu)(t) = Z ∞ g(t − τ )u(τ )dτ. (2.59) −∞ Therefore, for the considered H2 norm based interaction measure Σ2 , the issue of the H2 norm of not being an induced norm in the general case is of minor interest. Furthermore the way the interaction measure is supposed to be used does not mean that the norm necessarily has to satisfy the multiplicative property (2.56). As can be seen in Equation (2.58), the 2-∞ induced operator norm of the convolution operator can be interpreted as the maximum output amplitude that finite energy input signals give rise to (Antoulas, 2001). As pointed out by Wilson (1989), this norm can be useful in a control design performance criteria when the aim is to reduce the influence of finite energy input disturbances. Promising (simulation) results have been reported by Rotea (1993) and by Dharmasanam et al. (1997) where various mixed performance criteria based on the H2 norm, the 2-∞ induced operator norm of the convolution operator and the ∞-∞ induced operator norm of the convolution operator were used in the control design. This interpretation is also important for the considered interaction measure. 2.7 Examples The usefulness of the interaction measure Σ2 has previously been exemplified for a linearized model of a coal injection vessel by Birk and Medvedev (2003). In this section, further examples are given. 2.7.1 Example 1 In the first example the interactions present in a quadruple-tank system will be examined (see Johansson (2000) for a general description of this process). The considered linear minimum-phase model is given by the following state space matrices: B= −0.0159 0 0.159 0 0 −0.0159 0 0.02651 , A= 0 0 −0.159 0 0 0 0 −0.02651 0.05459 0 0 0.07279 , C = 1 0 0 0 , D = 0 0 . (2.60) 0 0.0182 0 1 0 0 0 0 0.03639 0 37 The steady-state transfer function is 3.4326 1.1442 G(0) = . 2.2884 4.5768 (2.61) If this matrix is normalized to make the sum of the magnitudes of all elements equal to 1 the following matrix is obtained: 0.3 0.1 G(0) = . (2.62) 0.2 0.4 G(0) can be seen as a rough measure of the relative importance of each input-output channel in steady-state. The condition number for G(0) is 2.6 and the minimized condition number is around 2.4 so there is no need to scale the system. The RGA, denoted Λ, for the system is 1.2 −0.2 Λ(G(0)) = (2.63) −0.2 1.2 and the Gramian based interaction matrices are 0.2866 0.1029 ΣH = , 0.2285 0.3821 0.2809 0.0364 Φ = , 0.1834 0.4994 0.3168 0.0320 Σ2 = 0.0880 0.5632 (2.64) (2.65) (2.66) where the HIIA matrix is denoted ΣH , the PM Φ and Σ2 is the H2 norm based interaction measure. To be able to make direct comparisons between the three considered Gramian based interaction measures, it is beneficial to calculate the square-root of the PM and of the Σ2 and then renormalize the measures. The following matrices then result: 0.2856 0.1028 Φ = , (2.67) 0.2308 0.3808 0.3146 0.1000 Σ2 = . (2.68) 0.1658 0.4195 In this way the three matrices are expressed in the same units: Recall that the HIIA can be interpreted as the gain between past inputs and future outputs whereas Σ2 can be interpreted in terms of energy (hence the need of the square root). Also recall that the HIIA and the PM differ both by the number of eigenvalues considered and by a square root, see (2.19), (2.32) and (2.33). This means that the PM rather is a measure that quantifies the interaction in terms of energy. 38 As can be seen, the Gramian based interaction quantifiers all rank the importance of the input-output channels in the same order and they advocate the same decentralized diagonal pairing as the RGA does. The product P Q has four eigenvalues: 9.2133, 1.2668, 0.0050 and 0.0016. Since the first one accounts for 88% of tr(P Q) there will only be minor numerical differences between ΣH and Φ as can be appreciated when comparing Equation (2.64) with (2.67). Neither does Σ2 in Equation (2.68) differ much from ΣH in Equation (2.64) nor Φ in Equation (2.67) with the exception of element (2,1). Furthermore, note the resemblance of ΣH , Φ and Σ2 to G(0). This is not surprising in view of the interpretation of the HIIA as being the gain between old inputs and future outputs. If the system is low-pass filtered the resemblance is even closer. Clearly, the considered system has most of its process dynamics in the lower frequencies. Similar arguments can be applied to explain the resemblance Σ2 shows. 2.7.2 Example 2 In the second example, the following 3 × 3-system is analysed G(s) = 0.4 (s+1)2 2 (s+2)(s+1) 6(−s+1) (s+5)(s+4) 4(s+3) (s+2)(s+5) 2 (s+2)2 4 (s+3)2 −2 s+4 1 s+2 8 (s+2)(s+5) . (2.69) Salgado and Conley (2004) compare different control structures based on the advice from the PM for this system. The steady-state gain and the normalized magnitudes G(0) are 0.4000 1.2000 1.0000 0.5000 G(0) = 0.3000 0.4444 0.0709 0.2126 0.1772 0.0886 G(0) = 0.0531 0.0787 −0.5000 0.5000 , 0.8000 0.0886 0.0886 . 0.1417 The interaction measures are −0.0831 0.9111 0.1720 Λ(G(0)) = 1.3809 −0.2745 −0.1064 , −0.2979 0.3634 0.9345 ΣH 0.0703 0.1663 0.0728 = 0.1728 0.0878 0.0728 , 0.1426 0.0781 0.1367 39 (2.70) (2.71) (2.72) Bode Diagram From: In(2) From: In(1) From: In(3) To: Out(1) 0 −20 −40 −60 −80 To: Out(2) Magnitude (dB) 0 −20 −40 −60 −80 To: Out(3) 0 −20 −40 −60 −80 −1 10 0 10 1 10 −1 10 0 1 10 10 Frequency (rad/sec) −1 10 0 10 1 10 2 10 Figure 2.1: Bode magnitude diagram for the non-filtered plant in Example 2. 0.0370 0.2018 0.0385 Φ = 0.2226 0.0578 0.0385 , 0.2193 0.0457 0.1389 0.0687 0.1604 0.0701 Φ = 0.1684 0.0858 0.0701 , 0.1672 0.0763 0.1331 (2.73) 0.0065 0.3545 0.0816 0.0316 0.2331 0.1119 Σ2 = 0.0544 0.0204 0.0408 , Σ2 = 0.0913 0.0559 0.0791 . 0.3429 0.0242 0.0746 0.2292 0.0609 0.1070 (2.74) The RGA, HIIA and PM all advocate the same decentralized pairing: u1 – y2 , u2 – y1 , u3 – y3 . Φ is close to HIIA but it emphasizes the importance of element (3,1) slightly more. Σ2 differs even more and suggests that u1 mostly affects y3 rather than y2 as indicated by the HIIA and the PM. The maximum eigenvalue of P Q amounts to 62% of tr(P Q) explaining the difference between HIIA and Φ. Next, consider the original system low-pass filtered with F LP (s) = 1 s+1 resulting in the interaction matrices 0.0740 0.1933 0.0797 0.1825 0.0908 0.0830 , ΣLP H = 0.0773 0.0787 0.1407 40 (2.75) (2.76) ΦLP 0.0429 0.2776 0.0472 0.0743 0.1889 0.0779 LP = 0.2596 0.0642 0.0517 , Φ = 0.1827 0.0909 0.0815 , 0.0574 0.0477 0.1517 0.0859 0.0783 0.1397 (2.77) 0.0215 0.3736 0.0718 0.0532 0.2216 0.0971 0.1595 0.0498 0.0598 , ΣLP 0.1448 0.0809 0.0887 . ΣLP 2 = 2 = 0.0718 0.0465 0.1458 0.0971 0.0782 0.1384 (2.78) Clearly, all of the Gramian based interaction measures now give the same decentralized pairing recommendation as the RGA and the G(0) do: Σ2 has aligned with the other measures. A high-pass filtering of the original system with the filter F HP (s) = s s+1 gives the following interaction measures: 0.0408 0.1873 0.0868 0.1197 0.0751 0.0724 , ΣHP H = 0.2058 0.0784 0.1336 ΦHP (2.79) (2.80) 0.0118 0.2861 0.0631 0.0400 0.1965 0.0923 HP = 0.1022 0.0402 0.0438 , Φ = 0.1175 0.0736 0.0769 , 0.2803 0.0438 0.1287 0.1945 0.0769 0.1318 (2.81) 0.0021 0.3489 0.0845 0.0192 0.2470 0.1216 HP ΣHP = 0.0235 0.0117 0.0352 , Σ2 = 0.0641 0.0453 0.0785 . 2 0.4227 0.0176 0.0537 0.2719 0.0555 0.0969 (2.82) Once again, all Gramian based interaction measures suggest the same decentralized control structure: u1 – y3 , u2 – y1 , u3 – y2 . This is the same pairing as suggested by Σ2 in the analysis of the original non-filtered system, see Equation (2.74). Recall that the aim is to find a pairing that yields a sum of the elements in the considered Gramian based interaction measure as large as possible. When a decentralized structure is sought-after, the pairing u3 – y2 has to be selected in favour of u3 – y1 and u3 – y3 (even though these corresponding elements is larger than the selected one) since the elements corresponding to u1 – y3 and u2 – y1 make a significant contribution to the sum and should therefore be included. To appreciate the dynamic behaviour of the plant a Bode magnitude diagram is provided in Figure 2.1. Particularly, note the resonance top in 41 the Bode magnitude diagram for u1 to y3 . Since the H2 norm is proportional to the integral of the magnitudes in the Bode diagram the shape of Σ2 is outlined by the Bode diagram. In particular, note the relatively high gain between u1 and y3 for high frequencies. This is reflected in Σ2 in (2.74) and (2.82). 2.7.3 Example 3 In the third example a discrete-time system with time delay is considered and the suggested pairing recommendations are tested in control simulations. The system is given by # " b G(z) = 0.5 (z−0.5) b21 (z−0.5)(z−0.8) 12 (z−0.8)z ℓ 0.3 (z−0.7) . (2.83) When the nonnegative integer ℓ > 0 there is a time delay present in the channel between u2 and y1 . The values of the parameters b12 and b21 are varied so that the static gain in the anti-diagonal channels (including the channel with time delay) are either relatively high or relatively low compared to the static gain of the diagonal channels. In the first case, b12 = 0.15 and b21 = 0.1 which corresponds to the low gain scenario. This system has previously been analysed by Salgado and Conley (2004) using the PM. Without the extra time delay in element G12 , i.e. with ℓ = 0 the following steady state transfer function G(0), normalized magnitudes G(0) and interaction matrices result: 1.0000 0.7500 0.2667 0.2000 G(0) = , G(0) = , (2.84) 1.0000 1.0000 0.2667 0.2667 4.0000 −3.0000 Λ(G(0)) = −3.0000 4.0000 0.2882 0.1801 ΣH = , 0.2774 0.2543 Φ= Σ2 = 0.3171 0.1239 0.3122 0.2469 0.5060 0.0949 0.1312 0.2679 , , Φ= Σ2 = , 0.2857 0.1786 0.2835 0.2521 0.3746 0.1622 0.1907 0.2725 (2.85) (2.86) , (2.87) . (2.88) All of the interaction quantifiers favour the diagonal pairing (ui – yi ) for decentralized control. The RGA also indicates that the anti-diagonal pairing should be avoided due to stability issues (the RGA has negative anti-diagonal elements). As pointed out by Salgado and Conley (2004) better control 42 performance could be expected if a sparse controller structure is designed which also includes the coupling between u1 and y2 . This can for instance be achieved by introducing a suitable feedforward. When an extra time delay, ℓ = 10, is introduced between u1 and y2 the HIIA and the PM change to 0.2631 0.2514 ΣH = , (2.89) 0.2533 0.2322 0.2193 0.3941 0.2372 0.3180 Φ= , Φ= . (2.90) 0.2159 0.1707 0.2354 0.2093 Recall that the Σ2 (and the RGA) remains unaffected by the time delay. As described in Section 2.6.5 the PM is able to detect time delays and this is what is seen in (2.90). Now the PM recommends the anti-diagonal pairing in contrast to both Σ2 and the RGA. It is hard to draw any clear conclusions from the HIIA in (2.89). To validate the relevance of these pairing recommendations decentralized, integrating, controllers were designed using a polynomial pole-placement methodology. The results are visualized in Figure 2.2. The upper plot shows the plant outputs for the system without the extra time delay controlled by a diagonal controller. The sampling time was set to 0.5 s in all of the control simulations. For each of the selected channels a SISO controller were designed with poles in z = 0.4. Unit step changes were applied at time 0 and 15 [s]. The lower plot shows control of the plant with the very same controller but when the extra time delay of ℓ = 10 was introduced. Furthermore, for the delayed system a controller with anti-diagonal pairing as suggested by the PM were designed. The poles were placed in different locations and the time delay was accounted for in the controller but stable control of the plant could not be obtained unless the two input-output channels that were not included in the controller (i.e. u1 – y1 and u2 – y2 ) were detached. In the second system the parameters b12 and b21 were increased to 0.8 and 0.4, respectively, so that the importance of the anti-diagonal elements in G in (2.83) increases. This way the influence of the time delay in G12 should be more prominent compared to the previous case. The following steady state transfer function G(0), normalized magnitudes G(0) and interaction matrices were then obtained: 1.0000 4.0000 0.1000 0.4000 G(0) = , G(0) = , (2.91) 4.0000 1.0000 0.4000 0.1000 −0.0667 1.0667 Λ(G(0)) = , (2.92) 1.0667 −0.0667 0.1103 0.3677 ΣH = , (2.93) 0.4247 0.0973 43 Diagonal control 1.4 Outputs 1.2 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 30 35 30 35 Time [s] Diagonal control of the time delayed plant 1.4 Outputs 1.2 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 Time [s] Figure 2.2: Plant outputs for decentralized diagonal control of the system (2.83) with b12 = 0.15 and b21 = 0.1 in Example 3. The upper plot shows the control of the plant with ℓ = 0 and the lower plot control of the delayed one (ℓ = 10). The solid lines show output 1 and the dashed ones output 2. Φ= Σ2 = 0.0349 0.3879 0.5500 0.0272 0.0908 0.4844 0.3767 0.0481 , , Φ= Σ2 = 0.1089 0.3629 0.4321 0.0961 0.1646 0.3802 0.3353 0.1198 , (2.94) . (2.95) Evidently, the higher gain in the anti-diagonal input-output channels made all interaction measures to recommend the anti-diagonal pairing for decentralized control in contrast to the previous low gain case. With a time delay of ℓ = 10 the HIIA and the PM become 0.0923 0.4706 , (2.96) ΣH = 0.3556 0.0815 0.0146 0.7446 0.0908 0.4844 Φ= , Φ= . (2.97) 0.2295 0.0113 0.3767 0.0481 As seen there are no changes in the decentralized recommendation from these measures when the time delay is introduced. Controllers for these cases were also designed in a similar manner as for the previously studied systems. Figure 2.3 shows the plant outputs for the 44 Anti−diagonal control 1.4 1.2 Outputs 1 0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 60 70 80 90 100 90 100 Time [s] Anti−diagonal control of the time delayed plant 1.4 1.2 Outputs 1 0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 60 70 80 Time [s] Figure 2.3: Plant outputs for decentralized anti-diagonal control of the system (2.83) with b12 = 0.8 and b21 = 0.4 in Example 3. The upper plot shows the control of the plant with ℓ = 0 and the lower plot control of the delayed one (ℓ = 10). The solid lines show output 1 and the dashed ones output 2. Note that the time scale is different from the scale in Figure 2.2. control simulations for the non-delayed system (upper plot) and the delayed system (lower plot). In both of these cases the anti-diagonal input-output pairing were selected with satisfactory control performance. Note that in this particular case it was necessary to include the time delayed channel in the controller. Due to the time delay the controllers had to be slow: The poles were placed in z = 0.9. A working controller based on the diagonal pairing was not found, and was not expected to be found due to the severe cross-couplings that would then be present. The control simulations indicate that the presence of a time delay by itself is not a reason enough to say that this particular input-output pair should be included in the controller when a decentralized controller structure is desired. This is, for this particular example, in contrast to the indications given by the PM, but in agreement with those of the Σ2 . 45 2.7.4 Example 4 Now consider the 2 × 2 process given by: " −40s G(s) = 5e 100s+1 −5e−4s 10s+1 e−4s 10s+1 5e−40s 100s+1 # . (2.98) This process has been extensively analysed by Mc Avoy et al. (2003) and Xiong et al. (2005) with the conclusion that the anti-diagonal pairing is preferred for decentralized control. One reason for this is that the antidiagonal pairing corresponds to faster elements in G. Mc Avoy et al. (2003) came to this conclusion using their dynamic relative gain array (DRGA) and verified it in a simulation study. Xiong et al. (2005) used the ERGA with the same result. The steady state transfer function G(0), the normalized magnitudes G(0) and the interaction matrices are: 5.0000 1.0000 0.3125 0.0625 G(0) = , G(0) = , (2.99) −5.0000 5.0000 0.3125 0.3125 Φ= Σ2 = 0.3289 0.3289 0.0806 0.8065 0.8333 0.1667 Λ(G(0)) = , 0.1667 0.8333 0.3125 0.0625 ΣH = , 0.3125 0.3125 0.0132 0.3125 0.0625 , Φ= , 0.3125 0.3125 0.3289 0.0323 0.1726 0.1091 , Σ2 = . 0.0806 0.5457 0.1726 (2.100) (2.101) (2.102) (2.103) The time delays have been approximated by third order Padé approximations. The RGA, the HIIA and the PM suggest the diagonal pairings for decentralized control. However, the recommendation from the Σ2 is the diagonal pairing which is in agreement with the findings by Mc Avoy et al. (2003) and Xiong et al. (2005). 2.7.5 Example 5 As a concluding example, consider the 3 × 3 process given by: 1 −4.19 −25.96 1−s G(s) = 6.19 1 −25.96 . (1 + 5s)2 1 1 1 (2.104) This process is used by Hovd and Skogestad (1992) as an example of when the RGA does not recommend the most desirable pairing. The steady-state 46 gain and the normalized steady-state gain are: 1.0000 −4.1900 −25.9600 G(0) = 6.1900 1.0000 −25.96000 , 1.0000 1.0000 1.0000 0.0149 0.0623 0.3857 G(0) = 0.0920 0.0149 0.3857 . 0.0149 0.0149 0.0149 The interaction measures are: 1.0009 5.0010 −5.0019 Λ(G(0)) = −5.0028 1.0009 5.0019 , 5.0019 −5.0019 1.0000 (2.105) (2.106) 0.0149 0.0623 0.3857 ΣH = 0.0920 0.0149 0.3857 , (2.107) 0.0149 0.0149 0.0149 0.0149 0.0623 0.3857 0.0007 0.0125 0.4784 Φ = 0.0272 0.0007 0.4784 , Φ = 0.0920 0.0149 0.3857 , 0.0149 0.0149 0.0149 0.0007 0.0007 0.0007 (2.108) 0.0007 0.0125 0.4784 0.0149 0.0623 0.3857 Σ2 = 0.0272 0.0007 0.4784 , Σ2 = 0.0920 0.0149 0.3857 . 0.0007 0.0007 0.0007 0.0149 0.0149 0.0149 (2.109) The RGA recommends the diagonal pairing. However, as found by Hovd and Skogestad (1992) this pairing is not suitable due to instability issues. Instead, they recommend the pairing that corresponds to the RGA elements with values near 5. The same pairing suggestion is found by He and Cai (2004) when considering loop-by-loop interaction energy. The HIIA, the PM and Σ2 all give the same pairing recommendation for decentralized control: u1 – y2 , u2 – y3 and u3 – y1 . For the HIIA this gives a sum of 0.54. However, if the rule of avoiding pairings corresponding to negative RGA elements is obeyed, the HIIA, the PM and Σ2 suggest the very same pairing as the one recommended by Hovd and Skogestad (1992) and He and Cai (2004). This gives for the HIIA a sum of 0.46. 2.8 Conclusions Theoretical arguments for including the H2 norm in an interaction measure were given in Section 2.6. As seen, the H2 norm can be given various useful energy interpretations, for instance it is proportional to the integral of the Bode magnitudes. Furthermore, as also investigated in Section 2.6, it 47 can be seen as a measure of the output controllability of the plant. Some fundamental properties of the H2 norm based interaction measure Σ2 were also derived. The Σ2 was found to be unaffected by time delays. The other Gramian based interaction quantifiers, the Hankel Interaction Index Array (HIIA) and the Participation Matrix (PM), are not. All of the Gramian based interaction measures (including the Σ2 ) are scaling dependent. Therefore a proper scaling of the considered systems is important. The condition number can be compared with the minimized condition number as a guidance in the search for the scaling matrices. In Section 2.7 different interaction measures, including the Σ2 , were compared in the analysis of different multivariable systems. It was found that often the Σ2 is similar to the HIIA and the PM. In other examples it was seen that Σ2 is potentially able to more clearly reveal interactions present for high frequencies than the HIIA and the PM are. However, it should be noted that this way important low frequency behaviour of the plant may be less prominent in the Σ2 . It is therefore of vital importance that the system is filtered in advance in order to focus on the interesting range of frequencies. Furthermore, it was found that the presence of a time delay in one of the input-output channels does not necessarily imply that this channel should be avoided – or selected – for decentralized control design. The impact of a time delay has to be evaluated in each separate case. It was also seen in one example that the Σ2 was able to select the proper pairings in contrast to the RGA, the HIIA and the PM. In the final example, the HIIA, the PM and the Σ2 proposed the correct pairings if their recommendations were combined with the use of the RGA rule of avoiding pairings that correspond to negative RGA elements. This indicates that it could be beneficial to consider several different interaction measures when solving the pairing problem. This is the approach in e.g. (He et al., 2006). However, to give general rules for how to design such a pairing algorithm is out of the scope of this thesis. 48 Chapter 3 Interaction Analysis in a Bioreactor Model In this chapter the well-known Relative Gain Array (RGA) and the more recently proposed Hankel Interaction Index Array (HIIA) are utilized to quantify the degree of channel interaction in a multivariable bioreactor model, an activated sludge process (ASP) configured for nitrogen removal. To be more precise, both the nitrification and the denitrification process in an ASP are studied. The considered model is a 2 × 2 system with the dissolved oxygen concentration set point in the aerobic compartment and the internal recirculation flow rate as input signals (control handles). The effluent concentration of ammoniun and of nitrate are the output signals (the controlled signals). The HIIA can deal with plant structures where the RGA fails and can furthermore also be used to evaluate multivariable controller structures. It was found that the RGA method was unable to give reasonable inputoutput pairing suggestions in some cases while the HIIA method provided useful information in all of the considered cases. 3.1 Introduction When comparing the HIIA to the RGA, there are some major differences. The most important one from the authors point of view is that the RGA assumes a decentralized control structure to be used, and therefore attempts to suggest the best possible input-output pairing. This is not the case with the HIIA, that rather considers the controllability and the observability of every sub-system in the plant separately. This measure can therefore be valuable when evaluating alternatives to decentralized control structures, i.e. multivariable control structures with reduced complexity. In (Conley and Salgado, 2000) and (Salgado and Conley, 2004), it is shown how to do this when employing the similar interaction measure the Participation Matrix (PM) (see Chapter 2 for a discussion of the PM). Another difference is that 49 when using the HIIA, the whole frequency range is taken into account, not only one frequency at the time. As shown in examples given by Wittenmark and Salgado (2002), the HIIA outperforms the RGA when dealing with systems that have interactions with non-monotonic frequency behavior. The reason for this is that the full dynamics of the system will be taken into account when using Gramians. If the objective is to study the interactions in a specific frequency range only, then the transfer functions can be filtered before the HIIA is calculated, see Wittenmark and Salgado (2002). There are also cases, see for instance Kinnaert (1995), where the RGA fails to suggest a proper pairing due to large off-diagonal elements or triangular structure in the plant. On the other hand, a drawback of the HIIA compared to the RGA is that it is scaling dependent. It is therefore of great importance that the system has been scaled in a physically relevant way in order for the HIIA to provide meaningful results. The Gramian based approach is further discussed in Salgado and Conley (2004) and in Section 2.5 in Chapter 2. In this chapter, the RGA and the HIIA will be employed in the selection of input–output signal pairings for a part of a MIMO bioreactor system: an activated sludge process configured for nitrogen removal. Modelling and control of the activated sludge process have been an intense research area in the last decade, see for example Olsson (1993), Lindberg and Carlsson (1996), Alex et al. (1999), Vanrolleghem et al. (1999), Samuelsson and Carlsson (2001), Yuan et al. (2002) and Jeppsson and Pons (2004). The results from the RGA analysis will be compared with those of the HIIA and with results obtained from physical insights of the considered system. It is also discussed what additional conclusions that can be drawn from the HIIA analysis. For a theoretical discussion and definition of the employed interaction measures, i.e. the RGA and the HIIA, see Section 2.4 and Section 2.5 in Chapter 2. 3.2 The bioreactor model In the complex process of wastewater treatment, many different cause-effect relationships exist, and therefore, there are many possible choices of input and output signals, see Olsson and Jeppsson (1994). Consequently, this can motivate the study of wastewater treatment plant models with respect to the selection of input and output signals. From a theoretical point of view, the bioreactor models are non-linear multivariable systems that may contain a significant degree of coupling. Hence, this also gives an interesting opportunity to test the performance of the methods for input-output pairing selection discussed in the previous section. The objective in this chapter is to find suitable control structures. If the couplings between the different control handles in the system are sufficiently 50 Influent, Q Effluent Anoxic Aerobic Internal recirculation, Qi Settler Sludge recirculation Excess sludge Figure 3.1: A basic activated sludge process (ASP) configured for nitrogen removal. low, then a controller selection involving several decoupled SISO controllers may be suitable. If this is not the case, a MIMO controller structure will provide a better solution. The MIMO solution will, however, generally be much more complex. Both the RGA and the HIIA method will be used in the sequel. The considered model is a simplified version of the IAWQ Activated Sludge Model No. 1 (ASM1) that models an activated sludge process configured for nitrogen removal. ASM1 is thoroughly described by Henze et al. (1987). In this study the bioreactor consists of two tanks of equal volume (one anoxic and one aerobic of 1000 m3 each) and a settler, see Figure 3.1. The influent flow rate, Q, is 18 446 m3 /day. The model is valid in the medium time-scale (i.e. hours to days). For a discussion of the model parameters, see Halvarsson (2003). Two different processes, nitrification and denitrification, are simultaneously being performed. To get an indication of how well these processes are being performed the effluent ammonium concentration (SNH,2 (t)) and the nitrate concentration (SNO,2 (t)), respectively, can be considered. Hence, these concentrations are selected as output signals. The considered input signals are the concentration of dissolved oxygen (DO set point, SO,2 (t)) in the aerobic compartment and the internal recirculation flow rate (Qi (t)). According to Ingildsen (2002) the denitrification is mainly influenced by Qi (t) (among the selected input signals) while the nitrification is mainly influenced by SO,2 (t). Hence, if the couplings between Qi (t) and SO,2 (t) are low, then the denitrification and the nitrification process may be considered separately when choosing controller structure and thus, SISO controllers may be selected. Three different operating points were selected1 . These correspond to the input signals: 1 These operating points do not necessarily correspond to feasible choices concerning an optimal operation of the plant. Instead, these are chosen in order to illustrate different interaction points. 51 • u1 = [10 000 m3 /day 2 mg/l]T , • u2 = [36 892 m3 /day 2 mg/l]T , • u3 = [50 000 m3 /day 2 mg/l]T . Since both the RGA and the HIIA are defined for linear models, the simplified ASM1 model was linearized around each operating point using the MATLAB function linmod. In a small neighbourhood of each operating point the linearized model will mimic the characteristics of the nonlinear system. Thus, the analysis in the following sections is strictly valid only in the above mentioned neighbourhoods. However, as can be seen in the lower part of Figure 3.2, the operational maps can be divided into two different regions where the process shows different stationary characteristics. It is therefore probable that each operating point describes the corresponding area fairly well. The obtained linear models can be represented in standard state-space form as: ∆ẋ(t) = A∆x(t) + B∆u(t), ∆y(t) = C∆x(t). (3.1) where x(t) is the state vector given by x(t) = [SNH,1 (t) SNH,2 (t) SNO,1 (t) SNO,2 (t) SS,1 (t) SS,2 (t)]T (3.2a) where the elements are the concentrations of ammonium (SNH,n ), nitrate (SNO,n ) and readily biodegradable substrate (SS,n ) in compartment n in the bioreactor. The operator ∆ refers to the deviation from the operating point. For a more thorough description see Halvarsson (2003). The input signal vector u(t) is given by: Qi (t) u(t) = (3.2b) SO,2 (t) and the output signal vector is: SNH,2(t) y(t) = (3.2c) SNO,2 (t) and 0 1 0 0 0 0 C= . (3.2d) 0 0 0 1 0 0 The steady-state operational maps for the model, are shown in Figure 3.2. The output signals, SNH,2 (t) and SNO,2 (t) are plotted against the two input signals SO,2 (t) and Qi (t). The operational maps in Figure 3.2 clearly indicate that different controller structures should be used in the different operating points, at least in the lower operating point, u1 , compared to the upper operating points, u2 and u3 . Note, however, that these operational maps can only be used to give an indication of the interactions in the system. 52 4 6 x 10 NH,2 14 14.5 15 4 [mg/l] 3 15.5 i S 16 Q [m3/day] 5 2 1 0 1 1.5 2 2.5 3 SO,2 [mg/l] 4 6 x 10 6 5 i 4 4.5 3 5 6 5.5 2 6 6.5 7 8 9 6.5 1 SNO,2 [mg/l] 5.5 Q [m3/day] 5 7 8 9 0 1 10 11 12 10 11 1.5 2 6.5 7 8 9 10 11 12 13 7 8 9 12 13 2.5 3 SO,2 [mg/l] Figure 3.2: Steady-state operational maps for the considered bioreactor model. The upper plot shows the level curves for the first output signal, the effluent ammonium concentration, SNH,2 , and the lower one shows the effluent nitrate concentration, SNO,2 . The operation points are indicated in the plots. 3.3 RGA analysis The steady-state RGA matrices for the linearized model in the three operating points are: 0.0055 0.9945 , (3.3a) RGA(Gu3 (0)) = 0.9945 0.0055 0.0051 0.9949 RGA(Gu2 (0)) = , (3.3b) 0.9949 0.0051 0.0041 0.9959 RGA(Gu1 (0)) = . (3.3c) 0.9959 0.0041 Apparently, the RGA suggests the anti-diagonal pairing SNH,2 (t)–SO,2 (t) and SNO,2 (t)–Qi (t) for all of the three operating points. This contradicts the results from the operational maps in Figure 3.2. 53 3.4 HIIA analysis The HIIA is a scaling dependent tool. This motivates a scaling of the systems before the HIIA is considered. A reasonable scaling procedure is to scale the systems so that the maximum deviation from the average point of the considered variables lies in the interval [−1, 1] (for a detailed description of this scaling procedure, see Halvarsson (2003)). If all of the three operating points are scaled in the same way the following steady state transfer function matrices are obtained: 0.0004 −0.7048 scaled , (3.4a) Gu3 (0) = 0.0748 0.6674 −0.0001 −0.7050 scaled , (3.4b) Gu2 (0) = −0.0135 0.6422 −0.0313 −0.7069 scaled Gu1 (0) = . (3.4c) −4.9176 0.4526 Furthermore, since the HIIA is a dynamic measure that considers all possible frequencies while the considered model is only valid in a limited frequency band it is also reasonable to perform a band-pass filtering before calculating the HIIA. This was carried out using a simple first-order low-pass filter, F (s), given by: 0.001 F (s) = (3.5) s + 0.001 where s is the Laplace-variable. This filter has a 3 dB cut-off frequency at approximately 10−3 rad/s which is reasonable since the considered bioreactor model is valid for frequencies ranging from approximately 10−5 rad/s up to 10−3 rad/s. Note also that this filter does not introduce any additional scaling in the steady state. The filtering can be expressed as: Gf iltered = GF. (3.6) If the systems are scaled in the suggested way and filtered using the lowpass filter F given in (3.5) before the HIIA is calculated, then the following HIIA matrices, ΣH , are obtained for the three operating points: 0.0003 0.4869 u3 ΣH = , (3.7a) 0.0517 0.4611 0.0001 0.5181 u2 , (3.7b) ΣH = 0.0099 0.4719 0.0052 0.1157 u1 ΣH = . (3.7c) 0.8050 0.0741 If a decentralized controller structure is to be used, the HIIA analysis suggests the same input-output pairings as the RGA, i.e the anti-diagonal 54 pairing in all of the considered operating points. However, since [ΣH ]22 is large for u2 and u3 this indicates that SO,2 affects both outputs, SNH,2 and SNO,2 . This in turn means that the suggested decentralized controller structure could be insufficient to provide good control performance. Instead, improved control performance can be expected if a (multivariable) triangular controller structure that also includes the impact SO,2 has on SNO,2 is used. In the lowest operating point, u1 , the HIIA also suggests a triangular controller structure, even though not as strongly as for u2 and u3 . In fact, in this operating point a decentralized controller may be good enough since the sum of the anti-diagonal HIIA elements is 0.9207 which is close to one. Concerning the scaling procedure, it was found that reasonable small changes in the scaling matrices (for instance, ±40% in the element that scales Qi ) do not alter the HIIA recommendations. 3.5 Discussion In the RGA analysis of the bioreactor model it was seen that the RGA method did not provide reasonable input-output pairings in all of the considered operating points. The reason for this can be found if the steady-state gain matrices for the considered systems are studied. Triangular systems will always give the same RGA, namely the identity matrix (under the assumption that the rows in the transfer function matrix are permuted to get nonzero elements along the diagonal before calculating the RGA). The transfer function matrices of the (scaled) model are almost right under triangular, see (3.4a)–(3.4c). Therefore, the structure of the RGA will be similar for all of them: almost the anti-identity matrix. The RGA matrices are given in equations (3.3a)–(3.3c), and evidently they are all very close to the anti-identity matrix. Obviously, the HIIA provides an interaction analysis that goes deeper than the RGA is able to. When considering the information given by the HIIA there is no longer any contradiction with the steady-state results in the operational maps in Figure 3.2. This can also be seen as a confirmation that the applied scaling procedure is reasonable. Note once again, that these steady-state operational maps can merely be used to give an indication of the interactions in the system, and what a reasonable controller structure may look like. Compared to the RGA, the HIIA possesses several advantages. Evidently, the HIIA is able to deal with special transfer function matrix structures such as the analysed nearly triangular ones. The HIIA does not require decentralized (diagonal) controller structures as the RGA does. Instead, the HIIA considers each subsystem in the model independently. Therefore, the HIIA can be used to suggest MIMO controller structures as seen in Section 3.4. The RGA method is unable to do this. 55 It was also observed that the HIIA method is scaling-dependent. This means that some effort must be spent on finding proper scaling matrices. However, this is not necessarily a drawback, since this gives an opportunity for the user to weight the considered signals according to his own choice. The RGA method is scaling-independent and does not offer this possibility. Based on the RGA results in this particular case, it should not be concluded that the couplings are low between the DO set point (SO,2 (t)) and the internal recirculation flow rate (Qi (t)) independent of operation point. Instead, the operational maps indicate that there are some couplings between the nitrification and the denitrification process. A MIMO controller structure can therefore be expected to give better control performance compared to a solution involving decentralized control. The HIIA analysis supports this view, and also suggests possible controller structure selections. 3.6 Conclusions The RGA method provides a simple way to decide how a set of input signals should be utilized to control a given set of output signals. Often this method performs well, but in the analysis of the considered bioreactor model, it was clearly seen that the RGA method does not work properly in all cases. The reason for this was found to be the almost triangular structure of the transfer function matrices. From this it can be concluded that the RGA should be used with care. It is advisory to include an examination of the structure of the considered transfer function matrices in the RGA analysis. Furthermore, the more recently suggested HIIA method was employed to quantify the level of interactions occurring between the inputs and outputs in the considered bioreactor systems. It was noted that for the HIIA method to give reasonable information in this particular case, the considered systems had to be both scaled in a physically relevant way and low-pass filtered. The filtering was performed to select the frequency band of interest. When treating the systems according to this procedure, the HIIA method suggested the same decentralized controller structure as the RGA, but it also gave suggestions on other controller structures that may perform better. The RGA is unable to give this extra information. 56 Chapter 4 Interaction Analysis and Control of the Denitrification Process This chapter once again considers the problem of channel interaction in multivariable bioreactor systems. Similar to the previous chapter (Chapter 3) nitrogen removal in the activated sludge process in a wastewater treatment plant is studied. Here, the focus is on controlling the denitrification process when an external carbon source is added. Thus, one of the two considered control handles (input signals) is the readily biodegradable organic substrate in the influent water (which has the same influence as an external carbon source would have). The other input signal is the internal recirculation flow rate. The nitrate concentration in the anoxic compartment and in the effluent are the two controlled signals (output signals). To evaluate the degree of channel interaction, two different tools are compared: the well-known Relative Gain Array (RGA) and the more recently proposed Hankel Interaction Index Array (HIIA). The results of the analysis are discussed from a process knowledge point of view, and are also illustrated with some control experiments. The main conclusion is that the HIIA gives a deeper insight about the actual cross couplings in the system. This insight is also used in order to design a suitable structured multivariable controller. 4.1 Introduction In this chapter, both the RGA and the HIIA are calculated for different operating points of a certain process in a bioreactor model. Cross couplings in other processes in such systems have previously been studied by Ingildsen (2002) and Halvarsson et al. (2005) (see Chapter 3). The bioreactor model studied here models a pre-denitrifying wastewater treatment plant. The results of the methods are compared and discussed. The aim is to illustrate 57 the different conclusions that can be drawn from the two measures. In particular, it is shown how the HIIA can be used to determine more elaborate control structures that may increase the closed loop performance compared to the decentralized control case. The chosen control strategy as well as a decentralized control strategy is also evaluated in a simulation study in order to point out the validity of the analysis. The structure of this chapter is as follows: The bioreactor model used to illustrate the RGA and HIIA methods is discussed in Section 4.2, and the presented model is analysed with respect to the RGA and the HIIA in Section 4.3. To illustrate the results of the analysis in Section 4.3, some control experiments are performed in Section 4.4. In Section 4.5, the results in the previous sections are discussed. The general conclusions are summarized in Section 4.6. The used interaction measures, i.e. the RGA and the HIIA, are defined and further discussed in Section 2.4 and in Section 2.5 in Chapter 2. 4.2 The bioreactor model The RGA and the HIIA will be used on a bioreactor model describing reduction of nitrate in wastewater (conversion of nitrate to nitrous oxide, so called denitrification). Generally, the bioreactor is connected to a clarifier, and the process consisting of these two parts is commonly called an activated sludge process (ASP), see Henze et al. (1995). In recent years, the control problems in this area have become more and more important due to increased demands on the effluent water quality, see for instance Olsson and Newell (1999), Ingildsen et al. (2002), Yuan et al. (2002) and Jeppsson et al. (2002). In an ASP configured for nitrogen removal, ammonium is oxidized to nitrate under aerobic conditions. This process is called nitrification. The nitrate formed by the nitrification process, in turn, is converted into gaseous nitrogen under anoxic conditions, this is the so called denitrification. Therefore, a multi-step configuration is generally needed in order to perform an efficient nitrogen removal. If the anoxic part of the process is placed before the aerobic, the process is said to be pre-denitrifying. In this chapter, a pre-denitrifying process is considered, see Figure 4.1. For the denitrification process to take place, a sufficient amount of organic substrate (readily biodegradable organic substrate) is needed as well as an anoxic condition, i.e. absence of dissolved oxygen. In a pre-denitrifying system, the access to nitrate in the anoxic part is achieved by recirculating nitrate rich water from the aerobic to the anoxic part of the plant. To ensure that enough readily biodegradable substrate is present, an external carbon source (for example ethanol) is often added to the anoxic part. It is thus natural to consider the flow rates of the internal recirculation and the addition of an external carbon source as control signals (manipulated variables) in the denitrification process, although the concentration of read58 ily biodegradable organic substrate in the influent water is here used as an input signal instead of an external carbon source. The natural output signals are the nitrate concentration in the last anoxic compartment and the nitrate concentration in the effluent water. Several papers in the literature deal with the above described control problem, see for instance Carlsson and Rehnström (2002). The probably most used mathematical model describing biological nitrogen removal is the IAWQ Activated Sludge Model No. 1 (ASM1), see Henze et al. (1987) for a full description. This is a rather complex nonlinear model including eight different processes describing biomass growth and decay together with a number of hydrolysis processes. Due to the complexity of the model, it is not very suitable for control purposes. Instead a somewhat simplified version of the ASM1 will be used in the analysis carried out in this chapter. In the analysis, this simplified model is linearized for different operating points, and the RGA and HIIA analyses are performed on the linearized models. The model used here describes a pre-denitrifying process with one anoxic and one aerobic compartment, see Figure 4.1. The compartments are assumed to be completely mixed. For a full derivation of this model and a discussion of the parameters therein, see Ingildsen (2002) or Samuelsson et al. (2004). Q Q−Qw Qi Qr Qw Figure 4.1: An ASP with one anoxic and one aerobic compartment and a clarifier. The influent flow rate is denoted Q, the internal recirculation flow rate Qi , the flow rate of recirculated sludge Qr and the excess sludge flow rate Qw . In the sequel, the internal recirculation flow rate is denoted Qi [m3 /h] and the influent concentration of readily biodegradable substrate SS,in [mg (COD)/l]. The nitrate concentration in the anoxic compartment is denoted an [mg (N)/l] and the nitrate concentration in the effluent water (aerobic SNO e compartment) is denoted SNO [mg (N)/l]. The input signal vector of the system can thus be defined as T u = Qi SS,in (4.1) and the output signal vector as an T e y = SNO SNO . 59 (4.2) 4.3 Analysis of the model In this section, the bioreactor model will be analysed using the RGA and the HIIA described in Section 2.4 and Section 2.5.3 in Chapter 2. One objective is to investigate the cross couplings in the system and to choose suitable control structures. The desired control structure may change with different operating points since the system is nonlinear. Another objective is to illustrate the performance of both methods, i.e. what different conclusions that can be drawn from each of the methods: For instance, how the HIIA can be used to determine a suitable sparse multivariable control structure, and what can be seen from the RGA in the corresponding case. A first impression of the possible (stationary) cross couplings in the system can be obtained from the steady state operational maps of the nonlinear model. Such operational maps are also used by, for instance, Ingildsen (2002) and Galarza et al. (2001) in order to analyse the bahaviour of bioreactors. In Figures 4.2 and 4.3 the level curves of the stationary nitrate concentrations of the anoxic and aerobic compartments respectively are plotted against constant values of the two inputs. From these operational maps it is clear that the system behaves nonlinearly, i.e. the stationary characteristics are different depending on the choice of operating point. The static gain in some of the channels actually changes sign between the different operating ranges (which could also be seen in a simple step response analysis). From Figure e for 4.3 it can be seen that in order to accomplish a change stationary in SNO low values of the input signals, the concentration of readily biodegradable substrate in the influent, SS,in should be used. For larger values of the input signal SS,in , the change seems to be best accomplished if the internal recire for culation flow rate Qi is altered. Note also the multiple equilibria of SNO a given value of SS,in . In Figure 4.2 it is seen that both inputs affects the an over the whole operating range, although the relative imporoutput SNO tance of the inputs depends on the choice of operating point. A conclusion that can be drawn is, however, that for high values of the input signal SS,in , an is low for both the input signals. the gain of SNO Although these plots contain only stationary values for the open loop system and therefore cannot be assumed to fully describe all cross couplings in the system, they will be used to roughly validate the results obtained from the linear analysis utilizing the RGA and the HIIA. As indicated above, Figure 4.3 implies that there are three different areas in which the process may show different cross couplings. In order to analyse this bahaviour, three different operating points will be considered. The three operating points are indicated by stars “∗” in the operational maps in Figure 4.2 and Figure 4.3. These are the ones corresponding to the constant input 60 San [mg/l] NO 4 1 2 3 0.5 7 6 10 12 5 9 13 5.5 11 8 6 x 10 0.1 4 5 7 0. 5 2 3 5 1 0. 5 0.0 8 2 05 0. 1 6 9 2.5 4 i 3 11 8 4 3.5 10 Q [m3/day] 4.5 7 1.5 1 5 20 6 3 1 2 4 40 0.03 0.5 1 0. 0.05 60 80 S S,in 0.03 100 [mg/l] 120 140 160 Figure 4.2: Stationary operational map for the nitrate concentration in the an . The stars show the locations of the three operatanoxic compartment, SNO ing points. signals given by T ū1 = 35 000 40 , T ū2 = 26 000 100 , T ū3 = 20 000 120 (4.3) where the units are m3 /day for the first input signal and mg(COD)/l for the other. The first operating point given by ū1 lies in the area where the second input signal, the concentration of readily biodegradable substrate in the influent water, SS,in , is low. The second operating point given by ū2 lies in the transition phase between the areas, and the third point is in the area where the concentration of readily biodegradable substrate in the influent water is high. 4.3.1 Linearization and scaling of the model Both the RGA and the HIIA are defined for linear models. In order to perform the analysis, the model needs to be linearized around each operating point. Since the process characteristics are clearly different in the three operating points, three different linearizations are needed in order to properly analyse the system, one for each operating point. In a small neighbourhood of each point, the linearized model will mimic the characteristics of the nonlinear system. Thus, the analysis in the following section is strictly valid only in the same operating points. However, as seen in Figure 4.3, the ope erational map for SNO can clearly be divided into three areas with the same 61 Se [mg/l] 4 5 7 12 10 8 13 15 14 17 18 5.5 16 6 6 NO 4 x 10 5 9 11 5 7 12 10 8 13 15 17 16 18 i 14 4 3.5 6 3 6 11 2.5 9 Q [m3/day] 4.5 7 2 1 20 12 13 17 16 1.5 14 15 8 10 8 9 11 40 60 80 S S,in 10 100 [mg/l] 120 9 140 160 Figure 4.3: Stationary operational map for the nitrate concentration in the e . The stars show the locations of aerobic compartment (effluent water), SNO the three operating points. gain characteristics, which motivates that the results from the analysis hold with good accuracy over larger neighbourhoods. This can also be verified via further linearizations or simulations. In practice, the linearizations were made using the MATLAB function linmod. From the linearization procedure, standard linear state-space models of the form ∆ẋ = A∆x + B∆u, ∆y = C∆x (4.4) are obtained. Here x is a 7×1 state vector containing concentrations of seven different compounds (for instance the output signals) and u is the input signal vector defined in (4.1) as T u = Qi SS,in . The symbol ∆ refers to deviation from the operating point so that ∆x = x−x̄ and ∆u = u − ū, where ū is the constant input signal vector that in steady state renders the operating point x̄. The output signal vector is defined in an S e ]T and ∆y = y − ȳ. The matrix A is a 7×7 matrix, B (4.2) as y = [SNO NO is a 7×2 matrix and C is consequently given by a 2×7 matrix that is independent of the chosen operating point. The corresponding transfer function matrix is then G(s) = C(sI − A)−1 B. 62 (4.5) As mentioned in Section 2.5.5, the HIIA is a scaling dependent tool. In order to be able to compare the different elements of the HIIA directly, the linearized model obtained by (4.4) and (4.5) must be properly scaled. A standard procedure (as described by for instance Skogestad and Postlethwaite (1996)) is to scale the model according to G(s) = Dy−1 Go (s)Du where the original input-output model is given by y o (t) = Go (p)uo (t) and the superscript “o ” denotes the original (or physical) variables. Thus, Go (p) denotes the original transfer function matrix between output y o (t) and input uo (t); Du and Dy are diagonal scaling matrices. There exist many different possibilities for choosing the scaling matrices Dy and Du depending on what the desired achievements are. In this chapter the model is scaled in such a manner that the maximum deviation from the average value of each signal lies in the interval [-1,1]. This is achieved here by choosing 60000 0 , Du = 0 160 3 0 Dy = 0 3 where the diagonal elements in Du are the maximum allowed value of each input signal and the elements in Dy states that a maximum deviation in the output of three units is accepted. 4.3.2 RGA analysis of the model To test the ability of the RGA to provide reasonable pairing suggestions, the stationary RGA was calculated for the linearized models for each of the chosen operating points. The results were 1.1979 −0.1979 , (4.6) RGA(Gū1 (0)) = −0.1979 1.1979 0.3327 0.6673 RGA(Gū2 (0)) = , 0.6673 0.3327 (4.7) 0.3263 0.6737 RGA(Gū3 (0)) = . 0.6737 0.3263 (4.8) 63 Clearly, since the anti-diagonal elements in the RGA matrix corresponding to the first operating point are negative and the diagonal elements are fairly close to one, the RGA in this case suggests a diagonal controller, an i.e. that the first input signal Qi should be used to control the output SNO e . The latter also and the second input, SS,in , should control the output SNO seems probable when comparing the results to the operational map in Figure 4.3, provided that a decentralized control structure is to be used. For both the other operating points, anti-diagonal control structures are suggested, although without any strong indication since the diagonal element of the RGA matrix is quite far from one in both cases. For the second operating point it is hard to evaluate the validity of this from the operational maps, since the operating point lies in a transition phase. The result for the third operating point, however, seems reasonable when considering the operational maps. 4.3.3 HIIA analysis of the model The HIIA is a measure that indicates the size of the impact of each input signal on each output signal. It is therefore interesting to compare the results from the HIIA analysis to the results from the RGA analysis. The HIIA as defined in Section 2.5 was calculated for the linearized and scaled models for each operating point. The obtained results were 0.1425 0.3596 ū1 ΣH = , (4.9) 0.0450 0.4530 ΣūH2 0.1252 0.1379 = , 0.4009 0.3361 (4.10) ΣūH3 0.0090 0.0053 = . 0.7441 0.2415 (4.11) For the first operating point, the HIIA indicates that the first output an , is about equally dependent on both input signals since the elsignal, SNO ements on the first row in the HIIA matrix have the same magnitude of orders. Furthermore, it can be seen from the second row that the second e , mostly depends on the second input signal, S output, SNO S,in . The corresponding HIIA element is also of the the same size as the elements in the first row. The first element in the second row is however about ten times smaller than the second element in this row. Thus, the second input should definitely be employed in the control of the second output. Intuitively, the first input should then be used in the control of the first output in order to obtain a satisfying result. As pointed out in Section 2.5.3, it could be hard to determine whether an entry in the HIIA matrix is large enough to be 64 relevant in the control design, and it is also beyond the scope of this thesis to provide any general rules for such a decision. However, provided that a reduction of control structure complexity is desirable together with the fact that the first element on the second row is clearly the smallest element in the matrix, it is most natural to ignore the interactions in the channel from the first input to the second output in an eventual design of a reduced order controller in this operating area. A summation of the elements of the HIIA matrix as described in Section 2.5.3 also gives at hand that the contribution of the first element on the second row is relatively small in the total sum. The interpretation of the HIIA matrix in this case is thus that a good option for controlling the process in this operating range may be a sparse multivariable controller taking the analysis results into account. The control structure selection would therefore in this case be F1 (s) F3 (s) U (s) = E(s) (4.12) 0 F2 (s) where F1 (s), F2 (s) and F3 (s) are the transfer functions of each sub controller, U (s) is the Laplace transform of the input signal vector as defined in (4.1) and E(s) is the Laplace transform of the control error vector, i.e. a vector an and S e . The validity of this choice of containing the control errors of SNO NO control structure is also verified in the simulations in Section 4.4. In the second operating point, corresponding to the transition phase between the operating ranges, all elements in the HIIA matrix are of the same magnitude of orders. This indicates that a full multivariable control structure should be chosen here. Finally, in the third operating point both elements in the first row of the HIIA matrix are small compared to the elements in the second row. This indicates that the first output signal could be difficult to control at all. A physical interpretation of this will be given in Section 4.5. Note that no pre-filtering of the transfer functions is performed before calculating the HIIA. A pre-filtering procedure was evaluated but made no difference in the results and was therefore omitted. 4.4 Control simulations In order to illustrate the findings in the previous section, some control experiments were performed. As an example, control of the nonlinear system in the neighbourhood of the first operating point given by the input signal ū1 in (4.3) is considered here. Both a decentralized control structure and a simple multivariable strategy are evaluated. The purpose is to compare the results from the linear analysis in the previous section to the actual results obtained when controlling the nonlinear system. In particular, it is shown how the information extracted from the HIIA matrix can be used to design a more elaborate control structure than decentralized control. 65 4.4.1 Decentralized control A decentralized control law with the input-output pairing recommended by both the RGA analysis in (4.6) and the HIIA in (4.9) was evaluated. The results in (4.6) and (4.9) both suggested that if a decentralized control structure was to be used, the natural pairing selection was to control the first an , by manipulating the first input signal, Q . Consequently, output signal, SNO i e , should be controlled by manipulating the second the second output, SNO input signal SS,in . Thus, the decentralized control law can be written as Qi (s) = F1 (s)E1 (s), (4.13) SS,in (s) = F2 (s)E2 (s) (4.14) where E1 (s) is the Laplace transformed control error of the first loop, i.e. an,sp an (t), and E (s) is the Laplace transformed control e1 (t) = SNO (t) − SNO 2 e (t) − S e,sp (t) since this process is error of the second loop, i.e. e2 (t) = SNO NO an,sp e,sp known to have negative gain. Here, SNO and SNO denote the set point values of the output signals, respectively. The controllers F1 (s) and F2 (s) were in this experiment chosen as ordinary PI-controllers and were tuned in order to achieve approximately the same rise time in both control loops to make later comparisons meaningful. The used PI-controllers were F1 (s) = 8000 + F2 (s) = 15 + 8 s 7000 , s (4.15) (4.16) where the large difference in size between the parameters in the controllers are explained by the large gain differences in the open loop systems. The decentralized control law (4.13)–(4.14) was then used to control the nonlinear system. Figure 4.4 shows the output responses of the system when an,sp a 10 % step change in the set point of the first output, SNO , is applied. In the same way, Figure 4.5 shows the output responses for a 10 % step change e,sp in the set point value of the second output, SNO . The conclusions from an , is affected by both the the HIIA in (4.9) were that the first output, SNO e input signals while the second output SNO is mainly affected by the second input signal, SS,in . Considering the control law (4.13)–(4.14) it is clear by an,sp definition that a change in the set point SNO causes a direct change in the first input signal, Qi while the second input signal is unaffected. In the same e,sp manner, a change in the other set point SNO causes an immediate change in the second input, SS,in , but leaves the first input signal unaffected. Combining this reasoning with the results of the HIIA, it can thus be an,sp expected that a step change in SNO will have a relatively small impact e,sp e on the output SNO , while a step change in SNO will have a larger impact 66 an SNO [mg/l] 11 10 9 8 40 45 50 55 45 50 55 60 65 70 75 80 60 65 70 75 80 15.4 NO Se [mg/l] 15.2 15 14.8 14.6 14.4 40 time [h] Figure 4.4: Decentralized control output responses of the system for a step an . Upper plot: Solid line shows change in the set point of the first output, SNO an . Dashed line shows the set point value. Lower the response of the output SNO e plot: The response of the second output SNO is plotted. in the first output channel. This is also confirmed by the simulation results in Figures 4.4 and 4.5. The disturbance response of the first output is e,sp is changed than vice versa. It considerably larger when the set point SNO should be noted that like the stationary operational maps, these simulations are merely strong indications that the HIIA in (4.9) provides a reasonable result. Of course, the performance of the closed loop system depends also on the choice of decentralized controller. However, using reasonable controllers tuned to achieve same rise time in both control loops should make the comparison above relevant. 4.4.2 Multivariable control Next, a simple multivariable control strategy is evaluated. The specific structure of the controller is determined using the HIIA analysis results. In Section 4.3.3, it was concluded from the HIIA analysis that in the neighbourhood of the first operating point given by the input signal ū1 in (4.3), a structured multivariable controller might be preferable to decentralized control. To choose a suitable control structure, first note that according to (4.9), the dependence of the first input signal on the second output signal is relatively low. One possibility to perform the control design could therefore be to approximate the nonlinear system with a triangular linear system 67 9.4 NO San [mg/l] 9.2 9 8.8 8.6 8.4 40 45 50 55 45 50 55 60 65 70 75 80 60 65 70 75 80 SeNO [mg/l] 16 15 14 13 40 time [h] Figure 4.5: Decentralized control output responses of the system for a step e . Upper plot: The response change in the set point of the second output, SNO an of the first output SNO is plotted. Lower plot: Solid line shows the response e . Dashed line shows the set point value. of the output SNO according to an SNO (s) G1 (s) G3 (s) Qi (s) e (s) = SNO 0 G2 (s) SS,in (s) (4.17) where the elements in the transfer function matrix are obtained by linearizing the nonlinear system in the neighbourhood of the operating point given by the input signal ū1 . e , is assumed to depend only on the second Since the second output, SNO input, SS,in , it is convenient to choose SS,in according to e,sp e SS,in (s) = F2 (s)(SNO (s) − SNO (s)). (4.18) an is affected by both input signals. A suitable The first output signal, SNO choice might be to take this into account in the control law, for instance by letting an,sp an Qi (s) = F1 (s)(SNO (s) − SNO (s)) + F3 (s)SS,in (s) (4.19) where the latter term can be seen as a feedforward part. Inserting the control signals (4.18)–(4.19) into the expression for the 68 linearized system (4.17) directly yields G1 (s)F1 (s) S an,sp (s)+ 1 + G1 (s)F1 (s) NO (G1 (s)F3 (s) + G3 (s)) + SS,in (s), 1 + G1 (s)F1 (s) −G2 (s)F2 (s) e,sp e SNO (s) = S (s) 1 − G2 (s)F2 (s) NO an SNO (s) = (4.20) (4.21) and it is seen that the system will be completely decoupled if the feedforward controller F3 (s) can be chosen as F3 (s) = −G3 (s) . G1 (s) (4.22) In the simulations, the controller F3 (s) was obtained in the way described above. The nonlinear system was linearized in a neighbourhood correspondan e ] = [9 15] which corresponds to the input signal values ing to [SNO SNO u0 = [39400 43]. Using (4.22) on the obtained linear model resulted in this case in a strictly proper linear feedforward controller. The controllers F1 (s) and F2 (s) used in the experiment were the same as before, see (4.15) and (4.16). The presented control law was applied to the nonlinear system. Figures 4.6 and 4.7 show the output responses for step changes in the set point of each output as in the previous simulations. Comparing Figures 4.5 and 4.7 it can be seen that the impact of the input signal SS,in on the an is reduced when the feedforward controller is included, output signal SNO e is applied, the magnitude of i.e. when a step change in the set point of SNO an the disturbance response in SNO is much lower. In the simulation example considered here, the decoupling above is only approximate since the system is nonlinear. Thus, the disturbance in the an in Figure 4.7 is not completely attenuated. Further, as menoutput SNO tioned, in the true nonlinear system the impact of the first input signal on the second output is not strictly zero as in the linear model example (4.17). e in Figure 4.6 is not strictly zero. Therefore, the disturbance response of SNO This example, however, shows how the HIIA can be used to determine an approximate decoupling control law for a nonlinear system in the neighbourhood of some operating point. 4.5 Discussion Here, the results in the previous section are further discussed. The results from the RGA and the HIIA analysis are also compared to each other and the relevance of the results are discussed from a process knowledge point of view. The validity of the results is also discussed from a more general point of view. 69 an SNO [mg/l] 11 10 9 8 40 45 50 55 45 50 55 60 65 70 75 80 60 65 70 75 80 15.4 NO Se [mg/l] 15.2 15 14.8 14.6 14.4 40 time [h] Figure 4.6: Feedforward control output responses of the system for a step an . Upper plot: Solid line shows change in the set point of the first output, SNO an . Dashed line shows the set point value. Lower the response of the output SNO e plot: The response of the second output SNO is plotted. For the first operating point, the RGA clearly suggested a diagonal pairan and that ing, i.e. the input Qi should be used to control the output SNO e . This seems reasonable when the input SS,in should control the output SNO considering the operational map in Figure 4.3 where the stationary value e of SNO seem to depend mostly on the stationary values of SS,in . From the other operational map in Figure 4.2 it is harder to draw any conclusions. The HIIA analysis also shows that if a decentralized controller is to be used, diagonal pairing is preferable. However, the HIIA also provides the inforan is also dependent on mation that in this operating point, the output SNO the input SS,in and thereby that a sparse multivariable controller according to (4.12) may be a better option. Thus, the HIIA adds valuable information about the cross couplings in this operating point. The results are also in line with the conclusions that can be drawn from general process knowledge. In the first operating point, SS,in is comparatively low, and there will be a lack of readily biodegradable substrate available for denitrification. Since there is not enough readily biodegradable substrate, the denitrification process in the anoxic compartment will be incomplete, which means that all of the rean will circulated nitrate is not denitrified. Thus, the nitrate concentration SNO depend both on how much readily biodegradable substrate is added and on how much nitrate that is recirculated from the aerobic compartment to the anoxic, i.e. on both input signals. In this situation the nitrate concentration e , depends mainly on the input S in the aerobic compartment, SNO S,in , because when the denitrification in the anoxic compartment is incomplete, the 70 9.4 NO San [mg/l] 9.2 9 8.8 8.6 8.4 40 45 50 55 45 50 55 60 65 70 75 80 60 65 70 75 80 SeNO [mg/l] 16 15 14 13 40 time [h] Figure 4.7: Feedforward control output responses of the system for a step e . Upper plot: The response change in the set point of the second output, SNO an of the first output SN O is plotted. Lower plot: Solid line shows the response e . Dashed line shows the set point value. of the output SNO internal recirculation only leads to an internal transport of nitrate that does not affect the effluent nitrate concentration of the system. In other words, for low values of SS,in , there is no meaning in increasing Qi since it does not e . To conclude, both the result affect the effluent nitrate concentration SNO from the RGA analysis and from the HIIA analysis therefore seem valid. The HIIA gives more information about the actual cross couplings in the system, and thereby gives an opportunity to design a better controller. The control simulations in Section 4.4 also confirm these conclusions. The particular decentralized control structure suggested by the RGA is also suggested by Ingildsen (2002), however not as a result of a cross coupling analysis but from an economical point of view. In the second operating point, the RGA gives an indication (however not very strong) that the input-output pairing now should be the reversed, i.e. an anti-diagonal pairing. The conclusion from the HIIA analysis is that a full multivariable controller should be used. It is hard to evaluate the relevance of the RGA analysis from the operational maps or from a physical reasoning. It is clear from the operational maps that in this transition phase, both outputs rely on both inputs, and the HIIA thus seems to provide a reliable result. In the third operating point, the RGA suggests the same pairing as in the second operating point with approximately the same order of magnitude on the RGA elements. Here, an interesting difference occur when considering 71 the HIIA analysis. Since the elements on the first row of the HIIA is close an is difficult to affect at all using to zero, it indicates that the first output SNO e , is mainly affected any of the two input signals. The second output, SNO by the first input signal, Qi , but also to some extent by the other input SS,in . A physical interpretation is that in this case the access of readily an takes very biodegradable substrate is sufficient, and the concentration SNO low values. This means that the denitrification in the anoxic compartment is complete. Since there is a good access to readily biodegradable substrate an takes low values, S an will not decrease and the nitrate concentration SNO NO further if more readily biodegradable substrate SS,in is added. If less readan will not increase as long as the ily biodegradable substrate is added, SNO addition is large enough for the denitrification to remain complete. In this operating point, when more nitrate is recirculated through an increase of the an will not be affected much as long as internal recirculation flow rate, Qi , SNO the transition phase is not passed, i.e. while the denitrification still is complete. If instead the internal recirculation flow rate is decreased, less nitrate an remains relatively unchanged. has to be denitrified and the stationary SNO Thus, the gains from both input signals are low. As mentioned, the second e , is in this operating point mainly affected by the internal reoutput, SNO circulation flow rate, Qi . If, for instance, more nitrate is recirculated, more e nitrate will also be denitrified and the nitrate concentration SNO is thus reduced. This bahaviour is at least reflected in the HIIA, although these conclusions are hard to draw directly from the HIIA without any prior process knowledge. In order to control the system in this operating range, i.e. so that a low effluent nitrate concentration is obtained, one possible strategy is to add sufficient amounts of readily biodegradable substrate to achieve complete denitrification in the anoxic compartment. The nitrate concentration in the aerobic compartment can then be moderated by adjusting the internal recirculation flow rate. Expressed in control terminology, the input an at a low set point. The input SS,in should be used to keep the output SNO e Qi should be used to control the output SNO to some desired value. This specific pairing is also what the RGA recommends, although as seen above, additional useful information can be extracted from the HIIA. The main conclusion that can be drawn from this study is that the HIIA is a more powerful tool than the RGA when it comes to evaluating channel interactions in general, and may therefore be used in order to determine more elaborate control structures than just decentralized controllers. It should, however, be noted that the RGA assumes a decentralized control law, while the HIIA rather investigates the controllability and observability of each partial input-output subsystem. The RGA certainly gives some indication of how large the channel interactions may be, but does not preserve the structure of the transfer function matrix. The HIIA gives a detailed information of the size of the interactions for each channel. One way of describing it is that the HIIA investigates things that the RGA cannot because of the 72 needed preassumption. Another difference is that the HIIA takes the whole frequency range into account. This could certainly increase the usefulness for many applications with high energy content at higher frequencies. In this particular study, however, this has very little to do with the different conclusions that can be drawn since most of the energy is located at low frequencies. For instance, the HIIA seemed insensitive to a a pre-filtering procedure cutting off higher frequency components. The results of the HIIA actually follows the conclusions that can be drawn directly from Figures (4.2)–(4.3), although higher frequency components also are taken into account in the HIIA. The most obvious drawback of the HIIA method may be that the results are scaling dependent. However, in the study in this chapter, the results did not seem very sensitive to changes in the scaling matrices, as long as the elements were chosen fairly reasonable in a physical sense. It should be noted that in reality, it might be sufficient to run the process only in the operating range described by the first operating point depending on the actual demands on effluent nitrate concentration. In the overall consideration, process economy should also be weighted into the choice of control structure and desired operating range, not just the goal to reduce nitrate concentration as much as possible. In a case where the process may run in different operating ranges, however, the use of different control structures in the different operating ranges may provide better control of the process. 4.6 Conclusions In this chapter, the cross couplings in a bioreactor model describing a predenitrifying wastewater treatment plant have been studied. Two different tools were used to evaluate the cross couplings, the Relative Gain Array (RGA) and the Hankel Interaction Index Array (HIIA). A general conclusion from the presented analysis is that both the RGA and the HIIA give reasonable results for the studied example. An important difference is, however, that the HIIA provides information that the RGA does not. The results from the HIIA analysis gives an understanding of the actual cross couplings in the system in terms of magnitude and character, and are thereby useful for suggesting suitable multivariable control structures. The validity of the results is also illustrated by means of some control experiments where the control structure suggested by the HIIA outperformed the decentralized control structure. 73 74 Chapter 5 Economic Efficient Operation of a Pre-denitrifying Activated Sludge Process In this chapter, the choice of optimum set points and cost minimizing control strategies for the activated sludge process are treated. Both the denitrification and the nitrification process are considered. In order to compare different criterion functions, simulations utilizing the COST/IWA simulation benchmark (BSM1) are considered. By means of operational maps the results are visualized. It is found that it is easy to distinguish set point areas where the process can be said to be efficiently controlled in an economic sense. The characteristics of these set point areas depend on the chosen effluent nitrate and ammonium set point as well as the distribution of the different operating costs. It is also discussed how efficient control strategies may be accomplished. 5.1 Introduction In recent years, cost minimization has become increasingly important in the control and operation of wastewater treatment plants. In order to run a plant economically, operational costs such as pumping energy, aeration energy and dosage of different chemicals should be minimized. At the same time, the discharges to the recipient should be kept at a low level. Of course, minimizing the operational costs and at the same time treat the wastewater properly may lead to a conflict of interest that must somehow be solved. The main problem is how to keep the effluent discharges below a certain pre-specified limit to the lowest possible cost, see Olsson and Newell (1999). Part of the answer is to design the control algorithms in such a way that the overall operational costs are minimized. This goal can be attained in different ways. As an example, the controller set points could be separately 75 optimized or the cost could be minimized online by some control strategy, for instance model predictive control (MPC). See Qin and Badgwell (2003) for how MPC can be used in this context. In some countries, the authorities charge according to effluent pollution. A possible way to formulate the on-line minimization criterion in such a case is to use a cost function that takes actual costs (energy and chemicals) into account and at the same time economically penalizes the effluent discharges. Over the years, much effort has been put in developing economically efficient control strategies for operation of wastewater treatment plants. An interesting cost function is presented in Carstensen (1994) where the effluent nitrogen is penalized using a piecewise linear discontinuous function. Effluent ammonium is penalized in a similar way. The papers (Yuan et al., 1997), (Yuan et al., 2002) and (Yuan and Keller, 2003) all consider efficient control of the denitrification process. The optimum set point for the nitrate concentration at the outlet of the anoxic zone is then found to be near 2 mg(N)/l or, at least, in the interval 1–3 mg(N)/l. In (Ingildsen et al., 2002), an optimization of the dissolved oxygen (DO) and nitrate set points is made. In (Galarza et al., 2001) steady-state operational maps are utilized to examine the feasible operating area for two activated sludge processes with emphasis on sensitivity analysis. Fuzzy control evaluated using multi-criteria cost functions is the subject of (Cadet et al., 2004). In (Vanrolleghem and Gillot, 2002), different multi-criteria control strategies are evaluated. In this chapter, the choice of optimum set points and cost minimizing control strategies for an activated sludge process configured for predenitrification are evaluated. Both the denitrification and the nitrification are treated. The manipulated variables (input signals) are the internal recirculation flow rate and the flow rate of an external carbon source and the controlled variables (output signals) are the effluent ammonium concentration, the nitrate concentrations in the last anoxic compartment and in the effluent. In order to compare the impact of different criterion functions, stationary simulations utilizing the COST/IWA simulation benchmark (BSM1), see Copp (2002), are considered. By means of operational maps the results are visualized. It is also discussed how efficient control can be accomplished. The organization of the chapter is as follows: In Section 5.2, the simulation model (BSM1) is briefly described together with the associated operational costs. In Section 5.3 simulation results are presented using operational maps. The simulation results are discussed and interpreted in Section 5.4. Finally, in Section 5.5 the general conclusions are drawn. 5.2 The model and the operational cost functions In the simulation study presented in this chapter, the COST/IWA simulation benchmark model number 1 (BSM1) is used, see Jeppsson and Pons (2004) 76 for a general survey and Copp (2002) for a more technical description. BSM1 is an important tool for simulation of the activated sludge process in various realistic wastewater treatment scenarios. In BSM1, five biological reactors are implemented using the IWA activated sludge model No. 1 (ASM1), see Henze et al. (1987). Despite being a fairly complex model, ASM1 has some limitations. For example, the pH does not affect the process rates. The pH of the wastewater should hence be near neutrality. The alkalinity is, however, calculated in ASM1 and may be used to detect pH related problems, see further Henze et al. (1987). The model plant is pre-denitrifying with two anoxic and three aerated compartments. A secondary settler is also implemented. To allow for consistent experiment evaluation, three dynamic data input files are defined, each describing different weather conditions. The purpose of the simulation benchmark is to provide an objective and unbiased tool for performance assessment and evaluation of proposed automatic control strategies. A great benefit of the benchmark is that it allows for comparison of many automatic control strategies given the same conditions. The aim here is to analyse the stationary operational costs of the activated sludge process, and in order to visualize the costs, these are presented in stationary operational maps together with the considered output signals. The output signals are the nitrate concentration in the last anoxic coman [mg(N)/l], the nitrate concentration in the effluent, S e partment, SNO NO e [mg(N)/l] and the ammonium concentration in the effluent, SNH [mg(N)/l]. The available control handles considered in this chapter are the internal recirculation flow rate Qi [m3 /day], the flow rate of an external carbon source, Qcar [m3 /day], and when the nitrification is studied, the concentration of dissolved oxygen in the aerated compartments, DO. To express the cost for controlling the denitrification process, a number of partial costs are taken into account: • Pumping costs due to the required pumping energy. • Aeration costs due to the required aeration energy (which varies due to the input load). Excessive use of an external carbon source has a large impact on the required aeration energy, and thus on the total cost. • External carbon dosage costs. • Possible fees for effluent nitrate discharge and for effluent ammonium discharge. In BSM1, the total average pumping energy over a certain period of time, T , depends directly on the internal recirculation flow rate Qi and is according to Copp (2002) calculated as Z 0.04 t0 +T EP = Qr (t) + Qi (t) + Qw (t) dt (5.1) T t0 77 expressed in units of kWh/day. In (5.1), Qr denotes the return sludge flow rate and Qw the excess sludge flow rate, both in units of m3 /day. The average energy in kWh/day required to aerate the last three compartments can in turn be written as 24 EA = T Z t0 +T t0 5 X 0.4032KL ai (t)2 + 7.8408KL ai (t) dt i=3 (5.2) where KL ai (t) is the oxygen transfer function in the aerated tank number i in units of h−1 . Further, assuming a prize kcar [EUR/m3 ] on the external carbon source and that an external carbon flow rate, Qcar (t) [m3 /day], is fed into the process during the time interval T , the cost per day of the external carbon flow rate is Z 1 t0 +T kcar Qcar (t)dt (5.3) Ccar = T t0 expressed in EUR/day. Normally, when nitrogen concentrations in the effluent water are economically charged, the fees consider the total nitrogen discharges in the effluent water. In this chapter, the first part only considers the denitrification process. Therefore, the effluent discharge fees used in this chapter are separated into one fee for the effluent nitrate and one for the effluent ammonium. 5.2.1 The nitrate fee A reasonable way to describe a fee for the effluent nitrate discharge is to let the fee depend on how large mass of nitrate that is discharged per time unit. This depends, of course, on the effluent flow rate, Qe (t), and the e (t). The nitrate discharge cost nitrate concentration in the effluent, SNO may be expressed as (in EUR/day) CNO = 1 T Z t0 +T t0 e fNO (Qe (t), SNO (t))dt (5.4) where fNO is some function describing the fee. Now, assuming a constant energy prize, kE , the total cost expressed (in EUR/day) can be calculated during a representative time interval T from (5.1)–(5.4) as Ctot = kE (EP + EA ) + Ccar + CNO . (5.5) The fee functions for the discharge of nitrate can have different forms. Normally, the fee functions are set-up by the legislative authorities. Here, three typical fee functions are considered: 78 1. No charge is added for the disposal of nitrate, i.e. e (t)) ≡ 0. In practice, even though not associated with fNO (Qe (t), SNO a direct fee, it is common to have legislative limits on the average effluent nitrate concentration in such a case. 2. Effluent nitrate is charged with a constant cost, say ∆αNO per discharged kg. Such a fee function is e e (t)) = ∆αNO SNO (t)Qe (t). fNO (Qe (t), SNO (5.6) 3. Effluent nitrate is charged with a constant cost per discharged kg, ∆αNO , up to a legislative discharge limit for the effluent concentration, αlimit,NO . Above this limit the cost of discharging additional nitrate is ∆βNO . Exceeding the limit also imposes an additional charge of β0,NO per volume effluent water. A mathematical description of this cost function is given in (5.7): e fNO (Qe (t), SNO (t)) = e e (t) ≤ α if SNO limit,NO ∆αNO SNO (t)Qe (t) = ∆αNO αlimit,NO Qe (t) + β0,NO Qe (t)+ e (t) − α e (t) > α +∆βNO (SNO if SNO limit,NO )Qe (t) limit,NO . (5.7) In the cost functions above, the effluent flow rate is calculated as Qe (t) = Qin (t) − Qw (t). The difference between the cost functions presented in (Carstensen, 1994) and the one in (5.7) is that here only the nitrate discharge is penalized, while in (Carstensen, 1994) the nitrate and ammonium concentrations are lumped together and charged. 5.2.2 The ammonium fee Effluent ammonium is penalized according a fee function that is similar to the fee for effluent nitrate. The fee function is described by: e fNH (Qe (t), SNH (t)) = e e (t) ≤ α if SNH limit,NH ∆αNH SNH (t)Qe (t) = ∆αNH αlimit,NH Qe (t) + β0,NH Qe (t)+ e (t) − α e (t) > α +∆βNH (SNH if SNH limit,NH )Qe (t) limit,NH . 79 (5.8) 5.3 Simulation results The considered cost functions for penalizing nitrogen discharge are evaluated utilizing BSM1. In the simulation study, the benchmark WWTP is fed with constant influent flow rate of wastewater, 18 446 m3 /day. Basically, the default values of BSM1 are used, with the following exceptions: • The influent is assumed to have a concentration of readily biodegradable substrate, SS,in , of 60 mg(COD)/l. • The external carbon source is ethanol with a COD of 1.2·106 mg (COD)/l. • No limit for the carbon dosage is assumed. • The last three compartments are aerated utilizing DO controllers with a fix set point of 2 mg/l in the first part of the study where only the denitrification is considered. The DO control is fast compared to the other control loops. In the simulations where also effluent ammonium is charged, various fix DO set points in the range of 0.5 mg/l and 5.0 mg/l have been selected. All of the other parameters adopt the standard values used in BSM1, these values can be found in, for instance, (Copp, 2002) or on the benchmark webpage (IWA, November 19, 2007). See Table 5.1 and Table 5.2 for other values describing energy prices, carbon source prices and fee functions. These values are in the sequel referred to as the nominal case. Parameter kE kcar Value 0.037 549 Unit EUR/kWh EUR/m3 Table 5.1: Nominal energy and carbon source prices used in the simulation studies. The energy price, kE , is based on yearly average prices from a major Swedish supplier. In each experiment, the benchmark has been run for 150 simulation days for a grid of constant input values. Only the last 100 simulation days were considered when evaluating the cost functions to avoid the influence of transients. 5.3.1 Simulation results for the denitrification process In Figures 5.1–5.5 the benchmark has been run for a grid of different values for the external carbon dosage, Qcar , and the internal recirculation flow rate, 80 Parameter Value Unit αlimit,NO β0,NO ∆αNO ∆βNO 8.0 1.4 2.7 8.2 mg(N)/l EUR/1000 m3 EUR/kg NO EUR/kg NO αlimit,NH β0,NH ∆αNH ∆βNH 1.5 2.7 4.1 12.3 mg(N)/l EUR/1000 m3 EUR/kg NO EUR/kg NO Table 5.2: Default parameter values of the nitrogen fee functions used in the simulation studies. Qi , in order to obtain stationary values of the nitrate concentrations. Qcar has been varied in steps of 0.1 m3 /day and Qi in steps of 2500 m3 /day. In Figure 5.1 the stationary operational map depicting the total cost is shown for the nominal case in Table 1 when no nitrate discharge fee is used. The level curves for the nitrate concentrations in the anoxic and aerobic an and S e , respectively, are plotted. From the figure, an compartments, SNO NO optimum nitrate set point in the anoxic compartment is easily found for any given effluent nitrate set point. Approximate locations for these optima are marked by ’X’ in the figures. If, for instance, an effluent set point of 13 mg (N)/l is desired it is seen from the figure that the corresponding optimum set point of nitrate in the anoxic compartment is around 2.5 mg (N)/l and that the corresponding optimum operational cost is around 400 EUR/day. With an effluent nitrate set point of 7 mg(N)/l, the optimum set point of nitrate in the anoxic compartment is between 1 and 1.5 mg(N)/l. If an operational map for a wider operational area is studied it is seen more clearly that the an at the cost optimal point decreases as S e value of SNO NO decreases. For e an near 0.3 mg(N)/l gives the cost-optimal instance, if SNO = 2 mg(N)/l, SNO operating point. Note from Figure 5.1 that the difference in the operational costs between using optimal and non-optimal set points of nitrate in the anoxic compartment may be significant. In order to illustrate the impact of changes in energy prices on the choice of optimum nitrate set point in the last anoxic compartment, Figure 5.2 shows simulations with the same settings as in Figure 5.1 except that the energy price is ten times as high (kE = 0.37 EUR/kWh). It is seen that the external carbon dosage now is less dominant in the total cost and that the optimum set point of nitrate in the anoxic compartment decreases to below 1 mg(N)/l for effluent nitrate set points less than 10 mg(N)/l, see Figure 5.2. Another interesting case is illustrated in Figure 5.3, where the cost for dosing an external carbon source is set to zero. This case is not unrealistic since the carbon source may be provided free of charge if, for instance, industrial 81 4 6 0.0 5 4 8 6 8 0.0 3 0.5 1 1.5 0.3 7 3 2 0.1 12 4 11 9 14 11 7 X 2000 1800 1600 1400 1000 800 600 9 1200 10 8 x 10 10 15 10 0.3 5 5 X 5 12 01 0. 3 0.0 0.0 5 0.1 0.3 2 1 1.5 10 114 15 14 2000 1800 1600 1400 6 6 8 7 X 5 00 0. 5 4 3 0. 3 X 7 9 8 X 01 13 X 10 X 3 2 1200 1000 7 7 X 800 5 6 0.5 3 8 X 600 i 6 400 Q [m3/day] 13 9 7 5 1. 5 0. 1 11 2 3 8 2000 1.5 Qcar [m3/day] 11 10 12 13 3 0.00 9 03 1800 1 12 1 0.0 5 0.00 0.0 1600 0.5 15 3 0.0 0.0 11 1400 14 0.05 1200 1000 13 0.3 0.1 05 10 800 600 0 0 15 400 1 0.0 0 0. . 12 0 14 03 0. 9 03 0. 5 1 0. 0.001 13 14 15 2 2.5 3 Figure 5.1: Stationary operational map for a grid of different values of Qcar and Qi for the case with no nitrogen fee. Solid lines show the total cost, dashe , and dotted lines dotted lines show the effluent nitrate concentration, SNO an . Approxishow the nitrate concentration in the anoxic compartment, SNO mate locations for the optimal nitrate set points in the anoxic compartment for given desired effluent nitrate concentrations are marked by ’X’. Nominal energy and carbon source prices according to Table 1 are used. by-products are available. The principal behaviour of the cost function in this case is the same as if a very high energy price is used. The conclusion that can be drawn from Figure 5.3 is that even though provided for free, the dosage of an external carbon source may have a large impact on the total cost. This is due to the impact of external carbon addition on the aeration costs. The cheaper the price of carbon source, the lower the optimum nitrate set point in the anoxic compartment becomes. Table 5.3 summarizes optimum set point values of nitrate in the anoxic compartment for a number of effluent set point values and operational costs. In the table, the tendencies described above is seen. The first part of the table shows the impact of a decreasing effluent nitrate set point on the optimum value of the anoxic nitrate set point, the second part of the table shows the impact of a higher energy price and the third part of the table shows some optimum values when the carbon source is provided without cost. 82 4 1 5 0.0 0.25 0.5 5 6 8 9 00 55 5 6 12 00 60 00 65 0.0 25 1.5 2 00 8 0.1 3 10 11 4 10 7 8 15 45 00 50 9 7 x 10 11 10 40 00 9 14 13 X 7 01 0. 25 0.5 0.0 0.1 1 1.5 2 5 0.2 5 6 6 0 550 00 40 8 7 X 05 0 12 0. 8 X 01 0. X1 11 5 0. 5 0 0. 8 9 5 02 0. 9 05 10 25 5000 2 1 .5 1 0. 0 450 10 X 3 2X 7 9 X 13 4 14 3 0 600 X 00 4 6 0.0 5 3 10 11 7 15 8 7 5 00 50 X 00 45 6 35 Qi [m3/day] 4 5 0.0 0. 2 0.00 11 12 10 13 3500 0.1 0 0 0.0 5 0.5 1 0.0 0.0 25 5500 14 11 12 4000 15 3000 1 13 14 15 5 0.00 2 0.00 15 1 1.5 3 12 131 0.00 2 14 2.5 3 Qcar [m /day] Figure 5.2: Stationary operational map with the same settings as in Figure 5.1 except that the energy price is 10 times as high. Next, the case with a constant cost per discharged kg effluent nitrate according to (5.6) is investigated. Figure 5.4 shows this case with a fee of 5.5 EUR/kg effluent nitrate, i.e. ∆αNO = 5.5 EUR/kg. The nominal energy and carbon source prices from Table 1 are used. The cost-optimum is now found at Qcar = 0.7 m3 /day and Qi = 52 500 m3 /day with a cost of 1620 an = 1.7 mg(N)/l and S e EUR/day corresponding to SNO NO = 8.8 mg(N)/l. The penalization of nitrate discharges creates a well-defined minimum in the total cost function and makes it less desirable to discharge more nitrate with the effluent. Consequently, the importance of the set point choice for an has become larger. The overall optimum nitrate set point is marked by SNO a star. The effluent nitrate fee certainly has an impact on the cost-optimal effluent nitrate set point. However, since there is no impact of the desired (set point) effluent nitrate concentration and in the discharge fee, it may be hard to relate the value of ∆αNO to the optimum operating point – or operating region – at (or below) a certain effluent nitrate concentration. The third investigated nitrate cost function given by (5.7) has a discontinuity at a certain predefined concentration of effluent nitrate, αlimit,NO . If the jump at the discontinuity, β0 , is sufficiently large it is easy to find 83 4 x 10 3 0 0.2 5 0.5 0. 0 10 0.1 1 1.5 25 0.0 5 2 6 4 01 46 0 42 3 8 X X 7 15 360 11 6 6 38 0.2 5 12 0 00 9 1 0 0. 0. 5 5 X 0 42 340 7 0 40 4 44 0 X8 5 0. 2 3 X 11 1 0.0 11 13 15 .5 0 2 0.00 380 360 5 0.2 10 12 14 11 300 13 5 0 0 1 0.0 9 05 10 12 0.1 8 0 X X 9 5 02 0. 0.0 0.5 25 0.0 14 15 5 0.00 400 5 1. 05 0. 1 0. 34 2 10 X 320 14 13 3 1 7 8 X 2 0.00 15 1 1.5 3 Q car 2 14 2.5 12 0.001 13 420 3 0.0 25 0.1 1.5 1 5 0 0.0 14 13 40 7 Qi [m /day] 5 0 380 5 48 44 7 6 0 0 9 9 12 8 6 X 400 6 8 50 11 11 420 10 9 7 4 0.0 5 15 2 46 5 10 4 44 0 7 8 10 3 [m /day] Figure 5.3: Stationary operational map with the same settings as in Figure 5.1 except that the carbon source is assumed to be provided for free. the optimum operating point. Also in this case, the nominal energy price is considered. Figure 5.5 shows the stationary operational map when using this cost function with parameter values given by Table 5.2 (nominal energy and carbon source prices in Table 5.1 were used). The optimum point is located at Qcar = 0.825 m3 /day and Qi = 63 000 m3 /day with a total cost an = 1.7 mg(N)/l and S e of 1227 EUR/day, SNO NO equals 8 mg(N)/l. In practice, when treating non-constant influents, it is of course advisory to choose an operating point that is located slightly below the legislative limit. The region of economic efficient operation, say operation with a total cost below 1300 EUR/day, is relatively large. The larger β0 , the sharper the limit at αlimit,NO becomes. To check the robustness of the results, the sensitivity to changes in model parameters on the location of the cost-optimal operational points was investigated. It was found that there were only minor differences between the obtained operational maps, and in fact, for many changes of the model parameters most of the optimum set points almost coincide. Since the conclusions from the sensitivity analysis were similar to the conclusions in the previous cases, the results are omitted here, see Samuelsson et al. (2005a) for details. 84 Case kE =0.037, kcar =549 kE =0.37, kcar =549 kE =0.037, kcar =0 e SNO [mg(N)/l] 10 7 5 2 10 7 10 7 an SNO [mg(N)/l] 1.8 1.3 0.9 0.35 1.0 0.6 0.6 0.4 Qi [m3 /d] 43 73 110 280 35 58 31 56 000 000 000 000 000 000 000 000 Qcar [m3 /d] 0.55 1.0 1.4 2.7 0.6 1.1 0.7 1.2 an , Q and Q Table 5.3: optimum values of SNO i car for a number of given values e of SNO are shown for the nominal case, the case in Figure 5.2 and the case in Figure 5.3. The energy price kE are in units of EUR/kWh and the price for carbon source in EUR/m3 . Furthermore, in (Samuelsson et al., 2005a), the case with time varying influents were also considered in order to study the effects of dynamic influents and different load situations. The time varying influents used were the ones specified by BSM1 to describe different weather conditions. These experiments did not lead to any essential changes in the results and are therefore omitted in this chapter. 5.3.2 Simulation results for the combined denitrification and nitrification process The aeration is a key parameter when considering the nitrification. Therefore, the simulations discussed in this section have been run for a range of different constant DO set points, from 0.5 mg/l up to 5.0 mg/l. In Figure 5.6 operational maps for the effluent ammonium concentration are shown for four different DO set points. A DO set point of 0.5 mg/l results in very high effluent ammonium concentrations and is therefore clearly not sufficient for good nitrification performance. The performance in terms of effluent ammonium is much better with DO set points of 2.0 mg/l, 3.5 mg/l and 5.0 mg/l, see Figure 5.6. For a comparison, Figure 5.7 shows four operational maps for the effluent nitrate concentration for the same DO set points [mg/l] as in Figure 5.6. As expected, the denitrification benefits from low DO set points. From Figure 5.8, where operational maps for the total cost are shown, it is clear that the selection of the DO set point is crucial for the total cost. When the DO set point is 0.5 mg/l the total cost is very high due to the large effluent ammonium concentrations. For the other three considered DO set points, note, for instance, how the operational area corresponding to a 85 4 0.0 3 2400 0.0 3 0.3 0.5 9 6 8 1700 16 50 8 1800 0.0 5 2200 1 1.5 0.1 2000 0.3 2 4 17 00 10 5 3 12 6 180 110 8 10 13 9 4 5 200 15 0 7 7 x 10 11 10 9 5 14 2600 0. 03 220 0 0.0 5 200 0 18 240 0 00 165 0 6 0.1 1 0.3 2 1.5 10 * 11 165 0 13 5 12 5 170 0 4 6 3 7 8 15 180 0 6 8 05 7 01 3 9 0. 03 0.5 17 00 0 0. 0. 0. 4 1650 3 6 7 Qi [m /day] 0. 01 7 7 3 10 2 2 1700 1 5 1. 3 0. 12 11 05 0 03 0. 0 20 0. 00 22 00 24 15 0 0 00 0 010 30 3 2200 5 3 0.0 0.5 280 14 5 0.00 0.00 1 3 1.5 Q car 0.00 12 0 260 15 2400 11 0.00 13 0 2000 0.0 9 03 0.0 1 12 0.0 3 0.0 14 0.1 05 0.0 11 0.5 0.3 10 8 28 9 1 0. 13 1 26 00 3 14 8 0.001 13 2 [m3/day] 15 2 14 2.5 3 Figure 5.4: Stationary operational map for a grid of different values of Qcar and Qi . Solid lines show the total cost including a constant nitrate-charge per kg effluent nitrate, dash-dotted lines show the effluent nitrate concentration, e , and dotted lines show the nitrate concentration in the anoxic compartSNO an . The star indicates the minimum-cost point. Here ∆α ment, SNO NO = 5.5 EUR/kg. total cost of less than 2000 EUR/day expands when the DO set point is decreasing from 5 mg/l to 2 mg/l. A DO set point near 2 mg/l appears to be close to optimal from a total cost point of view. Note also, that the impact of the ammonium fee on the total cost is rather limited in the case of sufficient aeration (i.e. in this study for the DO set points 2 mg/l, 3.5 mg/l and 5 mg/l). Therefore the optimum operating point does not change much when also charging effluent ammonium. This is further illustrated in Figure 5.9 that shows the operational map when the DO set point is 2.0 mg/l. The optimum point is located at approximately Qi = 58 000 m3 /day and Qcar = 0.75 m3 /day. 5.4 Discussion Cost efficient operation of the denitrification and nitrification processes was studied using operational maps. Considering the case when no charge on 86 4 x 10 0.0 3 1300 9 2000 1400 0.3 4 1800 1600 0.5 8 2200 0.0 5 0.3 1 1.5 2 11 7 3 0. 0 1 1400 0.3 10 2 1.5 0. 0.5 11 01 6 0. 7 0. 3 9 00 5 00 7 1800 1600 8 4 1600 14 8 14 20 10 13 3 5 0.1 3 6 5 12 15 8 18 7 00 1300 4 0.0 13 6 * 6 05 1 0. 0. 1 2 12 600 11 .01 0 0 180 3 03 8 0.0 9 0. 0. 5 0. 00 03 1 5 1. 2 2200 1600 5 7 i 0.1 7 3 10 5 6 12 8 1800 15 2000 10 14 9 7 Q [m3/day] 5 1300 8 3 1400 9 5 4 6 11 10 10 2000 220 0 9 0 240 11 10 12 1 0 0 15 13 1800 0.3 2200 14 2000 0.1 2200 5 0.0 3 0.0 2400 0.5 15 0.01 1 2400 2600 13 14 5 2600 3 0.00 0.00 2800 1.5 Qcar [m3/day] 2800 12 0.001 3000 15 2 11 13 14 2.5 3 Figure 5.5: Stationary operational map for a grid of different values of Qcar and Qi . Solid lines show the total cost including a nitrate-charge according e , and to (5.7), dash-dotted lines show the effluent nitrate concentration, SNO dotted lines show the nitrate concentration in the last anoxic compartment, an . The additional charge for exceeding the legislative discharge limit at SNO αlimit,NO = 8.0 mg(N)/l is here β0,NO = 1.4 EUR/1000 m3 . The star indicates the minimum-cost point. effluent nitrate is imposed it is seen that, in the nominal case, the cost for dosing an external carbon source dominates the total cost. It is also clear that the operational area with respect to effluent nitrate is divided in two parts with different gain characteristics, and that for each desired value (set e , there is a cost-optimal point in the operational map correpoint) of SNO sponding to a certain value of the nitrate concentration in the last anoxic an . This point naturally also corresponds to certain stationcompartment, SNO ary values of the input signals, Qcar and Qi . In contrast to what is described in, for instance (Yuan and Keller, 2003), the findings in this chapter are that the cost-optimal set point of nitrate in the anoxic compartment depends on the choice of effluent nitrate set point, as well as the specific operational an level decreases with decreasing S e costs. The optimum SNO NO levels and with increasing energy costs (or decreasing costs for external carbon). The location of the optimum set point choice is, however, not very sensitive to 87 SeNH with DOsp =0.5 4 10 34 34.5 Qi [m3/day] 31 32 30.5 30 6 4 1.2 6 0.9 4 0.8 1 0.9 31.5 0 0 0.5 1 1.5 3 2 2 0 0 2.5 3 1.1 1 0.7 0.8 33 35 33.5 2 x 10 8 32.5 8 SeNH with DOsp =2 4 Qi [m3/day] 10 x 10 0.6 0.7 0.5 1 Qcar [m /day] e 10 sp SNH with DO 4 x 10 e =3.5 10 0. x 10 85 Qi [m3/day] Qi [m3/day] 0.7 0.65 0.5 2 0.5 1 0. 65 6 0.5 4 0.6 0.45 0.5 5 2 0.6 0.5 5 0.55 0.45 0 0 0.5 7 0. 0.6 0.4 1.5 2 3 2.5 =5 6 8 6 0.55 2.5 0. 8 0.7 5 4 sp SNH with DO 4 0.7 0.6 5 2 Qcar [m /day] 0. 8 1.5 3 0 0 3 Qcar [m3/day] 0.5 0.5 0.4 0.35 0.45 1 1.5 2 2.5 3 Qcar [m3/day] Figure 5.6: Stationary operational maps for a grid of different values of Qcar and Qi showing the effluent ammonium concentrations for four different constant DO set points [mg/l]. variations in the ASM1 parameters. Simulations showed that when an appropriate level of aeration is employed the effluent ammonium concentration is kept low. Therefore, the location of the optimum operating region was not significantly changed when also effluent ammonium was charged. From the results in the previous section, some questions may arise. The first is which control structures that can be expected to give a good performance in terms of disturbance rejection and set point tracking. In different operating points different control structure selections may be suitable. As indicated by e.g. Figure 5.1, in the area of the operational map corresponde is mostly affected by the input signal ing to cost-efficient nitrate control, SNO an Qcar , while SNO is affected by both input signals in this area. This situation in particular is further discussed by Samuelsson et al. (2005c) (see Chapter 4) and also to some extent by Yuan and Keller (2004). The second question is how to design the actual control law in order to minimize operational costs. Below, some possibilities for this control design are discussed: • One approach is to use two different control loops (for instance PI88 SeNO with DOsp =0.5 4 4 13 11 Qi [m3/day] 0.15 6 6 4 7 8 9 2 2 0 0 0 0 13 14 1 1.5 2 2.5 3 0.5 =3.5 10 19 8 9 2 0 0 18 17 11 12 13 0.5 1 1.5 14 15 16 2 2.5 9 5 11 10 6 7 4 8 9 10 2 10 14 18 17 8 10 4 16 Qi [m3/day] 9 7 17 15 16 13 12 11 6 7 15 3 6 6 8 5 12 Qi [m3/day] 8 2.5 SeNO with DOsp =5 4 x 10 14 10 sp 2 Qcar [m /day] 15 14 13 e SNO with DO 1.5 3 Qcar [m /day] 4 13 14 1 3 x 10 12 15 16 17 8 0.5 10 11 12 7 0.2 Qi [m3/day] 6 5 7 8 6 8 8 10 14 10 SeNO with DOsp =2 4 x 10 10 9 4 x 10 0 0 3 Qcar [m3/day] 11 15 17 18 0.5 14 13 12 16 1 1.5 15 2 2.5 3 Qcar [m3/day] Figure 5.7: Stationary operational maps for a grid of different values of Qcar and Qi showing the effluent nitrate concentrations for four different constant DO set points [mg/l]. e an separately. Given the desired controllers) to control SNO and SNO e an can nitrate effluent concentration, SNO , the optimum set point of SNO be found from the operational maps depending on process conditions. e is chosen as 8 For instance, in the nominal case, if the set point for SNO an mg(N)/l, the optimum set point for SNO is close to 1.5 mg(N)/l. If dee and Q S an . centralized control is to be used, Qcar should control SNO i NO Other control structures may, however, yield a better performance, see Samuelsson et al. (2005c) (see Chapter 4). • Another possibility in the nominal case, see Figure 5.1, is to use a constant high internal recirculation flow rate Qi and to use only Qcar in order to control the effluent nitrate concentration. Since Qi has a much smaller impact on the total cost than Qcar , this would render a close to cost-optimal operation. This possibility has also been mentioned by Ingildsen (2002). This is, however, not a suitable strategy if the carbon source is inexpensive, see Figure 5.3. • To achieve a cost-optimal performance in the nominal case, the total 89 Total cost with DOsp =2 x 10 1 1.5 2 2.5 3 1600 2000 2200 0.5 1 Qcar [m3/day] sp Total cost with DO =3.5 2600 2200 1600 0 24 00 0 18 2 0 0 0.5 280 3000 2600 1 1.5 3 2.5 3 3200 2 3400 2.5 26 0 2600 00 20 4 0 0 0 3 24 00 00 6 2 0 2000 2200 2400 28 Qi [m3/day] 160 0 4 2 sp x 10 8 2000 220 0 0 200 6 1.5 Total cost with DO =5 4 10 180 0 Qi [m3/day] 8 2800 3000 Qcar [m3/day] 0 10 0 260 2400 22 00 4 x 10 1800 0 1400 1400 1600 1800 0 0 2800 0.5 1600 2400 0 0 4 2 8000 2 6 150 4 1500 Qi [m3/day] Qi [m3/day] 6 00 8 1800 13 7500 8 2400 4 10 2200 Total cost with DOsp =0.5 3000 x 10 2000 4 10 220 0.5 0 340 0 2400 2600 280 3000 00 32 1 1.5 3800 4000 3600 2 2.5 3 Qcar [m3/day] Qcar [m /day] Figure 5.8: Stationary operational map for a grid of different values of Qcar and Qi showing the total cost including a nitrate-charge according to (5.7) and an ammonium-charge according to (5.8) for four different constant DO set points [mg/l]. cost could be minimized on-line using quadratic criteria yielding for example LQG or MPC controllers. Such a criterion has the typical form Z T V = eT (t)Q1 e(t) + uT (t)Q2 u(t)dt (5.9) 0 where e is a column vector containing the control errors and u the input signals. The weighting matrix Q2 can be chosen to reflect the costs for the different input signals and Q1 can be seen as a performance weight. The difficulty with this criterion is how to weight control performance against cost minimization, i.e. how to choose the matrices Q1 and Q2 . From the prior knowledge obtained from Figure 5.1, in the nominal case, the elements of the matrix Q2 could be chosen in an intuitive manner. A rule of thumb could be to choose Q2 as a diagonal matrix with the element corresponding to Qcar significantly larger than the element corresponding to Qi , since the external carbon source is much more expensive than the pumping of the internal recirculation flow 90 rate in the nominal case and thereby dominates the total cost. Such a choice clearly penalizes a large value of Qcar in the criterion. • A simple grid search could be performed on-line until the optimum point is reached. This method is simple and has the advantage that no operational map and thereby no model is required. One such optimization algorithm is presented by Ayesa et al. (1998). This algorithm is employed to minimize a global penalty function combining effluent requirements and costs. For the case when the nitrate discharge is penalized with a constant charge per kg, see (5.6), it is seen from Figures 5.4 that this creates a minimum in the total cost function (5.5). The main drawback using such a cost function for automatic control is that it is hard to relate the location of the cost-optimal set point to the nitrate discharge fee, ∆αNO , and thereby hard to say which set point a certain fee results in. This also depends on the effluent flow rate, Qe . Due to the discontinuity, this problem can be overcome if instead the fee function according to Carstensen (1994) is implemented. The location of the discontinuity of the fee function (5.7) immediately coincides with the optimum set point for the effluent nitrate, e , if the discontinuity is sufficiently large. Using this fee in the total SNO cost is a convenient way to achieve cost optimality for a certain set point of e , see Figure 5.5. Minimizing this total cost function oneffluent nitrate, SNO line using some automatic control strategy would be a good way to impose the importance of good performance via penalizing the effluent discharge into the control design. If the discharge of nitrogen over a certain legislative limit is directly associated with a higher fee, this could clearly motivate the use of more advanced control strategies. The impact of the fee in the control design is also easy to understand even for people with a limited knowledge in automatic control, compared to the related matter of choosing a performance weight in some quadratic criterion. 91 4 1800 0. 01 01 5 00 0. 1 00 14 12 4 3 0. 15 00 7 2400 2200 130 i 11 10 9 1400 0 6 3 8 1 0. 5 5 .2 .1 0 1500 1 2000 14 2000 160.6 5 0.0 3 02 0.0 2 0.00 5 0.00 152800 2600 16 1 1.5 Q car [m3/day] 280011 2600 2400 15 2400 0.5 2 10 12 0.7 2200 14 0.0 400 11 13 1800 0.9 9 00 0522 0.00. 108 0.0 12 17 0.100 22 8 0. 0 180 11 0.7 03 0. 05 0 13 1600 10 00 20 9 0.8 5 1. 2 7 0.9 00 16 0 0 2200 0.0 0.0 0.1 0.2 5 5 1 0.9 0.8 1 0.0 3 2000 1600 3 0.5 1 1.5 2 5 1500 13 6 16 00 7 1237 X 8 2 5 1. 1 6 0.9 6 4 5 1800 1500 1400 4 1300 7 Q [m3/day] 0.1 0.25 0.5 1 1.5 1 3 12 8 8 5 0.0 9 150 10 0 5 1400 6 1600 14 1800 11 5 1.2 2400 13 8 2 6 1.1 0 9 4 130 1 9 7 7 x 10 10 10 13 14 2 0.7 3000 0.8 12 001 0. 3200 13 2.5 Figure 5.9: Stationary operational map for a grid of different values of Qcar and Qi . Solid lines show the total cost including a nitrate-charge according to (5.7) and an ammonium-charge according to (5.8), dash-dotted lines show e , dotted lines show the nitrate conthe effluent nitrate concentration, SNO an and dashed lines show the centration in the last anoxic compartment, SNO e effluent ammonium concentration, SNH . The additional charge for exceeding the legislative discharge limits at αlimit,NO = 8.0 mg(N)/l and αlimit,NH = 1.5 mg(N)/l is here β0,NO = 1.4 EUR/1000 m3 and β0,NH = 2.7 EUR/1000 m3 , respectively. X indicates the minimum-cost point. 92 3 5.5 Conclusions In this chapter a bioreactor model describing a pre-denitrifying wastewater treatment plant was studied from a process economic point of view. The impact of different nitrate and ammonium cost functions on the location of the cost-optimal operating point was examined. Given the desired value of the effluent nitrate concentration, the energy price and the price of carbon source, there is a corresponding optimum set point for the nitrate in the anoxic compartment. For low effluent set points, the optimum anoxic set point may be located well below 1 mg(N)/l. The simulations also show that the difference in the operational costs between an optimum and nonoptimum anoxic set point may be large. The locations of the optimum set point values are, however, not very sensitive to changes in the ASM1 parameters. In summary, it can be concluded that the approach presented in this chapter may give a valuable tool towards running a WWTP in a more cost effective way. Natural extensions and topics for further research include: • Evaluation of the suggested approach using live data from a full-scale WWTP. • Extending the criteria function with costs for the sludge handling. This will indeed penalize an excessive carbon dosage. Another interesting extension is to include chemical precipitation for phosphorous removal. In a pre-denitrifying plant, there is an interesting trade-off between removal of phosphorous and substrate where the latter is useful for the denitrification process. 93 94 Chapter 6 Aeration Volume Control in an Activated Sludge Process – Discussion of Some Strategies Involving On-Line Ammonium Measurements In this chapter the influence of the aeration on the efficiency of the nitrogen removal in an activated sludge process is studied. Different strategies for controlling the DO set point and the aerated volume are compared in terms of treatment efficiency in a simulation study. 6.1 Introduction In an activated sludge process (ASP) configured for nitrogen removal microorganisms (mainly bacteria) are employed to reduce the influent nitrogen. Microorganisms in the aerated basins convert ammonium into nitrate while consuming oxygen. This is the nitrification process. For this process to occur, the concentration of dissolved oxygen (DO) in the aerobic basins must be sufficiently high. In the anoxic basins, another type of bacteria is employed in the denitrification process where nitrate is converted into nitrogen gas. Most common today is to control the airflow rate to maintain a specific DO level. However, since the influent load situation may vary significantly the ammonium in the influent may also undergo large variations. Improvements in controlling the nitrification rate could therefore be expected if online-measurements of the ammonium concentration in, for instance, the last aerated basin, are utilized in the control of a time-varying DO set point 95 (see e.g. Lindberg (1997)). Supervisory control of the DO set point could then be implemented such that a specific concentration of ammonium in the effluent is aimed at. Furthermore, in DO control, it is of great importance that the aerated volume is sufficiently large to ensure that the desired DO concentration may be reached. If the aeration volume is too small, the airflow rate may saturate before the desired DO concentration is achieved. Also, a too high DO level is very expensive to maintain (regarding the energy cost for the aeration pumps) since the DO concentration in the water does not increase linearly with the airflow rate. On the contrary, a too low DO concentration not only reduces the nitrification but may also produce sludge with poor settling capability and often requires basins with mixers. In addition, nitrous oxide may be created. The need of aeration control has been investigated by several authors. Vrecko et al. (2006) present and compare different aeration controllers. Meyer and Pöpel (2003) use fuzzy control in the determination of DO set points and the ratio between aerated and non-aerated basins. Both of these papers evaluate the performance of the suggested strategies in pilot plants. Optimum aerobic volume control based on the oxygen uptake rate as an indicator of the nitrification performance is discussed by Svardal et al. (2003). Gerksic et al. (2006) presents a method for on-line estimation of the respiration rate. Brouwer et al. (1998) and Samuelsson and Carlsson (2002) discuss different strategies for aeration volume control. Recently, two strategies for aeration volume control have been developed by Ekman (2005) (also presented in (Ekman and Carlsson, 2005a) and (Ekman and Carlsson, 2005b)) and by Samuelsson (2005) (also presented in (Samuelsson et al., 2005b)). A great advantage of the approach presented in Ekman (2005) is that only standard DO sensors are needed. The key point of this strategy is that the DO consumption reflects the load situation in terms of influent ammonium. However, there may be situations when this is not entirely true. These occasions may for instance occur during the winter season when the water temperature is low. Therefore, if on-line ammonium sensors are available some alternative solutions (involving volume control) have been suggested to take care of this drawback. An example is the strategy presented by Samuelsson (2005) where the key idea is to feed back the actual treatment efficiency (in terms of ammonium concentration) and use supervisory DO set point control combined with a feedforward approach to control the aeration volume. Instead of estimating the reaction rate a tuning parameter is used. In this chapter, an improved version of the strategy developed by Samuelsson (2005) is presented. Also, another strategy, supervisory feedback aeration volume control, involving feedback from the effluent ammonium concentration, that share some similarity with the strategy by Ekman (2005) is discussed. For a more comprehensive background of the topic of aeration 96 volume control and DO set point control, see Samuelsson et al. (2005b) and Samuelsson (2005) and the references therein. 6.2 The simulation setup To illustrate and compare the suggested control strategies, a simulation study was performed using the COST/IWA simulation benchmark model number 1 (BSM1), see Jeppsson and Pons (2004) for a general survey and Copp (2002) for a more technical description of this simulation platform. The benchmark plant models pre-denitrifying wastewater treatment and consists of five basins. The first two basins are always anoxic and have a volume of 1000 m3 each. The volume of the last three basins is 1333 m3 each. The last two basins, number four and five, are aerated while the middle one, basin number three, may be aerated or anoxic, depending on the selected strategy. Three influent files based on the dry weather influent data, DRYINFLUENT, provided with BSM1, have been used to feed the simulation plant. The first data file, here called D, is the DRYINFLUENT in original form, whereas the other two data files, called D3 and D5, are the same as D except that the influent flow rate is decreased to a third and to the half of the flow rate in D (with an average value of 18 446 m3 /day), respectively. In this way three different load scenarios are simulated. For the aerated basins the maximum DO set point were limited to 5 mg/l max , to 360 day−1 . In all simulations and the maximum aeration intensity, KLa sp the set point for the effluent ammonium concentration, SNH , was selected to 3 mg/l. The DO controllers used in the experiment were ordinary PIcontrollers tuned to give a fast response in the DO concentration when the set points were changed. A further description of BSM1 and a schematic figure of the benchmark plant layout are given in Chapter 1. 6.3 6.3.1 Description of the proposed control strategies The reference aeration control strategies The suggested volume control strategies were compared with two strategies without volume control, the reference strategies. For both of these, the DO set points for basin four and five are the same and controlled supervisory such that the effluent ammonium concentration is kept at some pre-specified level, here 3 mg/l. The third basin is in the first reference case always anoxic while it is aerated in the same way as basin four and five in the second reference strategy. In short, the two reference strategies can be described as: 97 Reference strategy I: • DOsp 3 = 0. sp e • DOsp 4 =DO5 is controlled so that SNH = 3 mg/l. Reference strategy II: sp • DOsp 3 =DO4,5 . sp e • DOsp 4 =DO5 is controlled so that SNH = 3 mg/l. 6.3.2 Feedforward aeration volume control I and II In the volume control strategy initially suggested by Samuelsson (2005), feedforward control is combined with supervisory feedback control of the DO set points. The basic idea is to calculate the desired aeration volume, Vdes , from the simple relation in e Vdes (t) = Kv Q(t) SNH (t) − SNH (t) (6.1) where Kv is some positive constant parameter tuned by the user, Q is the in is the ammonium concentration in the influent and S e flow rate, SNH NH is the ammonium concentration in the effluent. If the desired aeration volume according to (6.1) is larger than the currently used aeration volume, then more basins are aerated until the aeration volume is at least as large as the calculated desired volume. Equation (6.1) is derived from the mass balance consideration for SNH in one completely mixed basin given by dSNH (t) Q(t) in e = RSNH (t) + SNH (t) − SNH (t) dt V (t) (6.2) where RSNH is the reaction rate of ammonium. The original strategy proposed by Samuelsson (2005) is here denoted feedforward aeration volume control I. Feedforward aeration volume control II is a slightly modified version of the previously discussed strategy where sp DOsp 3 is allowed to follow DO4,5 when basin three is aerated. In that way DOsp 3 will certainly be more smooth. As a second improvement a simple hysteresis rule is applied to Equation (6.1): Aeration of basin three with sp DOsp 3 =DO4,5 is switched on when the desired aerated volume exceeds 2666 3 m3 , but off, i.e. DOsp 3 =0, when this calculated volume is less than 2400 m . The two feedforward aertaion volume control strategies can for this simulation study be summarized as: 98 Feedforward aeration volume control I: • Calculate Vdes from (6.1). – If Vdes > 2666 m3 aerate basin 3 with DOsp 3 =3 mg/l. – Otherwise DOsp 3 =0. sp e • DOsp 4 =DO5 is controlled so that SNH = 3 mg/l. Feedforward aeration volume control II: • Calculate Vdes from (6.1). sp – If Vdes > 2666 m3 aerate basin 3 with DOsp 3 =DO4,5 . – If Vdes ≤ 2400 m3 and DOsp 3 > 0 switch off the aeration of basin 3, i.e. DOsp =0. 3 – Otherwise DOsp 3 =0. sp e • DOsp 4 =DO5 is controlled so that SNH = 3 mg/l. 6.3.3 Supervisory feedback aeration volume control An alternative strategy where the measurement of the influent ammonium concentration is not needed has also been developed. The idea is to aerate an extra zone when the difference between the DO set point and the measured DO concentration in the first aerated zone exceeds some selected threshold and as long as the effluent ammonium concentration exceeds the set point. A significant error here indicates that the current load situation is high and therefore, that a larger aeration volume is needed. In the strategy suggested by Ekman (2005) a similar idea is used to decide the DO set points. However, in the strategy presented here this idea is combined with supervisory feedback control of the DO set points for the last two basins such that a specific effluent ammonium concentration is aimed at. Hence, this strategy also involves on-line ammonium measurements. In the simulations, basin four and five have the same DO set point that is supervisory controlled by a PI controller with the effluent ammonium concentration as input and with a set point for this concentration of 3 mg/l. Basin three is aerated when DOsp 4 − DO4 − estart ≥ 0.5, where DO4 is the measured DO concentration in basin four and estart is a parameter tuned to reduce the abrupt start of the aeration; DOsp 3 is controlled by a PI controller sp with DO4 − DO4 − estart as input. Here estart is 0.4. The aeration of basin three is turned off when DOsp 4 − DO4 − estart ≤ 0.2. The strategy in this simulation study can be summarized as: 99 Supervisory feedback aeration volume control: sp • When DOsp 4 − DO4 − estart ≥ 0.5: DO3 is controlled by a PI controller sp with DO4 − DO4 − estart as input. sp • When DOsp 4 − DO4 − estart ≤ 0.2 and DO3 > 0: The aeration of basin sp 3 is turned off, i.e. DO3 = 0. • Otherwise DOsp 3 = 0. sp e • DOsp 4 =DO5 is controlled so that SNH = 3 mg/l. 6.4 6.4.1 Simulation results The reference aeration control strategies As can be seen in Figure 6.1 where the influent file is T, it is clear that the influent ammonium load is too heavy for the first reference strategy: Even though basin four and five are maximally aerated, i.e. DOsp 4,5 = 5 mg/l, the effluent ammonium concentration is high with a peak value of 24 mg/l and an average of 17 mg/l. It is obvious that a larger aeration volume is needed in order to improve the nitrification. With the influent files D3 and D5, the ammonium load is smaller compared to the previous case, and consequently, the aeration volume needed to get good nitrification performance is smaller and that is why the first reference strategy may perform well in these particular cases, see Table 6.1 for numerical results. In reference strategy II the aerated volume is larger since basin three is sp always aerated (DOsp 3 =DO4,5 ). As can be seen in Table 6.1 this strategy performs better than the first reference strategy, compare for instance the consumed energy per kg ammonium that is removed. 6.4.2 Feedforward aeration volume control I In the simulation experiments the parameter Kv was tuned to 0.01. Since basin three is aerated during periods of high influent load, this strategy achieves much better nitrification in the case of influent T compared to the first reference strategy, see Table 6.1. The peaks of the effluent ammonium concentration are clearly reduced (maximum peak value is 12.9 mg/l) and so is the average effluent concentration (4.6 mg/l). Compared to the second reference strategy, this strategy reduces the ammonium concentration e is 4.6 mg/l compared to 4.3 mg/l for reference strategy slightly less (SNH II) but at a higher efficiency – compare the consumed aeration energy per kg removed ammonium. Also, the sum of the effluent ammonium and the effluent nitrogen concentration is slightly lower. However, the performance of this strategy may be sensitive to the value of the selected parameter Kv . This can be reduced by introducing hysteresis. 100 Table 6.1: Numerical simulation results for the different strategies proposed. e e SNH is the effluent concentration of ammonium, and SNO is the corresponding value of nitrate. AE is the aeration energy, i.e. the electrical energy consumed in the aeration. The lowest values of effluent ammonium and nitrate concentrations as well as the most efficient strategies in terms of AE for each influent are emphasized. DOsp i is the DO concentration in basin i. f.b. is feedback. All concentrations in the table are average values. Strategy Influent Reference I Reference I Reference I Reference II Reference II Reference II Feedforward I Feedforward I Feedforward I Feedforward II Feedforward II Feedforward II Supervisory f.b. Supervisory f.b. Supervisory f.b. D5 D3 D D5 D3 D D5 D3 D D5 D3 D D5 D3 D e SNH [mg/l] e SNO [mg/l] e e SNH +SNO [mg/l] AE/day [kWh] 3.0 3.3 17.4 3.0 3.0 4.3 3.0 3.0 4.6 3.0 3.0 4.6 3.0 3.1 5.4 2.9 4.8 3.8 2.9 5.5 10.9 2.8 5.0 10.4 2.8 5.1 10.5 2.9 4.9 9.5 5.9 8.1 21.3 5.9 8.5 15.2 5.8 8.0 15.0 5.8 8.1 15.0 5.9 8.0 14.9 1869 4199 7849 1429 2592 8409 1645 2913 7429 1564 2629 7395 1890 4012 9539 101 AE/kg NH removed [kWh] 0.86 1.87 6.91 0.66 1.14 3.45 0.76 1.29 3.08 0.72 1.16 3.07 0.87 1.78 4.06 DOsp 3 [mg/l] DOsp 4,5 [mg/l] 0.0 0.0 0.0 0.38 0.74 2.6 0.3 1.3 2.3 0.1 0.6 2.7 0.0 0.03 0.9 1.1 2.6 5.0 0.38 0.74 2.6 0.7 0.8 2.8 0.7 0.9 2.8 1.1 2.3 3.9 Se [mg/l] 30 NH 20 10 0 0 2 4 6 Time [days] 8 10 12 2 4 6 Time [days] 8 10 12 2 4 6 Time [days] 8 10 12 2 4 6 Time [days] 8 10 12 DO3 [mg/l] 1 sp 0 −1 0 sp DO4,5 [mg/l] 6 5 4 3 0 Se NO [mg/l] 6 4 2 0 0 Figure 6.1: Effluent concentrations and DO set points for reference strategy I with influent D. Also, the aeration of basin three starts and ends rather abruptly since DOsp 3 is either 0 or 3 mg/l. As a typical illustration of this behaviour, consider Figure 6.2 where the influent load is D3. 6.4.3 Feedforward aeration volume control II Figures 6.3 and 6.4 show the performance of this strategy with influent files D3 and D, respectively. Note particularly that now DOsp 3 is smoother e is slightly higher but compared to the previous case in Figure 6.2. SNO the aeration energy consumption is reduced and the efficiency in terms of aeration energy consumed per kg ammonium removed is higher (i.e. lower values). 6.4.4 Supervisory feedback aeration volume control Figure 6.5 shows the performance of the strategy when the influent is D3. The strategy works as expected since basin three is only aerated during the load peaks. With input D basin three is aerated during longer periods, see Figure 6.6. Note that in the case of influent D5, only basin four and five 102 Se [mg/l] 6 NH 4 2 0 0 4 6 Time [days] 8 10 12 2 4 6 Time [days] 8 10 12 2 4 6 Time [days] 8 10 12 2 4 6 Time [days] 8 10 12 3 DOsp [mg/l] 4 2 0 0 4 4,5 DOsp [mg/l] 2 2 0 0 5 e SNO [mg/l] 10 0 0 Figure 6.2: Effluent concentrations and DO set points for the originally proposed version of the feedforward aeration volume control strategy (feedforward aeration volume control I) with influent D3. need to be aerated according the strategy, see Table 6.1. As can be seen from the numerical results in Table 6.1, this strategy is not as efficient in terms of aeration energy as most of the other strategies. However, the reduction of nitrate and ammonium is very good for all of the studied load situations, in particular for influent D3 and D5. Furthermore, this strategy is the only of the considered volume control strategies that achieves a totally anoxic third basin for influent D5. Note that a fine tuning of the other strategies employing volume control may as well result in an anoxic third basin for influent D5. 103 Se [mg/l] 6 NH 4 2 sp DO3 [mg/l] 0 0 sp 4 6 Time [days] 8 10 12 2 4 6 Time [days] 8 10 12 2 4 6 Time [days] 8 10 12 2 4 6 Time [days] 8 10 12 4 2 0 0 DO4,5 [mg/l] 2 4 2 0 0 5 Se NO [mg/l] 10 0 0 Figure 6.3: Effluent concentrations and DO set points for the strategy feedforward aeration volume control II with influent D3. 104 10 5 e SNH [mg/l] 15 0 0 2 4 6 Time [days] 8 10 12 2 4 6 Time [days] 8 10 12 2 4 6 Time [days] 8 10 12 2 4 6 Time [days] 8 10 12 DOsp [mg/l] 6 3 4 2 0 0 4 4,5 DOsp [mg/l] 6 2 0 0 10 e SNO [mg/l] 20 0 0 Figure 6.4: Effluent concentrations and DO set points for the strategy feedforward aeration volume control II with influent D. 105 S e NH [mg/l] 8 6 4 2 2 4 6 Time [days] 8 10 12 2 4 6 Time [days] 8 10 12 2 4 6 Time [days] 8 10 12 2 4 6 Time [days] 8 10 12 0.6 0.4 0.2 0 0 DO sp 4,5 [mg/l] 6 4 2 0 0 e NO [mg/l] 8 S sp DO3 [mg/l] 0 0 6 4 2 0 0 Figure 6.5: Effluent concentrations and DO set points for the supervisory feedback aeration volume control strategy with influent D3. 106 10 5 e SNH [mg/l] 15 0 0 2 4 6 Time [days] 8 10 12 2 4 6 Time [days] 8 10 12 2 4 6 Time [days] 8 10 12 2 4 6 Time [days] 8 10 12 DOsp [mg/l] 3 3 2 1 0 0 4 4,5 DOsp [mg/l] 6 2 0 0 e SNO [mg/l] 15 10 5 0 0 Figure 6.6: Effluent concentrations and DO set points for the supervisory feedback aeration volume control strategy with influent D. 107 6.5 Conclusions Five different aeration control strategies involving ammonium measurements have been evaluated in a simulation study. Three strategies in the study involve aeration volume control. All of those were able to dampen the process disturbances that occur in form of load variations in the influent wastewater. The suggested improved feedforward strategy performs overall very well and is the most efficient strategy in terms of aeration energy for the high load scenario (influent D). The second reference strategy is however more cost efficient when considering the two low load scenarios (influent D5 and D3). On the other hand, the reference strategies do not offer the feature of volume control, and it is hence crucial that the operator selects to aerate an appropriate number of basins. Otherwise, bad performance both in terms of wastewater treatment and in terms of aeration energy consumption may result. In this simulation study this is illustrated by the bad performance of the first reference strategy when the influent load is high. With aeration volume control the selection of the number of aerated basins is decided automatically. Clearly, this is an advantage both from a process economy point of view as well as when considering the treatment efficiency. 108 Appendix A The minimized condition number This appendix reviews the concept of minimized condition number. The text is based on the description of condition numbers in Halvarsson (2003). To obtain the minimized condition number the scaling matrices, S1 and S2 , are chosen according to γmin (G) = min γ(S1 GS2 ) S1 ,S2 (A.1) As shown by Grosdidier et al. (1985) γmin are closely related to the RGA. For the case of a 2 × 2 plant, G, Grosdidier et al. (1985) show that the minimized condition number is given by q γmin = kΛ(G)k1 + kΛ(G)k21 − 1 (A.2) where Γ is the RGA matrix and the 1-norm is defined as kΛk1 = max j m X |λij | (A.3) i=1 i.e. “the maximum column sum”. It can also be shown that γmin is bounded by kΛ(G)k1 according to γmin ≤ 2kΛ(G)k1 (A.4) with equality when kΛ(G)k1 → ∞. For larger quadratic systems the following conjecture is valid (Grosdidier et al., 1985): γmin ≤ 2 max(kΛ(G)k1 , kΛ(G)k∞ ) (A.5) where the ∞-norm is defined as kΛk∞ = max i 109 m X i=1 |λij | (A.6) i.e. “the maximum row sum”. The work of finding γmin by means of optimization theory is often rather tedious and therefore it is handy to first calculate the RGA and then use (A.2), (A.4) or (A.5). 110 Bibliography Alex, J., J.F. Beteau, C. Hellings, U. Jeppson, S. Marsili-Libelli, M.N. Pons, H. Spanjers and H. Vanhooren (1999). Benchmark for evaluating control strategies in wastewater treatment plants. In: Proceedings of the European Control Conference, ECC99’. Karlsruhe, Germany. Antoulas, A.C. (2001). Frequency domain representation and singular value decomposition. UNESCO EOLSS (Encyclopedia for the Life Sciences), Contribution 6.43.13.4. Ayesa, E., B. Goya, A. Larrea, L. Larrea and A. Rivas (1998). Selection of operational strategies in activated sludge processes based on optimization algorithms. Water Science and Technology 37(12), 327–344. Birk, W. and A. Medvedev (2003). A note on gramain-based interaction measures. In: Proceedings of European Control Conference, Cambridge, UK, September 2003. Bristol, E. H. (1966). On a new measure of interaction for multivariable process control. IEEE Trans. Automatic Control AC-11, 133–134. Brouwer, H., M. Bloemen, B. Klapwijk and H. Spanjers (1998). Feedforward control of nitrification by manipulating the aerobic volume in activated sludge plants. Water Science and Technology 38(3), 245–254. Cadet, C., J. F. Beteau and S. Carlos Hernandez (2004). Multicriteria control strategy for cost/quality compromise in wastewater treatment plants. Control Engineering Practice 12(3), 335–347. Carlsson, B. and A. Rehnström (2002). Control of an activated sludge process with nitrogen removal - a benchmark study. Water Science and Technology 45(4–5), 135–142. Carstensen, J. (1994). Identification of Wastewater Processes. PhD thesis. Institute of Mathematical Modelling, Technical University of Denmark. Chellaboina, V., W.M. Haddard, D.S. Bernstein and D.A. Wilson (1999). Induced convolution operator norms for discrete-time linear systems. In: 111 Proceedings of the 38:th Conference on Decision & Control. Phoenix, Arizona, USA. Conley, A. and M. E. Salgado (2000). Gramian based interaction measure. In: Proceedings of the 39th IEEE Conference on Decision and Control. Sydney, Australia. pp. 5020–5022. Copp, J. B., Ed.) (2002). EUR 19993 – COST Action 624 – The COST simulation benchmark – Description and simulator manual. European Communities. Luxembourg. Dharmasanam, S., R. Scott Erwin and D. Bernstein (1997). Synthesis of optimal generalized LQG and Hankel-norm controllers. In: Proceedings of the 1997 American Control Conference. Ekman, M. (2005). Modeling and Control of Bilinear Systems – Applications to the Activated Sludge Process. PhD thesis. Uppsala University. Uppsala, Sweden. Ekman, M. and B. Carlsson (2005a). Control of the Aeration Volume in an Activated Sludge Process using Supervisory Control Strategies. HIPCON Report number HIP05-54-v1-r. Uppsala University. Ekman, M. and B. Carlsson (2005b). Control of the Aeration Volume in an Activated Sludge Process using Supervisory Control Strategies. HIPCON Report number HIP05-49-v1-r. Uppsala University. Farsangi, M.M., Y.H. Song and Kwang Y. Lee (2004). Choice of facts device control inputs for damping interarea oscillations. IEEE Transactions on Power Systems 19(2), 1135–1143. Galarza, A., E. Ayesa, M. T. Linaza, A. Rivas and A. Salterain (2001). Application of mathematical tools to improve the design and operation of activated sludge plants. Case study: The new WWTP of Galindo-Bilbao, part ii: Operational strategies and automatic controllers. Water Science and Technology 43(7), 167–174. Gerksic, S., D. Vrecko and N. Hvala (2006). Improving oxygen concentration control in activated sludge process with estimation of respiration and scheduling control. Water Science & Technology 53(4-5), 283–291. Glad, T. and L. Ljung (1989). Reglerteknik. Grundläggande teori. Studentlitteratur. In Swedish. Glover, K (1984). All optimal Hankel-norm approximations of linear multivariable systems and their l∞ error bounds. Int. J. Control 39, 1115–1193. 112 Goodwin, G., M. Salgado and E. Silva (2005). Time-domain performance limitations arisng from decentralized architectures and their relationship to the RGA. International Journal of Control 78(13), 1045–1062. Grosdidier, P. and M. Morari (1987). The µ interaction measure. Ind. Eng. Chem. Res. 26(6), 1193–1202. Grosdidier, P., M. Morari and B. R. Holt (1985). Closed-loop properties from steady-state gain information. Ind. Eng. Chem. Fundam. 24, 221–235. Häggblom, K. E. (1997). Control structure analysis by partial relative gains. In: Proceedings of the 36th Conference on Decision and Control. San Diego, California, USA. pp. 2623–2624. Halvarsson, B. (2003). Applications of coupling analysis on bioreactor models. Master’s thesis. Uppsala University. Uppsala, Sweden. Halvarsson, B., P. Samuelsson and B. Carlsson (2005). Application of coupling analysis on bioreactor models. In: Proceedings of the 16th IFAC World Congress. Prague, Czech Republic. Hammer, M. and M. Hammer Jr. (2008). Water and Wastewater Technology. 6 ed.. Pearson Prentice Hall. He, M.-J. and W.-J. Cai (2004). New criterion for control-loop configuration of multivariable processes. Ind. Eng. Chem. Res. 43, 7057–7064. He, M.-J., W.-J. Cai and B.-F. Wu (2006). Control structure selection based on relative interaction decomposition. International Journal of Control 79(10), 1285–1296. Henze, M., C. P. L. Grady Jr., W. Gujer, G. v. R. Marais and T. Matsuo (1987). Activated sludge model no. 1. Scientific and Technical Report No. 1. IAWPRC, London. Henze, M., P. Harremoës, J. la Cour Jansen and E. Arvin (1995). Wastewater treatment, biological and chemical processes. Springer-Verlag, Berlin Heidelberg. Horn, R.A. and C.R. Johnson (1985). Matrix analysis. Cambridge University Press. Hovd, M. and S. Skogestad (1992). Simple frequency-dependent tools for control system analysis, structure selection and design. Automatica 28(5), 989–996. Ingildsen, P. (2002). Realising Full-Scale Control in Wastewater Treatment Systems Using In Situ Nutrient Sensors. PhD thesis. Lund Institute of 113 Technology. Dept. of Industrial Electrical Engineering and Automation, Lund, Sweden. Ingildsen, P., G. Olsson and Z. Yuan (2002). A hedging point strategy – balancing effluent quality, economy and robustness in the control of wastewater treatment plants. Water Science and Technology 45(4–5), 317–324. IWA (November 19, 2007). IWA Task Group on Benchmarking of Control Strategies for WWTPs: BSM1. http://www.benchmarkwwtp.org/. Jeppsson, U. and M.-N. Pons (2004). The COST benchmark simulation model–Current status and future trends. Control Engineering Practice 12(3), 299–304. Ed. Jeppsson, U., J. Alex, M. N. Pons, H. Spanjers and P. A. Vanrolleghem (2002). Status and future trends of ICA in wastewater treatment - a european perspective. Water Science and Technology 45(4–5), 485–494. Johansson, K.H. (2000). The quadruple-tank process: A multivariable laboratory process with an adjustable zero. IEEE Transactions on Control Systems Technology 8(3), 456–465. Kalman, R. E., Y. C. Ho and K. S. Narendra (1963). Controllability of linear dynamical systems. Contributions to Differential Equations 1, No. 2, 189–213. Kinnaert, M. (1995). Interaction measures and pairing of controlled and manipulated variables for multiple-input multiple-output systems: A survey. Journal A 36(4), 15–23. Kommunförbundet (1988). Introduktion till avloppstekniken. Sweden. In Swedish. Kreindler, E. and P.E. Sarachik (1964). On the concepts of controllability and observability of linear systems. IEEE Transactions on Automatic Control Volume: 9, Issue: 2, 129– 136. (Correction: Vol. 10, No. 1, p. 118, 1965). Lindberg, C-F. (1997). Control and Estimation Strategies Applied to the Activated Sludge Process. PhD thesis. Uppsala University. Dept. of Systems and Control, Uppsala, Sweden. Lindberg, C-F. and B. Carlsson (1996). Estimation of the respiration rate and oxygen transfer function utilizing a slow DO sensor. Water Science and Technology 33(1), 325–333. Lu, W. and G. Balas (1998). A comparison between hankel norms and induced system norms. IEEE Transcations on Automatic Control 43(11), 1658–1662. 114 Mc Avoy, T., Y. Arkun, R. Chen, D. Robinson and P. D. Schnelle (2003). A new approach to defining a dynamic relative gain. Control Engineering Practice 11(8), 907–914. Meyer, U. and H. J. Pöpel (2003). Fuzzy-control for improved nitrogen removal and energy saving in WWT-plants with predenitrification. Water Science and Technology 47(11), 69–76. Niederlinski, A. (1971). A heuristic approach to the design of linear multivariable interacting control systems. Automatica 7(6), 691–701. Olsson, G. (1993). Advancing ICA technology by eliminating the constraints. Water Science and Technology 28(11–12), 1–7. Olsson, G. and B. Newell (1999). Wastewater Treatment Systems. IWA publishing. London, UK. Olsson, G. and U. Jeppsson (1994). Establishing cause-effect relationships in activated sludge plants – what can be controlled. In: Proceedings of workshop modelling, monitoring and control of wastewater treatment plants. pp. 2057–2070. Qin, S. Joe and T. A. Badgwell (2003). A survey of industrial model predictive control technology. Control Engineering Practice 11(7), 733–764. Rotea, M. (1993). The generalized h2 control problem. Automatica 29(2), 373–385. Salgado, M. and D. Oyarzún (2005). MIMO interactions in sampled data systems. In: Proceedings of the 16th IFAC World Congress. Prague, Czech Republic. Salgado, M. E. and A. Conley (2004). MIMO interaction measure and controller structure selection. Int. J. Control 77(4), 367–383. Samuelsson, P. (2005). Control of Nitrogen Removal in Activated Sludge Processes. PhD thesis. Uppsala University. Uppsala, Sweden. Samuelsson, P. and B. Carlsson (2001). Feedforward control of the external carbon flow rate in an activated sludge process. Water Science and Technology 43(1), 115 – 122. Samuelsson, P. and B. Carlsson (2002). Control of the aeration volume in an activated sludge process for nutrient removal. Water Science and Technology 45(4–5), 45–52. Samuelsson, P., B. Halvarsson and B. Carlsson (2004). Analysis of the inputoutput couplings in a wastewater treatment plant model. Technical Report 115 2004-014. Div. of Systems and Control, Dept. of Information Technology, Uppsala University. Uppsala, Sweden. Samuelsson, P., B. Halvarsson and B. Carlsson (2005a). Cost–efficient operation of a denitrifying activated sludge process – an initial study. Technical Report 2005-010. Div. of Systems and Control, Dept. of Information Technology, Uppsala University. Uppsala, Sweden. Samuelsson, P., B. Halvarsson and B. Carlsson (2005b). Feedforward Aeration Volume Control in An Activated Sludge Process. HIPCON Report number HIP05-51-v1-r. Uppsala University. Uppsala, Sweden. Samuelsson, P., B. Halvarsson and B. Carlsson (2005c). Interaction analysis and control structure selection in a wastewater treatment plant model. IEEE Transactions on Control Systems Technology 13(6), 955–964. Skogestad, S. and I. Postlethwaite (1996). Multivariable Feedback Control. John Wiley & Sons. Chichester, UK. Svardal, K., S. Lindtner and S. Winkler (2003). Optimum aerobic volume control based on continuous in-line oxygen uptake monitoring. Water Science and Technology 47(11), 305–312. Takács, I., G. G. Patry and D. Nolasco (1991). A dynamic model of the clarification-thickening process. Wat. Res. 25(10), 1263–1271. Vanrolleghem, P. A. and S. Gillot (2002). Robustness and economic measures as control benchmark performance criteria. Water Science and Technology 45(4–5), 117–126. Vanrolleghem, P., H. Spanjers, B. Petersen, P. Ginestet and I. Takács (1999). Estimating (combinations of) activated sludge model no. 1 parameters and components by respirometry. Water Science and Technology 39(1), 195– 214. Vrecko, D., N. Hvala, Stare, Burica, Strazar, Levstek, Cerar and Podbevsek (2006). Improvement of ammonia removal in activated sludge process with feedforward-feedback aeration controllers. Water Science & Technology 53(4-5), 125–132. Weber, B. (1994). Rational Transmitting Boundaries for Time-Domain Analysis of Dam-Reservoir Interaction. PhD thesis. Swiss Federal Institute of Technology Zürich. Wilson, D. (1989). Convolution and hankel operator norms for linear systems. IEEE Transactions on Automatic Control 34, 94–97. 116 Wittenmark, B. and M. E. Salgado (2002). Hankel-norm based interaction measure for input-output pairing. In: Proc. of the 2002 IFAC World Congress. Barcelona, Spain. Wittenmark, B., K. J. Åström and S. B. Jörgensen (1995). Process Control. KFS i Lund AB. Dep of Automatic Control, Lund University, Lund, Sweden. Xiong, Q., W.-J. Cai and M.-J. He (2005). A practical loop pairing criterion for multivariable processes. Journal of Process Control 15, 741–747. Yuan, Z., A. Oehmen and P. Ingildsen (2002). Control of nitrate recirculation flow in predenitrification systems. Water Science and Technology 45(4– 5), 29–36. Yuan, Z. and J. Keller (2003). Integrated control of nitrate recirculation and external carbon addition in a predenitrification system. Water Science and Technology 48(11), 345–354. Yuan, Z. and J. Keller (2004). Achieving optimal conditions for nitrogen removal using on-line sensors and control. In: Proceedings of the 2nd IWA Leading-Edge Conference on Water and Wastewater Treatment Technologies. Prague, Czech Republic. Yuan, Z., H. Bogaert, C. Rosen and W. Verstraete (2001). Sludge blanket height control in secondary clarifiers. In: Proceedings of the 1st IWA Conference on Instrumentation, Control and Automation. Malmö, Sweden. pp. 81–88. Yuan, Z., H. Bogaert, P. Vanrolleghem, C. Thoeye, G. Vansteenkiste and W. Verstraete (1997). Control of external carbon addition to predenitrifying systems. Journal of Environmental Engineering 123(11), 1080–1086. Zhou, K., with J.C. Doyle and K. Glover (1996). Robust and Optimal Control. Prentice Hall. Zuo, L. and S.A. Nayfeh (2003). Structured h2 optimization of vehicle suspensions based on multi-wheel models. Vehicle System Dynamics 40, 351– 371. 117 Recent licentiate theses from the Department of Information Technology 2007-005 Mahen Jayawardena: Parallel Algorithms and Implementations for Genetic Analysis of Quantitative Traits 2007-004 Olof Rensfelt: Tools and Methods for Evaluation of Overlay Networks 2007-003 Thabotharan Kathiravelu: Towards Content Distribution in Opportunistic Networks 2007-002 Jonas Boustedt: Students Working with a Large Software System: Experiences and Understandings 2007-001 Manivasakan Sabesan: Querying Mediated Web Services 2006-012 Stefan Blomkvist: User-Centred Design and Agile Development of IT Systems 2006-011 Åsa Cajander: Values and Perspectives Affecting IT Systems Development and Usability Work 2006-010 Henrik Johansson: Performance Characterization and Evaluation of Parallel PDE Solvers 2006-009 Eddie Wadbro: Topology Optimization for Acoustic Wave Propagation Problems 2006-008 Agnes Rensfelt: Nonparametric Identification of Viscoelastic Materials 2006-007 Stefan Engblom: Numerical Methods for the Chemical Master Equation 2006-006 Anna Eckerdal: Novice Students’ Learning of Object-Oriented Programming 2006-005 Arvid Kauppi: A Human-Computer Interaction Approach to Train Traffic Control Department of Information Technology, Uppsala University, Sweden

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

### Related manuals

Download PDF

advertisement