Optimal Management and Design of a Wastewater Purification System

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1 Optimal Management and Design of a Wastewater Purification System Lino J. Alvarez-Vázquez 1, Eva Balsa-Canto 2, and Aurea Martínez 1 1 Departamento de Matemática Aplicada II. E.T.S.I. Telecomunicación, Universidad de Vigo, Vigo, Spain lino@dma.uvigo.es,aurea@dma.uvigo.es 2 Instituto de Investigaciones Marinas, C.S.I.C., Vigo, Spain ebalsa@iim.csic.es 1 Introduction Coastal areas are continually exposed to land-based sources of pollution resulting from domestic and industrial activities including oil spills, discharge of sewage and industrial effluents, among others. These contaminants arrive in the sea through wastewater discharges from sewage farms where contaminant concentrations are reduced by means of biological or chemical processes. Biological processes are commonly used to treat domestic or combined domestic and some specific industrial wastewater. The basic idea is to reproduce the same processes that would occur naturally in the receiving water (river, estuary, etc.), but under controlled conditions, so that the cleansing reactions are completed before the water is discharged into the environment. The objective is, therefore, to provide an optimum environment for the microbial population to decompose the organic matter. Microorganisms utilize the organic matter for the production of energy by cellular respiration and the manufacture of new cells. As the pollution increases, the Biochemical Oxygen Demand (BOD) also increases and, as a result, the Dissolved Oxygen (DO) decreases. This damages the marine life and causes the organic matter decomposition by means of anaerobic processes, which do not use oxygen but produce sulphide of hydrogen and methane, both having a nauseating smell. To avoid this problem, we have to guarantee a minimum level of DO and a maximum level of BOD in each region to be protected. Is, therefore, of the highest interest to implement a wastewater treatment system that is able to assure water quality standards, corresponding to pollution concentrations lower than a certain allowed value imposed by the regional legislation. Note, however, that the economic cost of the process may

2 796 L.J. Alvarez-Vázquez et al. be excessively large, depending on both the purification cost at each plant and the distance from the plant to the outfall (design cost). The authors have recently considered, both from the theoretical and the numerical point of view, two related optimization problems. The first problem was devoted to determine the optimal level of the oxygen discharges in order to minimize the global purification cost (see Martínez et al. [5, 6]), and the second was aimed to obtain the optimal locations for the wastewater outfalls, assuming constant oxygen discharges (see Alvarez-Vázquez et al. [1, 2]) while keeping, in both cases, the constraints on the water quality. This work addresses, for the first time to the authors knowledge, the combined design and operation optimization problem. This problem can be formulated as finding the optimal design (outfall locations) and the optimal operation conditions (that is, the optimal oxygen discharge levels) which minimize the total economic cost of the system while guarantying the above mentioned constraints on the water quality. Note that this is a two-objective problem: in one hand, one would be interested in minimizing the initial cost due to the installation of the outfalls and in other hand it is also desirable to minimize the expenses of the purification process which will be repeated along time. The simultaneous optimization of multiple, usually competing, objectives, deviates from the single objective case in that it does not admit an unique optimal solution. Instead, a number of solutions, the so-called Pareto-optimal solutions, may be found that must be considered equivalent in the absence of information concerning the relative importance of the different objectives. In contrast to single-objective optimization, multi-objective optimization problems require the involvement of a decision maker who has to select one Pareto solution from the set. This work proposes the use of a classical approach, the weighted sum method, to combine the two objectives in a single-objective with the aim of, from the theoretical point of view, obtaining the optimality conditions and demonstrating the existence of a solution, for any given set of weights. From the numerical point of view, a Pareto front is obtained for the two-objective case, using a control vector parametrization approach to approximate the control variables and an evolutionary algorithm to deal with the non-convex character of the resultant nonlinear programming problem. 2 Optimal Operation and Design: Problem Formulation A domain Ω R 2 occupied by shallow waters, for instance an estuary, is considered. The sewage is dumped into the domain Ω through N submarine outfalls, each of them located at a point b j Ω (that must be determined), and connected to a purification plant, located at a point a j, which discharges an amount m j (t) (also to be determined). Moreover, there are M areas in Ω, denoted by A i, for example beaches or fish nurseries, that must be protected guarantying that the levels of pollution are bellow previously fixed thresholds.

3 Optimal Management and Design of a Wastewater Purification System 797 As it was stated in the introduction two of the most important parameters used to control pollution levels are the dissolved oxygen (DO) and the biochemical oxygen demand (BOD). As the pollution increases, the oxygen demand also increases and, as a result, the dissolved oxygen decreases with undesirable environmental consequences. To avoid this problem, we have to guarantee a minimum level ζ i of DO and a maximum level σ i of BOD in each region Āi to be protected. The evolution of the concentrations of BOD (ρ 1 (x, t)) and DO (ρ 2 (x, t)) in the domain Ω along the time interval [0,T]is governed by a complex system of partial differential equations coupled with the shallow water equations (see, for instance, Martínez et al. [5]). The pair (m, b) formed by the set of optimal discharges m = (m j ) N j=1 and the set optimal locations b =(b j ) N j=1 are the control variables of the problem. Now, we assume that inside the domain Ω there are M protected zones Ā i where a maximum level of BOD and a minimum level of DO must be ensured, that is, ρ 1 Ā i [0,T ] σ i, ρ 2 Ā i [0,T ] ζ i, i =1,...,M. (1) Moreover, taking into account technological limitations, the j-th discharge must verify that m j m j (t) m j, t (0,T), and the j-th outfall must be placed in a suitable region U j, where U j Ω\ M i=1 Āi is a compact, convex, polyhedral set representing all the admissible points to locate the outfalls. Thus, the optimal pair (m, b) mustverifym j m j (t) m j, b j U j, j = 1,...,N. If we define U ad = {m [L (0,T)] N : m j m j (t) m j, t (0,T) j =1,...,N} Πj=1 N U j, technological constraints can be written in the simpler way: (m, b) U ad. (2) Consider now that the purification cost in each plant is given by J 1 (m, b) = N T j=1 0 f j(m j (t)) dt, where function f j represents the cost of the purification in the j-th plant; and that the design cost depends on the distance from the farm to the outfall in the following manner J 2 (m, b) = 1 N 2 j=1 b j a j 2. In contrast to the single objective cases where an unique objective was pursued, the aim in this contribution is to solve the two-objective case, finding good compromises between the two different objectives. The notion of optimal solution is substituted by the notion of Pareto-optimal solution: a solution is said to be Pareto-optimal if there exists no feasible solution which would decrease one objective without causing a simultaneous increase in the other. Under certain conditions the weighted sum method can be used to obtain the set of Pareto-optimal solutions. This method transforms the original multi-objective case into a single-objective optimization problem in which the objective function is a weighted sum of the original objectives. In practice, the objectives are usually scaled and combined to form a composite objective: N T J(m, b) = f j (m j (t)) dt + α N b j a j 2, (3) 2 j=1 where α>0 is a weight parameter. 0 j=1

4 798 L.J. Alvarez-Vázquez et al. Therefore the problem, denoted by (P), consists of finding the time varying discharges m j (t), j =1,...,N, and the points b j, j =1,...,N, to minimize the cost function J subject to the system dynamics, the state constraints (1) and the control constraints (2). This is a parabolic optimal control problem with non-convex pointwise state constraints, which makes difficult its analysis and resolution. By using minimizing sequences we can prove that the optimal control problem has, at least, one solution. We are also able to obtain, introducing the adjoint state, a first order optimality system satisfied by the solutions of the optimal control problem (see the details in Alvarez-Vázquez et al. [3]). 3 Numerical Solution The usual approach to solve optimal control problems is to transform the original infinite dimension problem into a finite dimension nonlinear programming (NLP) problem. With this aim, the control vector parametrization approach proceeds by dividing the duration of the process [0,T] into a reduced number of non-equidistant intervals and approximating the control variables (m j )using low order Lagrange polynomials within each interval. As a result, a NLP problem is obtained where the vector of decision variables includes the coefficients in the polynomials, the switching times, and the time independent parameters (outfall locations, in our case). We must remark that the calculation of the objective function requires the solution of the state system, that in this work is approached using a characteristics-finite element method. As it was stated in previous sections the weighted sum method transforms the original two-objective case into a single-objective optimization problem in which the objective function is a weighted sum of the original objectives. The Pareto front can then be generated by varying the weight α in the objective expression (3) and solving the corresponding nonlinear programming problems with a suitable technique. The optimization literature offers a large number of methods to solve nonlinear programming problems: local and global strategies, deterministic and stochastic methods...(cf. [4] and the references therein). In this work, a evolutionary global optimization method, Differential Evolution (DE) [7], will be used due mainly to its efficiency in solving real valued multimodal objective functions. DE is basically a parallel, population-based, direct search algorithm. In addition to its good convergence properties some of its main advantages are its conceptual simplicity and ease of use. Numerical results (Pareto front and optimal control profiles for a particular Pareto solution) for a realistic problem posed in the ría of Vigo (Spain) are shown, respectively, in Figs. 1 and 2. More details can be found in [3].

5 Optimal Management and Design of a Wastewater Purification System 799 Location cost 2.2 x x 10 4 Purification cost Fig. 1. Front of Pareto solutions m 1 (t) 80 m j (t) t m 2 (t) Fig. 2. Optimal control profiles for previous circled Pareto solution Acknowledgements The research contained in this work was partially supported by Project MTM of Ministerio de Educación y Ciencia (Spain). The authors also thank the help of C. Rodríguez and M.E. Vázquez-Méndez. References 1. Alvarez-Vázquez, L.J., Martínez, A., Rodríguez, C., Vázquez-Méndez, M.E.: Mathematical Analysis of the optimal location of wastewater outfalls. IMA J. Appl. Math., 67, (2002). 2. Alvarez-Vázquez, L.J., Martínez, A., Rodríguez, C., Vázquez-Méndez, M.E.: Numerical optimization for the location of wastewater outfalls. Comput. Optim. Appl., 22, (2002). 3. Alvarez-Vázquez, L.J., Balsa-Canto, E., Martínez, A.: Optimal design and operation of a wastewater purification system. Submitted (2006). 4. Balsa-Canto, E., Vassiliadis. V.S., Banga. J.R.: Dynamic Optimization of Singleand Multi-Stage Systems using a Hybrid Stochastic-Deterministic Method. Ind. Eng. Chem. Res., 44, (2005).

6 800 L.J. Alvarez-Vázquez et al. 5. Martínez, A., Rodríguez. C., Vázquez-Méndez. M.E.: Theoretical and numerical analysis of an optimal control problem related to wastewater treatment. SIAM J. Control Optim., 38, (2000). 6. Martínez, A., Rodríguez. C., Vázquez-Méndez. M.E.: A control problem arising in the process of waste water purification. J. Comp. Appl. Math., 114, (2000). 7. Storn, R., Price, K.: Differential Evolution - a Simple and Efficient Heuristic for Global Optimization over Continuous Spaces. J. Global Optim., 11, (1997).