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

doi:10.1016/j.jfranklin.2013.11.005

Journal of the Franklin Institute

Volume 351, Issue 3, March 2014, Pages 1356-1371

https://doi.org/10.1016/j.jfranklin.2013.11.005 Get rights and content

Highlights

•
A new control-based technique for determining the optimal location of a new industrial plant is presented.
•
Both economic and ecological aspects are considered.
•
A Pareto-optimal front for the decision maker is achieved.

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.

Introduction

Mathematical modelling and numerical simulation based on partial differential equations (PDE) are fundamental tools nowadays in the control of environmental pollution. Scientific literature on numerical modelling of contaminant dispersion is very extensive [35], [2], [16], [25], [26], [32], and there also exists a large amount of software packages (both free and commercial), devoted to this purpose [30], [31], [20], [36].

Mathematical simulation is the basis for the application of optimal control techniques and, consequently, the number of works on optimal control of environmental management has experimented a large development in last decades. If we focus our interests into the analysis of optimal locations, we can mention here several examples studied by the authors during past years within very different fields: the optimal location of sampling points for the monitoring of water quality in a river section [9], [29], the optimal placement of antennas for a surveillance system inside an airport [18], the optimal location of transmitters in a mobile wireless system [1], or the optimal location of wastewater outfalls in a sewage treatment plant [8]. For a more extensive study on the location analysis, the interested reader is directed to the references therein.

However, although this kind of problems involves numerous aspects both from economic and ecological viewpoints, most published works deal with problems using only one objective function. Within this framework we can cite, for instance, Skiba et al. [33], where a method for determining the optimal position for a new enterprise in a region is described, but taking into account only ecological purposes. In the same way, in a previous work of the authors [7] a related problem for marine pollution is also analyzed but, although economic and ecological aspects are considered, a prioritization of the latter leads us to the formulation of the problem in terms of the (constrained) minimization of a unique cost function. As far as we know, the multi-criteria analysis of this particular type of PDE optimal control problems has not been previously addressed in the mathematical literature, although we can cite here the previous attempts of the authors to analyse a multi-objective control problem related to wastewater management, both within a cooperative framework (Pareto-optimal solutions [3], [4]) and a non-cooperative one (Nash equilibria [19]).

Within the context of atmospheric contamination, in this work we study the problems derived from the construction of a new industrial plant. We take into account economic and ecological objectives, and we look, not only for the optimal location of the plant, but also for the optimal management of its pollutant emission rates. For the numerical simulation of the atmospheric pollution (Section 2) a model based on a parabolic system of PDE with Neumann boundary conditions and Dirac measures, of nonlinear type and strongly coupled, is presented. Starting from this model, and bearing in mind a clear distinction between economical and ecological objects, the problem is formulated (Section 3) as an original nonconvex multi-objective optimal control problem, and we deal with it from a cooperative point of view, seeking its Pareto-optimal solutions (Section 4). Introducing the adjoint model (Section 5), we propose a new and useful alternative formulation, and present a complete algorithm for its numerical resolution (Section 6). Finally, some numerical results for a very simplified case in the region of Galicia (NW Spain) and concluding remarks are shown in last sections.

Section snippets

Mathematical modelling of air pollution

Let $Ω \subset R^{2}$ be a bounded domain where several industrial plants, located at points $p_{i} \in Ω, i = 0, \dots, N$ , discharge pollutants into the atmosphere: sulphur dioxide, nitrogen dioxide, carbon monoxide, particulate matter, or even toxic metals (like lead or mercury). We suppose that these pollutants are transported through the atmosphere by air masses and turbulent diffusion, and the kinetics of all reactions taking place in the process can be expressed quantitatively by a rate law (where the reaction rate is

The multi-objective optimal control problem

Our main objective consists of determining the optimal location and management of a new industrial plant. We suppose that at the present moment there exist N already built plants working in the domain $Ω$ (i.e., points $p_{1}, \dots, p_{N} \in Ω$ and rates $Q_{1}^{j} (t), \dots, Q_{N}^{j} (t)$ , for $j = 1, \dots, N_{S},$ are known), and that a new industrial plant is to be built in a point $p_{0} \in Ω$ , which has to be determined. This new plant will be working for a time $(0, T)$ , and during this period of time its emission flow rates will be given by a

Pareto-optimal solutions

Obviously, economic and ecological objectives are contradictory and, consequently, it will not be possible to find an ideal (or utopic) element $(p_{0}, {\vec{Q}}_{0}) \in X_{ad} \times Q_{ad}$ minimizing J_E and J_k^j, for $k = 1, \dots, N_{Z}$ , $j = 1, \dots, N_{S},$ simultaneously. In this sense, the problem (MOC) (as it is usual in many multi-objective optimization problems) is ill-posed. Anyway, some distinguished elements of the admissible set can be extracted for examination. Such vectors are those where none of the components can be improved

An alternative formulation: the adjoint model

It is worthwhile remarking here that, although it is not explicitly shown in expression (6), the ecological cost functionals J_k^j depend implicitly on the controls p₀ and ${\vec{Q}}_{0}$ via the solution $ϕ (x, t)$ of the state system (1), (2), (3), (4). In the standard theory of optimal control of PDE, an usual technique in order to establish a direct relation between the controls and the objective functionals is the use of adjoint models [22], [5], [27], [19], [3], [10]. With this purpose in mind we

Problem (A)

As remarked in the above section, if $α_{j}^{i} = δ_{j}^{i}$ then $h_{j} (ϕ_{1}, \dots, ϕ_{N_{S}}) = κ_{j}$ , and the adjoint systems (7), (8), (9), (10) are similar for all $j = 1, \dots, N_{S}$ , $k = 1, \dots, N_{Z}$ . Due to this fact, and for the sake of simplicity, in this section we will suppress the indices k and j in the adjoint systems (7), (8), (9), (10).

In order to solve the problem (A) (i.e., adjoint systems (7), (8), (9), (10) with $h (ϕ_{1}, \dots, ϕ_{N_{S}}) = κ$ ) we use a method which combines characteristics for the time discretization with a Lagrange P₁ finite

Computational results

Although we have developed many computational experiences, for the sake of conciseness, in this section we present only one numerical test for a simple example posed in the region of Galicia (NW Spain). So, we take as domain $Ω = [0, 240] \times [0, 240]$ , a square area of $57 600 {km}^{2}$ (see Fig. 3). In this case, we assume the existence of N=3 industrial plants located in the points $p_{1} = (62, 165)$ , $p_{2} = (161, 203)$ and $p_{3} = (111, 45)$ and only one sensitive area (N_Z=1) occupying the right upper quadrant of $Ω$ . We are

Conclusions

This work addresses the theoretical and numerical solution of the problem related to the optimal location and management of a new industrial plant. The problem was regarded as a multi-objective optimization problem and the solution approached using the classical weighted sum method combined with the ɛ-constraint algorithm. From the theoretical point of view the optimal control problem was mathematically well posed, and an alternative formulation – in terms of the adjoint system – was

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^☆: This work was supported by Project MTM2012-30842 of M.E.C. (Spain). The first author thanks the support from Sistema Nacional de Investigadores SNI-52768 (Mexico). The authors are grateful to the interesting suggestions of the anonymous referees.

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

Highlights

Abstract

Introduction

Section snippets

Mathematical modelling of air pollution

The multi-objective optimal control problem

Pareto-optimal solutions

An alternative formulation: the adjoint model

Problem (A)

Computational results

Conclusions

Appl. Numer. Math.

J. Comput. Appl. Math.

J. Comput. Appl. Math.

Math. Comput. Simul.

Appl. Numer. Math.

Appl. Numer. Math.

Environ. Model. Software

Ecol. Model.

Math. Comput. Model.

Environ. Software

Environ. Model. Software

Ecol. Model.

Appl. Numer. Math.

Environ. Model. Software

Optimization methods for optimal transmitter locations in a mobile wireless system

IEEE Trans. Veh. Technol.

Multi-objective Pareto-optimal controlan application to waste-water management

Comput. Optim. Appl.

The water conveyance problemoptimal purification of polluted waters

Math. Models Methods Appl. Sci.

Mathematical analysis of the optimal location of wastewater outfalls

IMA J. Appl. Math.