M. E. Vázquez-Méndez

Theoretical and Numerical Analysis of an Optimal Control Problem Related to Wastewater Treatment

In this work we deal with the design and management of wastewater treatment systems, mainly the disposal of sea outfalls discharging polluting effluent from a sewerage system. This problem is formulated as a pointwise optimal control problem with state and control constraints. The main difficulties arise from the lack of regularity of the second member in the state system and from the pointwise constraints on the state variables. We develop the theoretical analysis of the problem, we propose an algorithm for its numerical resolution, and finally, we give results for a realistic problem posed in the ria of Vigo, Spain.

Numerical Optimization for the Location of Wastewater Outfalls

In this paper we solve a constrained optimal control problem related to the location of the wastewater outfalls in a sewage disposal system. This is a problem where the control is the position and the constraints are non-convex and pointwise, which makes difficult its resolution. We discretize the problem by means of a characteristics-Galerkin method and we use three algorithms for the numerical resolution of the discretized optimization problem: an interior point algorithm, the Nelder-Mead simplex method and a duality method. Finally, we compare the numerical results obtained by applying the described methods for a realistic problem posed in the ría of Vigo (Galicia, Spain).

Multi-objective Pareto-optimal control: an application to wastewater management

This work treats, within a multi-objective framework, of an economical-ecological problem related to the optimal management of a wastewater treatment system consisting of several purifying plants. The problem is formulated as a multi-objective parabolic optimal control problem and it is studied from a cooperative point of view, looking for Pareto-optimal solutions. The weighting method is used here to characterize the Pareto solutions of our problem. To obtain them, a numerical algorithm—based in a characteristics-Galerkin discretization—is proposed and numerical results for a real world situation in the estuary of Vigo (NW Spain) are also presented.

Optimal location of sampling points for river pollution control

Common methods of controlling river pollution include establishing water pollution monitoring stations located along the length of the river. The point where each station is located (sampling point) is of crucial importance and, obviously, depends on the reasons for the sample. Collecting data about pollution at selected points along the river is not the only objective; must also be extrapolated to know the characteristics of the pollution in the entire river. In this work we will deal with the optimal location of sampling points. A mathematical formulation for this problem as well as an efficient algorithm to solve it will be given. Finally, in last sections, we will present numerical results obtained by using this algorithm when applied to a realistic situation in a river mouth.

Numerical convergence for a sewage disposal problem

Abstract The management of sewage disposal and the design of wastewater treatment systems can be formulated as a constrained pointwise optimal control problem. In this paper we study the convergence of the numerical resolution for the corresponding state system by means of a characteristics Galerkin method. The main difficulty of the problem is due to the existence of Radon measures in the right-hand side of the state system. Finally, we present numerical results for a realistic problem posed in a r i a of Galicia, Spain.

THE WATER CONVEYANCE PROBLEM: OPTIMAL PURIFICATION OF POLLUTED WATERS

In this work we deal with the optimal purification of polluted areas of shallow waters by means of the injection of clear water in order to promote seawater exchange. This problem can be formulated as a control constrained optimal control problem where the control is the velocity of the injected water, the state equations are the shallow water equations together with that modelling the contaminant concentration, and the cost function measures the total amount of injected water and the fulfilment of the water quality standards. We analyze the solutions of the optimal control problem and give an optimality condition in order to characterize them. We also discretize the problem by means of a characteristics-mixed finite element method, focusing our attention on both the discrete and the discretized adjoint systems, and propose an algorithm for the numerical resolution of the discrete optimization problem. Finally, we present numerical results for some computational experiments.

A control problem arising in the process of waste water purification

Abstract In this paper we state and solve an optimal control problem arisen from the management of the sewage disposal which is dumped into the sea through submarine outfalls. Firstly, we fix oxygen and amount of organic matter as water quality indicators and we state a partial differential equations model to simulate them in a domain occupied by shallow waters. Constraints about water quality and economic objectives lead us to a pointwise optimal control problem with state and control constraints. (The theoretical analysis of the problem has been developed by the authors in (Martinez et al., C. R. Acad. Sci. Paris, Serie I 328 (1999) 35.) (Martinez et al., Preprint, Dept. Matematica Aplicada, Univ. Santiago de Compostela, Spain, 1998.)). We deal with the problem by using time and space discretizations and we propose two algorithms for the numerical resolution of the discretized problem. Finally, we give numerical results obtained by applying the described techniques for a realistic problem posed in the ria of Vigo (Spain).

On optimal location and management of a new industrial plant: Numerical simulation and control☆

Abstract Within the framework of numerical modelling and multi-objective control of partial differential equations, in this work we deal with the problem of determining the optimal location of a new industrial plant. We take into account both economic and ecological objectives, and we look not only for the optimal location of the plant but also for the optimal management of its emissions rate. In order to do this, we introduce a mathematical model (a system of nonlinear parabolic partial differential equations) for the numerical simulation of air pollution. Based on this model, we formulate the problem in the field of multi-objective optimal control from a cooperative viewpoint, recalling the standard concept of Pareto-optimal solution, and pointing out the usefulness of Pareto-optimal frontier in the decision making process. Finally, a numerical algorithm – based on a characteristics/Galerkin discretization of the adjoint model – is proposed, and some numerical results for a hypothetical situation in the region of Galicia (NW Spain) are presented.

Optimal location of green zones in metropolitan areas to control the urban heat island

In this paper we analyze and numerically solve a problem related to the optimal location of green zones in metropolitan areas in order to mitigate the urban heat island effect. So, we consider a microscale climate model and analyze the problem within the framework of optimal control theory of partial differential equations. Finally we compute its numerical solution using the finite element method, with the help of the interior point algorithm IPOPT, interfaced with the FreeFem++ software package.

Optimal shape design for fishways in rivers

Fishways are hydraulic structures that enable fish to overcome obstructions to their spawning and other migrations in rivers. In this paper we first introduce a mathematical formulation of the optimal design problem for a vertical slot fishway, where the state system is given by the shallow water equations determining the height of water and its velocity, the design variables are the geometry of the slots, and the objective function is related to the existence of rest areas for fish and a water velocity suitable for fish leaping and swimming capabilities. We also obtain an expression for the gradient of the objective function via the adjoint system. From the numerical point of view, we present a characteristic-Galerkin method for solving the shallow water equations, and an optimization algorithm for the computation of the optimal design variables. Finally, we give numerical results obtained for a standard 10 pools channel.