Model-based optimisation of Wastewater Treatment Plants design

Available online at www.sciencedirect.com This paper presents the mathematical basis and some illustrative examples of a model-based decision-making method for the automatic cal- culation of optimum design parameters in modern Wastewater Treatment Plants (WWTP). The starting point of the proposed methodology is the mathematical modelling of the main processes inside a plant’s units. The procedure for the automatic calculation of the design parameters is then based on expressing the optimum WWTP design problem as a Mathematical Programming (Optimisation) Problem that can be solved using a non-linear optimisation algorithm (GRG2). The paper shows how the proposed methodology is able to achieve optimum WWTP design using either a steady-state or dynamic mathematical model of the plant and a set of constraints associated with the permitted operational ranges and the required water quality in the effluent. As an illustrative example to show the usefulness of the proposed methodology, the optimum design of the Step-Feed process for nitrogen removal (Alpha) has been analysed by considering two different problems: the optimum plant dimensions, es- timated at critical temperature for effluent requirements (Problem 1), and the optimum selection of facultative volumes, fractions of the influent flow-rate and the values of oxygen set-points for long-term plant operation (Problem 2). The proposed decision-making method is intended to facilitate the task of the engineers involved in the design of new WWTP, especially when the complexity of the plant requires a systematic pro- cedure for the selection of the main design parameters. � 2007 Elsevier Ltd. All rights reserved. Keywords: Wastewater Treatment Plants; Mathematical modelling; Optimum design 1. Introduction Over recent decades, many advanced processes have been de- Software availability Name of software: DAISY 3.0. Developer: Environmental Engineering Department of CEIT (http://www.ceit.es/environment/index.htm). Contact person: Dr. Ion Irizar, Tel.: þ34 943 21 28 00; fax: þ34 943 21 30 76. E-mail: [email protected]. Program language: C/Cþþ and Visual Basic for Applications�. Hardware required: Personal Computer. Intel Pentium�. Software required: Microsoft Excel 2000� and Premium Solver Platform�. Availability: The specific implementation called Daisy 3.0 is owned by the Spanish engineering company CADA- GUA S.A. Therefore, it is only for internal use and not available as commercial application. Other re- lated web-based implementations of the proposed methodology are currently in development for a broader access and use. Model-based optimisation of Was A. Rivas a,*, I. Ir a TECNUN, University of Navarra, Manuel de b CEIT and TECNUN, University of Navarra, Man Received 26 September 2005; received in revis Available online Abstract Environmental Modelling & Softw * Corresponding author. Tel.: þ34 9432 19877; fax: þ34 9433 11442. E-mail address: [email protected] (A. Rivas). 1364-8152/$ - see front matter � 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.envsoft.2007.06.009 tewater Treatment Plants design izar b, E. Ayesa b Lardiza´bal 13, 20018 San Sebastia´n, Spain uel de Lardiza´bal 15, 20018 San Sebastia´n, Spain ed form 11 June 2007; accepted 15 June 2007 13 August 2007 are 23 (2008) 435e450 www.elsevier.com/locate/envsoft veloped to improve the efficiency of the removal of organic mat- ter and nutrients by Wastewater Treatment Plants (WWTP). These new processes are usually based on complex biological treatment configurations that include combinations of aerobic, anoxic and anaerobic reactors, and internal recycling between costs, etc.) also increases the complexity of the problem, such that the selection of the most appropriate plant design becomes a very difficult task, even for experienced designers. A general framework for the formulation and analysis of an overall decision support index is discussed in Vanrolleghem et al. (1996). The use of WWTP mathematical modelling and simulation as a tool for the selection of appropriate design parameters has become increasingly popular since its introduction in the mid-1990s (Dupont and Sinkjaer, 1994; Suescun et al., 1994; Dudley and Chambers, 1995; Vanrolleghem and Jeppsson, 1995; Gabaldon et al., 1998; Gernaey et al., 2004). Several factors have contributed to this increasing popularity, such as: the publication of new mathematical models for the differ- ent unit-processes of plants (Henze et al., 2000; Batstone et al., 2002); the development of new computational platforms with efficient methods for the numerical analysis of models (Copp, 2002), and the progressive elaboration of systematic procedures for the experimental calibration of the most com- mon plant models, like the protocols proposed by STOWA (Hulsbeek et al., 2002), HSG (Langergraber et al., 2004), WERF (WERF, 2003) or the BIOMATH Group (Petersen et al., 2003). The conventional model-based selection of the design attention to those parameters, such as the maximum nitrifica- tion rate or the fraction of non-biodegradable solids in the in- fluent, that can significantly affect the final design. Therefore, if detailed statistical descriptions of the expected influent load, the environmental conditions, or the main model parameters were available, this information about problem uncertainty could be used to the optimum plant design for different quan- titative risk criteria. Analysing the influence of uncertainty is a central theme in engineering design and especially in the modelling of environ- mental systems, confirmation of which is reflected in the number of recent works devoted to dealing with the subject (Poch et al., 2004; Jolma and Norton, 2005; Martı´n et al., 2006). In fact, spe- cific approaches combining Monte Carlo trials and deterministic mathematical models have been proposed for the statistical analysis of uncertainty in model-based WWTP design and con- trol (Rousseau et al., 2001; Bixio et al., 2002; Benedetti et al., 2005; Garcı´a-Sanz et al., 2006). However, the limited informa- tion commonly available to the plant designer concerning the load variations expected in the influent of the proposed plants and, particularly, the biochemical activity of the new processes significantly limits the practical applicability of these methodol- ogies. For this reason, in spite of its enormous potential, model- based WWTP design using a statistical description for load, the different unit-processes. Significantly, traditional design rules used by engineers are frequently too limited in the case of modern configurations, because the number of degrees-of- freedom demanded is steadily increasing. The expectation of having to simultaneously satisfy a variety of objectives (such as effluent requirements, safety, investment costs, operational Vector of plant’s states e Instantaneous values. _e Time derivatives.be Steady-state values. e Time-averaged mean values. e0 Values at the beginning of the period. eD Values at the end of the period. d Vector of plant’s parameters selected by the designer. l Vector of plant’s parameters fixed externally. M RHS of the mathematical model of the plant. SRT Solids Retention Time (day). HRT Hydraulic Retention Time (hour). qIN Influent flow-rate (m 3/h). T Temperature (�C). h Norm of M. R Vector of constraints. 4 Objective function. fV Vector of volume fractions of biological reactors (% total volume). fV k Volume fraction of the k reactor (% total volume). fQ Vector of fractions of the influent flow-rate (% qIN). 436 A. Rivas et al. / Environmental Model parameters in a new WWTP is based on a sequential procedure that starts with the initial selection of the plant layout and model, including kinetic and stoichiometric parameters and influent wastewater characterisation. Wherever possible, the plant model is previously calibrated from an experimental campaign in a pilot-plant (Concha and Henze, 1996); other- wise, its model parameters and influent wastewater character- isation must be selected conservatively, paying special fQ k Fraction of the influent flow-rate to the k reactor (% total volume). Nitrate concentration in the k reactor (mgN/l)bSkNO Steady-state value. SkNO Time-averaged mean value. Ammonia concentration in the k reactor (mgN/l)bSkNH Steady-state value. SkNH Time-averaged mean value. Mixed-liquor suspended solids concentration in the k reactor (mg MLSS/l)bck Steady-state value. ck Time-averaged mean value. fFZ Volume fraction of a facultative zone (% total volume). ZF Number of facultative zones working under anoxic conditions (dimensionless). DOSP Vector of oxygen set-points in the aerobic biological reactors. ling & Software 23 (2008) 435e450 model parameters or effluent requirements is not currently a common procedure for Spanish engineering companies. The second step in the selection of the design para- meters is the definition of all the constraints of the design problem, including, amongst others, the effluent requirements, the volume of units physically available and the maximum cost. Finally, model predictions are used by the plant designers to repeatedly select the most appropriate combination of de- sign and operational parameters until the constraints of the problem are fulfilled. Normally, critical conditions at low tem- peratures and high loads are considered in order to establish the minimum volume required in the reactors and settlers. Other less restrictive conditions should also be explored in or- der to define the basic operational rules of the new plant. Sometimes, a minimum global cost, obtained by combining construction and operational cost, can also be investigated (Rivas, 2000; Rivas et al., 2001). This sequential procedure usually requires several simula- tions of the mathematical model and, also a comprehensive knowledge of the process needed to modify the design param- eters in order to determine the optimum design. As the num- ber of design parameters increases and more complex optimum design criteria have to be taken into account, this sequential design procedure can be very tedious and even unmanageable. This paper presents a model-based decision-making method that carries out the optimum WWTP design automati- cally. The presented methodology proposes to formulate the optimum WWTP design as a mathematical optimisation problem and to solve this problem by combining the conven- tional techniques of simulation in WWTP (steady-state calcu- lation and dynamic simulation) with non-linear optimisation methods. The practical implementation of the proposed methodology focused mainly on the selection of those design parameters of biological process (i.e. Hydraulic Retention Time, Solids Retention Time, volume fractions, influent flow rate fraction- ing, oxygen set points, etc.) which minimise the volume of the reactors and fulfil certain effluent requirements. This meth- odology has been implemented as a model-based decision sup- port tool by several engineering companies and has been successfully used over the last few years to optimise the design of twenty or so new plants in Spain. The optimum design of the Alpha process (Step-Feed bio- logical treatment for nitrogen removal) has been selected as a case study for the verification of the proposed methodology. The motivation to select the Alpha process has been twofold. First, it is a design with a difficult structure and a large num- ber of design parameters, which can therefore provide a good test of the methodology. Second, it is an interesting process from the technical point of view, and for this reason the results and conclusions derived from the optimum design of the Alpha process can be useful for plant designers and operators. To illustrate the methodology’s capabilities, some analyses of sensitivity show quantitatively the influence of either A. Rivas et al. / Environmental Mode effluent requirements or operational strategies in optimum plant design of the Alpha process. 2. Problem formulation 2.1. Mathematical modelling of WWTP The dynamic behaviour of a WWTP can be represented by the states of the plant’s units, such that the processes happening in those units are expressed as mathematical relationships be- tween the plant states (e) and parameters. Certain of these plant parameters (d) can be selected by the designer, and include: reactor volumes, sludge wastage or SRT, possible influent flow fractioning, and the degree of aeration in the biological reactors, amongst others. Other parameters (l), such as those of influent load, temperature or the values of the kinetic and stoichiometric model parameters are fixed externally. In order for a mathematical model to be useful in the design or operation of a WWTP, the complexity of the model must be diminished; therefore it is usual to consider the process units of the plant as being fully mixed. Under this assumption, the spatial variations of the plant’s hydraulics and the states inside the units due to diffusion and convection phenomena are not taken into account. This simplification allows the dynamic behaviour of the WWTP to be expressed by a system of ordi- nary differential equations: _e¼M½eðtÞ;d;l� ð1Þ A steady-state be is then the solution of the plant’s steady mathematical model which consists of the following system of non-linear equations: M �be;d;l�¼ 0 ð2Þ To simulate either the dynamic or steady behaviour of the WWTP, it is necessary to solve the mathematical models expressed by Eqs. (1) and (2). Different techniques can be applied to obtain an analytical solution from steady-state mathematical models in such a way that it is possible to express the steady states as explicit functions of parameters (Ekama et al., 1984). However, the complexity of the most common mathematical models of WWTP precludes these simplistic relationships, there being no choice but to simulate complex processes (i.e. to solve their mathematical models) using numerical techniques and computation. 2.2. Mathematical background 2.2.1. Dynamic simulation Standard methods described in the bibliography (Press et al., 1992) can be used to integrate Eq. (1) numerically. Its application to biological processes is shown by Billing and Dold (1988a) (Brenner et al., 2005). The processes taken into account in recent mathematical models of WWTP are usually so complex that better numerical efficiency can be obtained by applying higher order integration algorithms or, when needed, more sophisticated implicit or stiff integration methods (Press et al., 1992; Seppelt and Richter, 2005). Although WWTP design based on a steady-state model is the conventional procedure (German ATV-DVWK-A 131E, 437lling & Software 23 (2008) 435e450 2000), the dynamic behaviour of a WWTP can differ significantly from that obtained considering steady conditions because of the non-linearity of processes modelled in Eq. (1). As will be shown later in this paper, this difference leads to designs based on dynamic models being more conservative than those obtained using a steady-state model. In order to take this difference into account, some wastewater engineers apply appropriate safety factors to the most sensitive parame- ters of the model (thus achieving a steady-state design) or, alternatively, check the dynamic response of the design under certain standard profiles in the influent load or temperature. In the latter procedure, the mean state value of the plant model, ðeÞ, obtained for cyclic changes in the wastewater characteris- tics of the plant’s influent is frequently used to verify the plant’s behaviour under expected daily or seasonal variations (Suescun et al., 1994). In some cases, as mentioned earlier, the mean value e can differ significantly from the steady-state model solution. Of course, actual variations in influent characteristics of the plant have to be considered when available (Ayesa et al., 2001; Butler and Schu¨tze, 2005; Belanche et al., 1999). 2.2.2. Steady-state simulation As in the case of dynamic simulation, the bibliography contains numerous methods to solve the non-linear system of equations that express the steady-state condition of the WWTP models (Billing and Dold, 1988b). Newton-Raphson or Quasi-Newton methods have been used successfully in several WWTP simulators (Suescun et al., 1994). Although the non-linear optimisation algorithm mentioned later in this paper is used specifically as a tool in the proposed model- based optimisation, it has also been used to solve Eq. (2). Steady-state simulation of a WWTP can be expressed as a Mathematical Programming Problem (Rivas, 2000). Eq. (2) can then be derived from the condition necessary to mini- mise function h, defined as: h¼ 1 2 M$M ð3Þ Thus, the solution of Eq. (2) is also a solution of the follow- ing Mathematical Programming Problem: min e^ � h;M �be�¼ 0� ð4Þ 2.3. Model-based optimisation of WWTP design Mathematical modelling and simulation are useful tools that can aid engineers to optimise WWTP designs. Undoubt- edly, a design engineer can solve a simple design problem using a WWTP simulator. However, to obtain the optimum values (according to a selected criterion) of design parameters involved in the design of configurations with more complex processes (such as the Alpha process), or the consideration of a larger number of design parameters and other complex optimum design criteria remains a difficult task, even after using a WWTP simulator. 438 A. Rivas et al. / Environmental Model The methodology presented in this paper proposes first to express mathematically the WWTP optimum design criterion. This criterion is composed by the objective function 4 and the constraints R� 0. The optimum criterion requires the objective function to be maximum or minimum and the con- straints fulfilled. Normally, the objective function is the total volume of the plant, but could equally be an economic objec- tive, such as the minimisation of the global cost (construction and exploitation) of the plant (Rivas et al., 2001; Doby et al., 2002). Some of the constraints are imposed by the designer, for example the effluent quality, and others are necessary for the design to be physically feasible. Both the Objective and the constraints are functions of the design parameters d, but are usually not simple and explicit functions; therefore, the numerical solution of the mathematical model of the plant, (1) or (2), is required. Once the optimum design criterion has been expressed mathematically, the problem of seeking the optimum values of design parameters is reduced to that of solving a Mathemat- ical Programming (Optimisation) Problem. Various optimisa- tion algorithms have been applied in WWTP design bibliography (Tyteca, 1985; Rivas and Ayesa, 1997; Ayesa et al., 1998; Rivas, 2000; Doby et al., 2002). In this paper an enhanced version of the Generalized Reduced Gradient algorithm (Powell, 1978; Abadie and Carpentier, 1969), called GRG2 (Lasdon et al., 1978, Lasdon and Waren, 1979) is pro- posed. Although the methodology is capable of being imple- mented using other optimisation algorithms, the selection of GRG2 is based on the fact that it fits the problem (i.e. in terms of its mathematical formulation and number of variables) has been applied successfully in several fields of engineering (Lansey and Mays, 1989; Gupta et al., 1993; Frangopoulos, 1994; Gupta and Gupta, 1999; Sakarya and Mays, 2000; Vijayaraghavan, 2003; Ge and Boufadel, 2006; Ostfeld, 2005; Rivas et al., 2006), and has already provided good results in the optimum design of several new WWTP (Rivas et al., 2001). Additionally, several robust commercial imple- mentations of the algorithm are available (Fylstra et al., 1998) for integration into the design tool. 2.3.1. Steady-state model-based optimisation The basis of the proposed methodology for optimum WWTP design using steady-state models has been previously presented (Rivas et al., 2001). Optimum WWTP design can be expressed as a Mathematical Programming Problem expressed as: min e^;d � 4 �be;d�;M�be;d�¼ 0;R�be;d�� 0� ð5Þ The steady-state condition of the mathematical model is also added as an additional constraint. 2.3.2. Dynamic model-based optimisation Differences between designs based on steady-state models and those based on dynamic models are mentioned in Section 2.2.1 and will be shown later in Sections 4 and 6. Because of these differences, the proposed methodology also covers optimum WWTP design based on dynamic models under ling & Software 23 (2008) 435e450 cyclic influent load. Mathematically, the problem of together following the procedure shown in Fig. 1. that in which the simulator possesses several public functions 2.4. Methodology implementation The methodology described in the previous section has been implemented taking into account the design procedures employed by the Spanish wastewater engineering companies which are using the tool. Moreover, the case study has been solved using this implementation. Microsoft Excel� was the selected platform because of its connectivity with dynamic model simulators and with other applications used by the engineering companies for the dimensioning of the plants. Additionally, Microsoft Excel was chosen because it is capable of incorporating Premium Solver Platform�, which possesses an enhanced and robust implementation of the GRG2 algorithm (Nenov and Fylstra, 2003). Of course, the proposed methodology could be implemented in other platforms or WWTP simulation frameworks. Specifically, the implementation requires that the function- alities of the dynamic simulator can be accessed from a client application written in Microsoft Excel using Visual Basic for Applications�. This client application implements an interface between Microsoft Excel and the dynamic simulator, in such Fig. 1. Algorithmic procedure in the dyn which can be called by the client application. Another more sophisticated way is that in which the simulator has imple- mented interfaces COM (Rivas, 2000; Ayesa et al., 2001; WEST) for its functionalities, which are represented by objects. In such a case, the client application controls the sim- ulator by means of these objects. 3. Case study: the Alpha process 3.1. Description of the Step-Feed process for nitrogen removal In order to demonstrate the real capabilities of solving optimum WWTP design using a methodology based on mathe- matical modelling, the proposed approach is specifically applied to the activated sludge Step-Feed process (Alpha pro- cess). Nitrogen removal is achieved in this process by using two or three stages of denitrificationenitrification serial units (Fig. 2). Such a disposition does not require utilisation of an internal recycle of mixed-liquor, as is used in conventional continuous flow denitrificationenitrification processes (DeN process). In addition, the influent flow distribution to the optimising WWTP design under these conditions can be expressed in a similar way to Eq. (5), as follows: min e0;d � 4ðe;dÞ;e0 � eD ¼ 0;Rðe;dÞ � 0� ð6Þ where the constraint e0 � eD ¼ 0 replaces the steady-state model, and is added to achieve time-periodicity of the states of the plant. By implicitly adding the dynamic mathematical model of the plant, this constraint, in fact, completes the def- inition of problem (6). The values of the WWTP states at the beginning and end of the simulation period are, respectively, e0 and eD. A numerical method should be used to obtain eD by integrating the dynamic mathematical model of the WWTP, (1). To solve problem (6) a numerical integrator of (1) and the optimisation algorithm were connected to work a way that the client application undertakes the following task; reading data (i.e. plan’s configuration) from several specific cells of Excel’s worksheets, communicating these data to the simulator, calling the simulator’s integrator and filling in the mentioned cells with the results of the simulation. Once the communication between Microsoft Excel and the dynamic simulator is done, the optimisation problem is defined in the Premium Solver Platform which considers as the objective function and the constraints those cells of the worksheet where the data of the simulator are located. Run- ning the Solver, the two codes work together following the scheme shown in Fig. 1 until an optimum is obtained. Two ways have been used in which the simulator exposes its functionalities to be accessed (API Application Program- ming Interface) from Microsoft Excel. The simplest one is 439A. Rivas et al. / Environmental Modelling & Software 23 (2008) 435e450 amic model-based design optimisation. this work the optimum design of a WWTP based on the Alpha process has been obtained using the proposed methodology described earlier. The mathematical model selected to describe the biological process has been the ASM1 model proposed by IWA (Henze et al., 2000). A simple-model approach, based on the instantaneous set- tling of solids and a stationary mass balance, has been selected for the secondary settler. Associated with this approach, in all cases the mixed-liquor total suspended solids concentration (MLSS) in the last aerobic reactor has been considered to have a value of 3500 mg/l. Fixing the MLSS ensures that mini- also been analysed using, in this case, a dynamic model (Case B). The same average influent load of organic matter and nitrogen has been applied in Case A and Case B to enable a comparison of results. Typical ratios found in urban wastewa- ter have been chosen to establish the ASM1 state values corresponding to the influent wastewater concentrations, and the default values proposed by the model for the kinetic and stoichiometric coefficients have been used (Henze et al., 2000). 3.2.1. Problem 1 e Case A: steady-state design optimisation of the Alpha process Minimum volume has been adopted as the optimum design anoxic reactors gives rise to a suspended solids’ gradient through the reactors; as a consequence, the total mass of solids and the Solids Retention Time (SRT) of the process can be increased. The Alpha process has been found to offer relevant advan- tages for both new and existing plants. By simulation, Lesouef et al. (1992) found that the Step-Feed process with three stages might reduce the HRT needed for the conventional DeN process by about 20%. Regarding existing plants, Schlegel (1992) showed efficient nitrogen removal in five plants using the Step-Feed process with two stages. Kayser et al. (1992) found successful results in a large plant using a three-stage process, while Carrio et al. (1992) showed the advantages of the Step-Feed process over the conventional process through studies in four full-scale plants. However, the optimum design and operation of the process are difficult because of the complexity introduced as a result of the arrangement of the biological reactors and the influent flow distribution. Ayesa et al. (1998), using mathematical modelling and an optimisation algorithm, estimated the opti- mum dimensions and operating strategies of an Alpha process. Larrea et al. (2001) used mathematical modelling and simu- lation to establish criteria to select the optimum design parameters and rules for efficient operational strategies of the Step-Feed process. 3.2. Analysis of the optimum design of the Alpha process Optimum design of the Alpha process is a complex task because there are a large number of design parameters whose influence in the process behaviour is difficult to quantify. In Fig. 2. Alpha p 440 A. Rivas et al. / Environmental Model mising the volume of the biological reactors, the sum of the latter plus the settler’s volume will be minimised (Ekama et al., 1997). Using this approach does not mean that the processes in the settler (i.e. clarification, thickening, biodegradation and sludge storage) do not play an important role in the WWTP perfor- mance, nor that they can be neglected. However, the thicken- ing and clarification functions of the settler are governed, once the sludge mass-flow is fixed, by the characteristics of the sludge and the flow. The latter is influenced to a great extent by the geometry and internal features of the settler. Due to the fact that these factors are complex to model and experi- ment, as the authors know well, a mathematical model does not exist which includes a full and precise description of all phenomena occurring in the secondary settler. Nonetheless, several dynamic models of the settler have been proposed which include the settleability characteristics of the sludge as model parameters and a simplified (1D) hydraulics (Vazquez-Sa´nchez, 1996). However, a simpler model has been preferred since the problems analysed in the case study have been defined in such a way that employing these dynamic models does not change significantly the results. Two problems have been analysed: the optimisation of plant dimensions, when the plant works at the critical temperature (13 �C in Spain) for effluent requirements (Problem 1); and the optimisation of plant’s long-term operation, when the tem- perature varies from 13 �C to 22 �C (Problem 2). For Problem 1, both the effect of influent wastewater characteristics and the effect of effluent requirements have been studied separately us- ing a steady-state model (Case A). Additionally, in order to quantify variations in the optimum design solution when con- sidering hourly variations of flow and composition in influent wastewater (Fig. 3), the effect of effluent requirements has rocess layout. ling & Software 23 (2008) 435e450 criterion in this problem. Mathematically the objective func- tion 4 to be minimised is the volume of biological reactors s ll reactors is controlled to 2.0 mg OD/l (as commonly assumed for design purposes) and no aeration is considered in the three anoxic reactors. The solution of problem (5) must meet several constraints, the most important of which are related to the effluent water quality requirements. European legislation (European Direc- tive 91/271/CEE) imposes limits on the values of effluent total nitrogen concentration of flow-proportional or time-based 24-h composed samples taken during the year. In this study, following common design rules and the values proposed by the directive, maximum values of the nitrate ðbSN3NOÞ and ammonia ðbSN3NHÞ concentrations in WWTP effluent, obtained from the steady-state model, are imposed as constraints. Mathematically: bSN3NH �bSmaxNH � 0 ð10Þ bSN3NO �bSmaxNO � 0 ð11Þ To limit the solids flux to the settler, the value of mixed- liquor suspended solids concentration bcN3 in the biological reactor prior to the settler has been constrained so that it f iQ � 0 ð16Þ 3.2.2. Problem 1 e Case B: dynamic design optimisation of the Alpha process The objective function and the design parameters are the same as when the steady-state model is used, except that SRT is replaced by the sludge waste flow fraction, fQw, thus: d¼ �HRT fQw fV fQ � ð17Þ In this problem constraints are functions of design parame- ters and the time-averaged values of the state e. Therefore, in this formulation effluent quality constraints become: SN3NH � SmaxNH � 0 ð18Þ SN3NO � SN3NO � 0 ð19Þ cN3 � cmax � 0 ð20Þ where the maximum values are those adopted when the or, equivalently, the Hydraulic Retention Time (HRT). The design parameters, whose optimum values are obtained as solutions of problem (5), are the HRT, the Solids Retention Time (SRT), the six volume fractions of biological reactors: fV ¼ � f D1V f N1 V f D2 V f N2 V f D3 V f N3 V � ð7Þ and the three fractions of the influent flow: fQ ¼ n f D1Q f D2 Q f D3 Q o ð8Þ The design parameters to be obtained in the problem (5) are: d¼ �HRT SRT fV fQ � ð9Þ The dissolved oxygen concentration in the three aerobic 0% 20% 40% 60% 80% 100% 120% 140% 160% 180% 0 4 8 ( % M e a n V a l u e ) SNH q DQO Fig. 3. The variation of influent hourly profiles with re A. Rivas et al. / Environmental Mode is less than the fixed maximum value (3500 mg/l) suggested by the solids flux limitation to the secondary settlers. Mathematically: bcN3 � bcmax � 0 ð12Þ In order for the design to be physically feasible, the reactor volumes and the three fractions of influent flow have to fulfil the following conditions:X i f iV � 1 ¼ 0 ð13Þ f minV � f iV � 0 ð14Þ X i f iQ � 1 ¼ 0 ð15Þ 12 16 20 24 Hour pect to the steady mean values. Problem 1 (Case B). 441ing & Software 23 (2008) 435e450 steady-state model is used (Table 1). Constraints to ensure the feasibility of the design are the same as those when the steady-state model is used, Eqs. (13)e(16). 3.2.3. Problem 2. Steady-state optimisation of long-term plant operation of the Alpha process In this problem the optimum guidelines for the long-term operation of the plant have been obtained when the tempera- ture varies from 13 �C to 22 �C. No other perturbations, like changes in load or hydraulics, have been considered. The value of HRT adopted in this problem is 9.3 h, obtained as the optimum solution in the design problem when the plant works at 13 �C. Specifically, it is the optimum solution, impos- ing 8.0 mg N/l and 1.0 mg N/l of nitrate and ammonia concen- tration as effluent requirements, respectively. At each working temperature, T, an optimisation problem similar to (5) has been solved where the objective function selected to be mini- mised is the nitrate plus ammonia concentration of the efflu- ent. Mathematically: 4ðTÞ ¼ bSN3NOðTÞ þbSN3NHðTÞ ð21Þ Solids Retention Time (SRT) and the three fractions of in- fluent flow, fQ, have been considered as design parameters in this problem. Results from a previous work (Ayesa et al., 1998) show that the optimum values of f V D1, f V N1, f V D2 and f V N2 obtained at different temperatures present little change (less than 3% of total volume) with respect to those obtained at 13 �C. On the other hand, fV D3 and fV N3 show significant varia- tion. This result indicates that when the change of volume fractioning is adopted as an operational strategy, it is only nec- essary to arrange certain zones in the N3 reactor, called facul- tative zones, which can work under anoxic conditions by switching off their aeration. If 5% of total volume is adopted as the volume of each facultative zone fFZ the optimum num- ber of facultative zones which work under anoxic conditions ZF must be automatically estimated at each temperature. Math- ematically, the problem can be expressed as: f D3VðTÞ ¼ f D3VðT¼13 �CÞ þ ZFðTÞfFZ ð22Þ f N3VðTÞ ¼ f N3VðT¼13 �CÞ � ZFðTÞfFZ ð23Þ The remaining volume fractions are then fixed at their optimum value when the plant is working at 13 �C. In addition to, or instead of, facultative zones (depending on the case stud- ied), the values of oxygen set-points in the aerobic biological reactors, DOSP, have also been included in the design parameters.n o Table 1 Maximum values used in Problem 1 SmaxNH (mg N/l) 0.5, 1.0, 1.5 and 2.0 SmaxNO (mg N/l) 6.0, 7.0, 8.0, 9.0 and 10.0 f minV (% volume) 4.0 cmax (mg MLSS/l) 3500.0 442 A. Rivas et al. / Environmental Model DOSPðTÞ ¼ DON1SPðTÞ DON2SPðTÞ DON3SPðTÞ ð24Þ The constraints of the problem and maximum values allowed are those expressed in Eqs. (10)e(16). Additionally, when the dissolved oxygen concentration set-points are included as problem variables, their values are limited to avoid unrealistically high or low values of dissolved oxygen concen- tration in the aerobic reactors. DOminSPðTÞ � DOiSPðTÞ � DOmaxSPðTÞ i ¼ N1; N2; N3 ð25Þ Another interesting alternative would have been to solve Problem 2 minimising the total operational cost of the plant (i.e. associated to sludge treatment and disposal and aeration). The biggest difficulty of this alternative is to possess realistic and reliable cost functions to calculate the operational costs. The operational costs have huge variations among WWTPs, even those working with the same biological process, and the results of the optimisation would be very much dependent on the cost functions employed. Moreover, when engineering companies are asked about the operational cost, it is difficult to find this data sorted and rationalised so as to be employed in a model-based design framework. Nevertheless, the pro- posed methodology has been utilised in the case of DeN and ReDeN processes to optimise their long-term operation minimising the total operational cost (Rivas, 2000). In this case, data about operational cost were available from an engi- neering company, which allowed the author to build a cost function. 4. Results: optimum design of the Alpha process 4.1. Problem 1. Optimisation of plant dimensions under critical conditions (minimum temperature) As an illustrative example of the broad potential of the pro- posed methodology, the influences on the optimum design pa- rameters of both effluent requirements and influent characteristics have been analysed. These analyses were car- ried out using both a steady-state solution of the mathematical model (Case A) and the averaged values of the dynamic model under cyclic influent load (Case B). To study the influence of effluent requirements on the values of optimum design parameters, Problem 1 has been solved for different values of ammonia and nitrate concentra- tions in the effluent. From the results of this analysis, optimum design charts have been constructed in which the solutions of problems (5) and (6) are depicted as functions of the nitrate concentration in the effluent and parameterised with the value of the ammonia concentration in the effluent. Some of these optimum design charts are shown as examples in Figs. 4 and 5. To analyse the influence of influent wastewater characteris- tics on the optimum value of plant volume, the values of nitrate and ammonia concentrations in the effluent have been fixed to 7.0 mg N/l and 1.0 mg N/l, respectively, and the optimum values of design parameters have been obtained for ling & Software 23 (2008) 435e450 different values of CODT and CODT/TKNT in the influent wastewater. The results of this analysis are presented in Fig. 6 p Q F SNHN3=0.5 (mgN/l) 1.0 1.5 2.0 SNHN3=0.5 (mgN/l) 1.0 1.5 2.0 SNHN3=0.5 (mgN/l) 1.0 1.5 2.0 0.0% 10.0% 20.0% 30.0% 40.0% 50.0% 60.0% 70.0% 80.0% 90.0% 100.0% 5.75 6.25 6.75 7.25 7.75 8.25 8.75 9.25 9.75 10.25 S NO N3 (mgN/l) ( % q I N ) 15% 19% 23% 27% 31% 35% 39% 43% 47% 51% 55% ( % V ) fQD1 fQD1+fQD2 fVD Fig. 5. The influence of effluent requirements on the optimum values of the volume fractions in the biological reactor and on the influent flow fractioning. Problem and Fig. 7, where the optimum values of HRT are depicted as a function of CODT and CODT/TKNT in the influent. 4.2. Problem 2. Optimisation of long-term plant operation Three case studies have been considered in solving Prob- lem 2: Case A considers SRT, f and Z as design parameters, Fig. 4. The influence of effluent requirements on the o 1 (Case B). but not the dissolved oxygen concentration set-points, their values being fixed at 2 mg OD/l. Contrarily, in Case B, volume fractions have been maintained fixed and DOSP are included in the design parameters together with the SRT and fQ. Finally, Case C analyses the optimum solutions which have been obtained when both the number of facultative zones and the values of dissolved oxygen concentration set-points are included as design parameters. In order to facilitate the timum values of HRT and SRT. Problem 1 (Case A). SNHN3=0.5 (mgN/l) 1.0 2.0 1.5 SNHN3=0.5 (mgN/l) 1.0 1.5 2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 5.75 6.25 6.75 7.25 7.75 8.25 8.75 9.25 9.75 10.25 S NO N3 (mgN/l) S R T ( d a y s ) 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 H R T ( h o u r s ) SRT HRT 443A. Rivas et al. / Environmental Modelling & Software 23 (2008) 435e450 comparison of results, all the cases presented in the paper have used the steady-state solution of the plant model. However, the results obtained using averaged values of the dynamic model under cyclic conditions suggest similar operational strategies. Table 2 and Figs. 8e10 present the optimum values of design parameters and the nitrate and the ammonia concentrations in the effluent for the three cases analysed in this example. 5. Discussion Prior to discussing the case study results in detail, attention is drawn to the usefulness of the proposed methodology for obtaining the optimum WWTP design automatically. Specifi- cally, the analyses undertaken in solving Problem 1 and Prob- lem 2 provide a demonstration of the methodology’s validity and power. The results obtained have very positive implica- tions for plant designers seeking to select optimum designs Fig. 7. CODT/TKNT for the least volum easily, analyse trends or estimate security limits. It should be noted that the application of numeric values extracted from the charts is limited to those values of model parameters and wastewater characterisation used in the specific analysis. Clearly, it would be necessary to construct other charts if the values of these parameters were different, thereby providing the possibility of analysing their influence on the optimum de- sign values. 5.1. Problem 1. Optimisation of plant dimensions at minimum temperature 5.1.1. Influence of effluent requirements The minimum value of HRT logically increases as the ammonia and nitrate concentrations in the effluent decrease, exactly as is shown in Fig. 4. It can also be observed that the SRT of the plant exhibits a trend, with respect to effluent 0.0 10.0 20.0 30.0 40.0 50.0 60.0 5.5 6.5 7.5 8.5 9.5 10.5 11.5 COD T /TKN T H R T ( h o u r s ) 400 375 350 325 300 275 250 225 CODT=200 mgCOD/l Fig. 6. The influence of influent characteristics on the optimum value of HRT. Problem 1 (Case A). 444 A. Rivas et al. / Environmental Modelling & Software 23 (2008) 435e450 e optimum plant. Problem 1 (Case A). 13.5% 11.6% 24.0%4% 27.0% 20.0% 0% 10% 20% 30% 40% 5 13 14 15 16 17 18 34.7% 42.0% 31.1% 37.2% 33.3% 44.0% 32.5% 42.5% 36.9% 48.9% ( ( T ° C ) 13.5% 11.6% 24.0%4% 13.5% 11.6% 24.0%4% 13.5% 11.6% 24.0%4% 13.5% 11.6% 24.0%4% 13.5% 11.6% 24.0%4% Fig. 8. Optimum volume fractions and infl 0% 60% 70% 80% 90% 100% 23.4% 31.7% 22.7% 25.0% 15.1% %V) 17.0% 30.0% 17.0% 22.0% 30.0% 25.0% 22.0% 25.0% 27.0% 20.0% requirements, similar to that of the HRT. This occurs as a con- sequence of the constraint imposed over the mixed-liquor sus- pended solids concentration (12) and (20). Fig. 5 shows how influent flow fractioning is affected by quality requirements in the effluent. As the ammonia concentration requirement in the effluent becomes stricter (i.e. as lower concentrations are demanded) the influent flow to the last anoxic reactor, f Q D3, is reduced, even to the point at which it disappears. The reduction of f Q D3 flow is caused because a low value of ammo- nia concentration in the effluent cannot be guaranteed when influent wastewater with high ammonia concentration is going into the last anoxic reactor. One of the most remarkable results arising from Problem 1 is the significant difference found between the optimum values of HRT obtained with the steady-state model, and those obtained with the dynamic model. Fig. 11 shows the increase (expressed as a percentage) of the HRT required to fulfil the effluent requirements when the cyclic influent load is consid- ered instead of the steady-state condition. A similar difference is also shown by the SRT values. Although SRT and HRT show similar trends with respect to the quality requirements in the effluent in both cases, optimum values obtained with the dynamic model are much higher than the optimum solution Table 2 Results of Problem 2 T (�C) 13 14 15 16 17 18 19 20 21 22 SRT (days) Case A 10.9 11.2 11.3 11.5 11.7 11.8 11.9 12.0 12.1 12.2 Case B 10.9 11.7 11.4 11.8 11.5 11.9 11.4 11.8 12.0 12.2 Case C 11.0 11.3 11.2 11.0 11.3 11.5 11.3 11.5 11.7 11.9 SNO N3 þ SNHN3 (mg N/l) Case A 9.0 8.3 7.7 7.3 7.0 6.8 6.7 6.6 6.5 6.4 Case B 9.0 8.8 7.9 7.7 6.9 6.8 5.9 5.8 5.7 5.6 Case C 8.9 8.2 7.6 7.1 6.6 6.3 5.8 5.5 5.3 5.2 19 20 21 22 35.3% 48.1% 16.6% 41.8% 56.4% 1.8% 40.1% 55.8% 4.1% 39.8% 55.5% 4.6% 40.2% 55.7% 4.1% 13.5% 11.6% 24.0%4% 13.5% 11.6% 24.0%4% 13.5% 11.6% 24.0%4% 13.5% 11.6% 24.0%4% 32.0% 15.0% 32.0% 15.0% 32.0% 15.0% 32.0% 15.0% 445A. Rivas et al. / Environmental Modelling & Software 23 (2008) 435e450 uent flow fractioning. Problem 2 (Case A). ( T from the steady-state model for fixed values of quality require- ments in the effluent. Furthermore, this difference appears more marked for low ammonia concentration requirements and hardly changes with the nitrate concentration requirement. It is important to remark that this result would have lessened when the spatial gradient of concentrations was modelled by dividing each biological reactor into three or more serial reac- tors. An appropriate and realistic description of the hydraulic properties of the reactors is, therefore, crucial for guaranteeing the accuracy of the results when comparing the plant behav- iour under steady-state and cyclic influent load conditions. Nevertheless, these results show that a careful analysis of dynamic plant behaviour under cyclic influent load to obtain a realistic estimation of effluent quality and for appropriate plant dimensioning is relevant. With regard to optimum values of fQ and fV obtained using the steady-state model and the dynamic model, both numerical 5.1.2. Influence of influent characteristics Other interesting conclusions can be drawn from the anal- ysis of the influence of the influent wastewater characteristics on optimum design. If the influent CODT value is fixed, it can be observed that as CODT/TKNT increases the optimum values of HRT and SRT decrease, reflecting both the effect of the ammonia load in the influent and the relevancy of the carbon supply required for the denitrification. Fig. 6 shows a steep slope for low values of CODT/TKNT which becomes smoother for CODT/TKNT values approximately greater than 7.5. In Fig. 7 it can be seen that for a value of CODT an opti- mum value of CODT/TKNT exists that minimises the optimum volume of the plant. For a given value of ammonia concentra- tion in the influent, an optimum value of influent CODT exists, and any shortage of this optimum CODT would slow down denitrification and, consequently, require higher reactor volumes. Contrarily, because the total suspended solid concen- 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 13 14 15 16 17 34.8% 42.5% 22.7% 35.0% 41.3% 23.7% 36.3% 39.9% 23.8% 37.6% 38.5% 23.8% (%V) 2.0 (mgOD/l) 2.0 (mgOD/l) 2.0 (mgOD/l) 2.0 (mgOD/l) 2.0 (mgOD/l) 0.7 (mgOD/l) 0.6 (mgOD/l) 1.4 (mgOD/l) 1.3 (mgOD/l) 2.0 (mgOD/l) 1.3 (mgOD/l) 0.9 (mgOD/l) 2.0 (mgOD/l) 1.5 (mgOD/l) 1.5 (mgOD/l) Fig. 9. Optimum DOSP and influent flow fractioning. Problem 2 (Case B). 18 19 20 21 22 38.9% 37.9% 40.6% 39.0% 42.2% 40.1% 43.8% 41.1% 45.5% 42.2% 45.5% 42.2% 0.5 (mgOD/l) 0.5 (mgOD/l) 0.5 (mgOD/l) 1.2 (mgOD/l) 1.2 (mgOD/l) 1.1 (mgOD/l) 0.5 (mgOD/l)1.1 (mgOD/l) 0.5 (mgOD/l)1.1 (mgOD/l) ° C ) 446 A. Rivas et al. / Environmental Model values and trends with respect to effluent quality requirements are similar. 23.2% 20.5% 17.8% 15.1% 12.4% 12.4% 2.0 (mgOD/l) 2.0 (mgOD/l) 2.0 (mgOD/l) 2.0 (mgOD/l) 2.0 (mgOD/l) ling & Software 23 (2008) 435e450 tration in the reactors is limited, an excess of influent CODT produces an excess of heterotrophic biomass at the expense of the autotrophic biomass, with a resulting reduction of nitrification activity. As can be seen in Fig. 7, the optimum value of CODT/TKNT in this particular example is between 9.0 and 11.0. 5.2. Problem 2. Optimisation of long-term plant operation The results obtained in all the cases analysed show that SRT hardly changes with temperature and that its value at each temperature is defined by the suspended solids concentra- tion constraint. The results of Case A reveal a relationship between the number of facultative zones and the change of influent flow fractioning. As temperature increases and a new facultative zone switches to anoxic conditions, part of the influent flow entering reactor D3 is transferred to reactor D2 and, to a lesser where the volume fractions have not been changed, influent fractioning shows only slight variation. This result highlights the fact that, logically, the optimum influent flow fractioning is clearly conditioned by the operation of the facultative volumes. As temperature increases, the values of optimum set-points decrease when dissolved oxygen set-points are designated as design parameters (Case B and Case C ). This fact augments the simultaneous denitrification in reactors N1 and N2. The results show that a small improvement in the total nitrogen concentration in the effluent is obtained when opera- tional strategies based on facultative zones and dissolved oxy- gen set-points are combined, as in Case C (Fig. 10). 6. Conclusions The formulation of the optimum dimensioning of the bio- 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 13 14 15 16 17 34.6% 42.2% 23.2% 35.3% 41.0% 23.7% 36.7% 45.5% 17.8% 41.0% 50.8% 8.2% 39.0% 50.3% 9.9% (%V) 13.5% 1.7 (mgOD/l) 11.6% 17.0%4% 11.6% 17.0%4% 11.6%4% 11.6%4% 22.0% 27.0% 11.6%4% 27.0% 24.0% 2.0 (mgOD/l) 30.0% 2.0 (mgOD/l) 13.5% 1.5 (mgOD/l) 24.0% 1.2 (mgOD/l) 30.0% 2.0 (mgOD/l) 13.5% 1.5 (mgOD/l) 24.0% 1.7 (mgOD/l) 25.0% 2.0 (mgOD/l) 13.5% 2.0 (mgOD/l) 24.0% 2.0 (mgOD/l) 20.0% 2.0 (mgOD/l) 13.5% 1.4 (mgOD/l) 24.0% 1.7 (mgOD/l) 20.0% 2.0 (mgOD/l) ( T ° C Fig. 10. Optimum volume fractions and influent flow fractioning. Problem 2 (Case C ). 18 19 20 21 22 41.5% 50.1% 44.4% 55.6% 44.8% 55.2% 46.2% 53.8% 47.7% 52.3% 11.6%4% 11.6%4% 11.6%4% 11.6%4% 11.6%4% 13.5% 1.3 (mgOD/l) 24.0% 1.3 (mgOD/l) 13.5% 1.5 (mgOD/l) 24.0% 2.0 (mgOD/l) 13.5% 1.2 (mgOD/l) 24.0% 1.9 (mgOD/l) 13.5% 1.2 (mgOD/l) 24.0% 1.7 (mgOD/l) 13.5% 1.1 (mgOD/l) 24.0% 1.6 (mgOD/l) ) A. Rivas et al. / Environmental Mode extent, to D1 (Fig. 8). On the other hand, in Case B (Fig. 9), 8.4% 27.0% 32.0% 32.0% 32.0% 32.0% 20.0% 2.0 (mgOD/l) 15.0% 2.0 (mgOD/l) 15.0% 2.0 (mgOD/l) 15.0% 2.0 (mgOD/l) 15.0% 2.0 (mgOD/l) Facultative Zone Oxic Zone Anoxic Zone 447lling & Software 23 (2008) 435e450 logical treatment in WWTP as a mathematical optimisation number of independent parameters to be selected increases, as has been corroborated by numerous design engineers. In this respect, it is important to remark that the proposed meth- odology formulates the WWTP design problem consistent with the standard procedure commonly used by Spanish engineer- ing companies in the design and dimensioning of new plants. The graphical representation of the optimisation results for different ranges of effluent requirements, wastewater charac- teristics or water temperature provides very valuable quantita- tive information about the crucial factors affecting plant design. Similar analyses could easily be carried out for other factors like biomass activity, level of oxygenation, or solids flux to the secondary settler. A PC-computer implementation of the proposed methodology in Microsoft Excel would be able to calculate all the points required for a complete analysis of the optimum plant design in a few hours, with the resulting optimum design charts containing all the information about every plant configuration required by the designer. The Alpha process (Step-Feed for nitrogen removal) has been selected as a complex case study by which the potential of the proposed methodology has been demonstrated. The obtained conclusions complete and widen significantly those presented in previous works (Ayesa et al., 1998; Larrea et al., 2001). In Problem 1 (the optimisation of plant dimen- sions at minimum temperature) two analyses have been carried plant operation) show that it is possible to minimise the total nitrogen concentration in the effluent by several means: namely, by arranging facultative zones in the N3 reactor; by modifying the values of the dissolved oxygen concentration set-points in oxic reactors, or by combining both these strate- gies to obtain less total nitrogen concentration in the effluent. The optimisation of long-term operation allows engineers to take into account the plant’s operation in the design phase. This is very important with respect to design and to introduce real-time control strategies in the plant (Galarza et al., 2001; Ayesa et al., 2006). The proposed methodology is able to calculate optimum design parameters for steady-state or dynamic conditions of the plant model. The results show that a significant increase in the required plant volume can be obtained when dynamic conditions are considered in the optimisation problem, espe- cially if the effluent requirements are very strict regarding ammonia concentration and the reactors are completely mixed. Therefore, the appropriate selection of plant hydraulics (the number of reactors required for each zone to model the possi- ble plug-flow characteristics in a realistic way) is a crucial fac- tor in the design of model-based WWTP. Moreover, the methodology can be used to optimise those processes in which a steady-state is not possible and using a dynamic model is problem offers a very powerful methodology for the selection of the most appropriate design parameters under different ef- fluent requirements or wastewater characteristics. The advan- tages of the proposed decision-making method are especially relevant for the design of new plant configurations where the 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 5,75 6,25 6,75 7,25 7,75 S NO 1 0 0 · ( H R T B - H R T A ) / H R T A Fig. 11. Problem 1. Comparison of optimum HRT values obtained using the stead averaged values of the dynamic model under cyclic influent load (HRTB). 448 A. Rivas et al. / Environmental Model out in which the influence on the results of effluent require- ments and influent wastewater characteristics has been studied. Interesting results obtained from those analyses are the shift of influent flow-rate towards the anoxic reactors D1 and D2 for low ammonia concentration in the influent, and the existence of a particular value of CODT/TKNT in the influ- ent for which the optimum value of HRT is minimum. The results of Problem 2 (the optimisation of long-term SNHN3=0.5 (mgN/l) 1.0 1.5 2.0 8,25 8,75 9,25 9,75 10,25 N3 (mgN/l) y-state solution of the mathematical model (HRTA) and those obtained using the ling & Software 23 (2008) 435e450 imperative, such as the discontinuous process (e.g. Sequential Batch Reactors). l Further research in the field of model-based WWTP design is directed along several complementary lines. Optimum design that considers different uncertainty sources and statisti- cal criteria for effluent quality is being studied by adapting the Monte Carlo theory to the WWTP design problem. In particu- lar, uncertainty-based calibration and description of the crucial model parameters in WWTP models are currently under study. Model-based integrated design and operation of the water and sludge lines in WWTP are also being carried out through the introduction of new plant-wide models. Other optimisation algorithms are being tested to explore if they can give better performance when the methodology is extended in the future. Acknowledgments The authors wish to acknowledge the Antonio Aranzabal Foundation-Universidad de Navarra Thermal Engineering and Fluid Flow Chair; the companies, Cadagua S.A. and ATM S.A., and the Basque Government for its financial sup- port in this field of research. References Abadie, J., Carpentier, J., 1969. Generalization of the Wolfe reduced gradient method to the case of non-linear constraints. In: Fletcher, R. (Ed.), Optimi- sation. Academic Press, London, pp. 37e47. Ayesa, E., Goya, B., Larrea, A., Larrea, L., Rivas, A., 1998. Selection of operational strategies in activated sludge processes based on optimisation algorithms. Water Sci. Technol. 37 (12), 327e334. Ayesa, E., Garralon, G., Rivas, A., Suescun, J., Larrea, L., Plaza, F., 2001. New simulators for the optimum management and operation of wastewater treatment plant. Water Sci. Technol. 44 (2e3), 1e8. Ayesa, E., De la Sota, A., Grau, P., Sagarna, J.M., Salterain, A., Suescun, J., 2006. Supervisory control strategies for the new WWTP of Galindo- Bilbao: the long run from the conceptual design to the full-scale experimental validation. Water Sci. Technol. 4 (5), 193e201. Batstone, D.J., Keller, J., Angelidaki, I., Kalyuzhnyi, S.V., Pavlostathis, S.G., Rozzi, A., Sanders, W.T.M., Siegrist, H., Vavilin, V.A., 2002. Anaerobic Digestion Model No. 1. In: Scientific and Technical Report, No. 13. IWA Publishing, London. Belanche, L.A., Valde´s, J.J., Comas, J., Roda, I.R., Poch, M., 1999. Towards a model of input-output behaviour of wastewater treatment plants using soft computing techniques. Environ. Model. Softw. 14, 409e419. Benedetti, L., Bixio, D., Vanrolleghem, P.A., 2005. Assessment of WWTP de- sign and upgrade options: balancing costs and risks of standards’ exceed- ance. In: 10th International Conference on Urban Drainage, Copenhagen/ Denmark, 21e26 August. Billing, E., Dold, P.L., 1988a. Modelling techniques for biological reaction systems 3. Modelling of the dynamic case. Water SA 14 (4), 207e218. Billing, E., Dold, P.L., 1988b. Modelling techniques for biological reaction systems 2. Modelling of the steady-state case. Water SA 14 (4), 193e206. Bixio, D., Parmentier, G., Rousseau, D., Verdonck, F., Meirlaen, J., Vanrolleghem, P.A., Thoeye, C., 2002. A quantitative risk analysis tool for design/simulation of wastewater treatment plants. Water Sci. Technol. 46 (4e5), 157e167. Brenner, A., Shacham, M., Cutlip, M.B., 2005. Applications of mathematical software packages for modeling and simulations in environmental engi- neering education. Environ. Model. Softw. 20, 1307e1313. Butler, D., Schu¨tze, M., 2005. Integrating simulation models with a view to optimal control of urban wastewater systems. Environ. Model. Softw. 20, A. Rivas et al. / Environmental Mode 415e426. Carrio, L.A., Sharpe, R., Bizzarri, R.E., Wilson, T.E., 1992. Advanced BNRe BNR for large plants. Presented at the 65th WEF Annual Conference in New Orleans. Concha, L., Henze M., 1996. Advance design and operation of municipal wastewater treatment plants. RDN and DRDN processes. Science Research and Development Technologies for Environmental Protection, Report 10 EUR 16869, EN Environment and Climate Programme; European Com- mission Directorate General XII. Copp, J. (Ed.), 2002. The COST Simulation Benchmark. Description and Sim- ulator Manual. Office for Official Publications of the European Commu- nity, Luxembourg, ISBN 92-894-1658-0. Doby, T.A., Loughlin, D.H., de los Reyes III, F.L., Ducoste, J.J., 2002. Opti- mization of activated sludge designs using genetic algorithms. Water Sci. Technol. 45 (6), 187e198. Dudley, J., Chambers B., 1995. Dynamic modelling of wastewater treatment processes using STOAT. WaPUG Meeting, Blackpool. Dupont, R., Sinkjaer, O., 1994. Optimisation of wastewater treatment plants by means of computer models. Water Sci. Technol. 25 (6), 181e190. Ekama, G.A., Marais, G.v.R., Siebritz, I.P., Pitman, A.R., Keay, G.F.P., Buchan, L., Gerber, A., Smollen, M., 1984. Theory, Design and Operation of Nutrient Removal Activated Sludge Processes. University of Cape Town, City Council of Johannesburg and the National Institute for Water Research of CSIR, Pretoria. Ekama, G.A., Barnard, J.L., Gu¨nthert, F.W., Krebs, P., McCorquodale, J.A., Parker, D.S., Wahlberg, E.J., 1997. Secondary Settling Tanks: Theory, Mod- elling, Design and Operation. IAWQ Scientific and Technical Report N� 6. Frangopoulos, C.A., 1994. Application of the thermoeconomic functional ap- proach to the CGAM problem. Energy 19 (3), 323e342. Fylstra, D., Lasdon, L., Watson, J., Waren, A., 1998. Design and use of the Microsoft Excel Solver. Interfaces 28 (5), 29e56. Gabaldon, C., Ferrer, J., Seco, A., Marzal, P., 1998. A software for the inte- grated design of wastewater treatment plants. Environ. Model. Softw. 13, 31e44. Galarza, A., Ayesa, E., Linaza, M.T., Rivas, A., Salterain, A., 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 Sci. Technol. 43 (7), 167e174. Garcı´a-Sanz, M., Eguinoa, I., Ayesa, E., Martı´n, C., 2006. Non-diagonal multi- variable robust QFT control of a wastewater treatment plant for simultaneous nitrogen and phosphorus removal. Accepted for the Fifth IFAC Symposium on Robust Control Design ROCOND 2006, Toulouse, France, July 5e7. Ge, Y., Boufadel, M.C., 2006. Solute transport in multiple-reach experiments: evaluation of parameters and reliability of prediction. Journal of Hydrol- ogy 323 (1e4), 106e119. German ATV-DVW-Standards. DVWK-A 131E., May 2000. Dimensioning of Single-Stage Activated Sludge Plants. ISBN: 3-935669-96-8. Gernaey, K.V., Van Loosdrecht, M.C.M., Henze, M., Lind, M., Jørgensen, B., 2004. Activated sludge wastewater treatment plant modelling and simula- tion: state of the art. Environ. Model. Softw. 19, 763e783. Gupta, I., Bassin, J.K., Gupta, A., Khanna, P., 1993. Optimization of water dis- tribution system. Environ. Softw. 8, 101e113. Gupta, I., Gupta, A., 1999. Genetic Algorithm for optimization of water distri- bution systems. Environ. Model. Softw. 14 (5), 437e446. Henze, M., Gujer, W., Mino, T., van Loosdrecht, M., 2000. Activated Sludge Models ASM1, ASM2, ASM2d and ASM3. In: Scientific and Technical Report, No. 9. IWA Publishing, London. Hulsbeek, J.J.W., Kruit, J., Roeleveld, P.J., van Loosdrecht, M.C.M., 2002. A practical protocol for dynamic modelling of activated sludge systems. Water Sci. Technol. 45 (6), 127e136. Jolma, A., Norton, J., 2005. Methods of uncertainty treatment in environmen- tal models. Environ. Model. Softw. 20, 979e980. Kayser, R., Stobbe, G., Werner, M., 1992. Operational results of the Wolfsburg wastewater treatment plant. Water Sci. Technol. 25 (4e5), 203e209. Lansey, K., Mays, L., 1989. Optimization model for water distribution system design. J. Hydraul. Eng. 115 (10), 1401e1418. Lasdon, L.S., Waren, A.D., Jain, A., Rather, M., 1978. Design and testing of a generalized reduced code for nonlinear programming. ACM Trans. 449ling & Software 23 (2008) 435e450 Math. Softw. 4 (1), 34e50. Lasdon, L.S., Waren, A.D., 1979. Generalized reduced gradient software for linearly and nonlinearly constrained problems. In: Greenberg, H. (Ed.), Design and Implementation of Optimization Software. Sijthoff and Noordhoff Publishers. Larrea, L., Larrea, A., Ayesa, E., Rodrigo, J.C., Lo´pez-Carrascoa, M.D., Cortacans, J.A., 2001. Development and verification of design and opera- tion criteria for the step feed process with nitrogen removal. Water Sci. Technol. 43 (1), 261e268. Langergraber, G., Rieger, L., Winkler, S., Alex, J., Wiese, J., Owerdieck, C., Ahnert, M., Simon, J., Maurer, M., 2004. A guideline for simulation stud- ies of wastewater treatment plants. Water Sci. Technol. 50 (7). Lesouef, A., Payraudeau, M., Rogalla, F., Kleiber, B., 1992. Optimizing nitro- gen removal reactor configurations by on-site calibration of the IAWPRC activated sludge model. Water Sci. Technol. 25 (6), 105e123. Martı´n, C., Cardona, C.M., Salterain, A., Ayesa, E., Eguinoa, I., Garcı´a-Sanz, M., 2006. Uncertainty based calibration of a new river water model. Ac- cepted for the Seventh International Conference on Hydroinformatics (HIC 2006), Nice. Nenov, I., Fylstra, D., 2003. Interval methods for accelerated global search in the Microsoft Excel Solver. Reliab. Comput. 9 (2), 143e159. Ostfeld, A., 2005. Optimal design and operation of multiquality networks under Rivas, A., Ayesa, E., Galarza, A., Salterain, A., 2001. Application of mathe- matical tools to improve the design and operation of activated sludge plants. Case study: the new WWTP of Galindo-Bilbao. Part I: optimum de- sign. Water Sci. Technol. 43 (7), 157e167. Rivas, A., Go´mez-Acebo, T., Ramos, J.C., 2006. The application of spread- sheets to the analysis and optimization of systems and processes in the teaching of hydraulic and thermal engineering. Comput. Appl. Eng. Educ. 14 (4), 256e268. Rousseau, D., Verdonck, F., Moerman, O., Carrette, R., Thoeye, C., Meirlaen, J., Vanrolleghem, P.A., 2001. Development of a risk assessment based technique for design/retrofitting of WWTPs. Water Sci. Technol. 43 (7), 287e294. Sakarya, A.B.A., Mays, L.W., 2000. Optimal operation of water distribution pumps considering water quality. J. Water Resour. Plann. Manag. 126 (4), 210e220. Schlegel, S., 1992. Operational results of waste water treatment plants with bi- ological N and P elimination. Water Sci. Technol. 25 (4e5), 241e247. Seppelt, R., Richter, O., 2005. ‘‘it was an artefact not the result’’: a note on systems dynamic model development tools. Environ. Modell. Softw. 20, 979e980. Suescun, J., Rivas, A., Ayesa, E., Larrea, L., 1994. A new simulation program 450 A. Rivas et al. / Environmental Modelling & Software 23 (2008) 435e450 steady conditions. J. Water Resour. Plann. Manag. 131 (2), 116e124. Petersen, B., Gernaey, K., Henze, M., Vanrolleghem, P.A., 2003. Calibration of activated sludge models. A critical review of experimental design. In: Agathos, S.N., Reineke, W. (Eds.), Biotechnology for the Environment: Wastewater Treatment and Modelling, Waste Gas Handling. Kluwer Aca- demic Publishers, Dordrecht, The Netherlands, pp. 101e186. Poch, M., Comas, J., Rodrı´guez-Roda, I., Sa´nchez-Marre`, M., Corte´s, U., 2004. Designing and building real environmental decision support systems. Environ. Model. Softw. 19, 857e873. Powell, M.J.D., 1978. Restart procedures for the conjugate gradient method. Math. Program. 14 (1), 41e72. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1992. Numeri- cal Recipes in C. The Art of Scientific Computing, 2a ed. Cambridge University Press. Rivas, A., Ayesa, E., 1997. Optimum design of activated sludge plants using the simulator DAISY 2.0. In: Zannetti, P. (Ed.), Computer Techniques in Environmental Studies V. Computational Mechanics Publications, South- ampton, Boston, pp. 507e515. Rivas, A., 2000. Optimisation of WWTP Design and Operation Through Mathematical Modelling and Simulation. Ph.D. dissertation, TECNUN, University of Navarra (in Spanish). oriented to the design of complex biological processes for wastewater treatment. In: Zanetti, P. (Ed.), Computer Techniques in Environmental Studies V. Computational Mechanics Publications, Southampton, Boston, pp. 209e215. Tyteca, D., 1985. Mathematical models for cost-effective biological wastewa- ter treatment. In: Mathematical Models in Biological Wastewater Treat- ment. Elsevier Science, Amsterdam, pp. 737e802. Vanrolleghem, P.A., Jeppsson, U., 1995. Simulators for Modelling of WWTP COST 682. Report 1992e95, Luxemburgo, pp. 67e78. Vanrolleghem, P.A., Jeppsson, U., Carstensen, J., Carlsson, B., Olsson, G., 1996. Integration of wastewater treatment plant design and operationeA system- atic approach using cost functions. Water Sci. Technol. 34 (3e4), 159e171. Vazquez-Sa´nchez, E., 1996. Dynamic Modelling of the Secondary Clarifier in the Activated Sludge Process: Analysis and Development of a New Extended 1-D Model. Master Thesis of the University of London. Vijayaraghavan, S., 2003. Thermodynamics Studies on Alternate Binary Working Fluid Combinations and Configurations for a Combined Power and Cooling Cycle. Ph.D. dissertation, University of Florida. WERF-Water Environmental Research Foundation, 2003. Methods for Waste- water Characterisation in Activated Sludge Modeling. WERF Report 99- WWF-3, co-published by WERF and IWA, 575 pp. WEST�. HEMMIS homepage. Available from: http://www.hemmis.com. Model-based optimisation of Wastewater Treatment Plants design Introduction Problem formulation Mathematical modelling of WWTP Mathematical background Dynamic simulation Steady-state simulation Model-based optimisation of WWTP design Steady-state model-based optimisation Dynamic model-based optimisation Methodology implementation Case study: the Alpha process Description of the Step-Feed process for nitrogen removal Analysis of the optimum design of the Alpha process Problem 1 - Case A: steady-state design optimisation of the Alpha process Problem 1 - Case B: dynamic design optimisation of the Alpha process Problem 2. Steady-state optimisation of long-term plant operation of the Alpha process Results: optimum design of the Alpha process Problem 1. Optimisation of plant dimensions under critical conditions (minimum temperature) Problem 2. Optimisation of long-term plant operation Discussion Problem 1. Optimisation of plant dimensions at minimum temperature Influence of effluent requirements Influence of influent characteristics Problem 2. Optimisation of long-term plant operation Conclusions Acknowledgments References