Sistema de costos ABC

In the natural gas network optimization problems, the references on multiob-

jective optimization are rarer than in the monobjective case. Surry et al.[1995]

and Surry and Radcliffe[1997] have developed the COMOGA method for sol-

ving monobjective constrained optimization problem by means of a multiobjec- tive genetic algorithm; the procedure is illustrated by a gas network pipe-sizing

problem. However this application is only related to monobjective case. Babon-

neau et al.[2009] solved the biobjective optimization of investment and energy

in a gas transmission network. As the problem was formulated in a convex form,

convex solvers presented byAbbaspour et al.[2005] were used.

4

Conclusion

The modelling equations presented in Section 2 will be used in Chapter 5 for

modelling a didactic network[Abbaspour et al.,2005]. These equations will be

also used to take into account hydrogen injection into natural gas transmission network.

4 Conclusion 39

Chapter

2

Nowadays, most of optimization studies in process engineering have to be performed within a multiobjective framework, where some objectives related to environmental impacts, security, etc., must be simultaneously optimized with classical economic or technical criteria. In natural gas network optimization problems a lack of published works on multiobjective optimization can be ob- served, and this thesis aims at filling this gap. So this topic will be the main pur- pose of the present study. In the following chapter, the most commonly used ap- proaches in multiobjective optimization (scalarization and evolutionary proce-

dures) are reviewed and three specific algorithms (Weighted-sum, "-constraint

and Genetic algorithm) are detailed. On the basis of two mathematical pro- blems and four multiobjective chemical engineering problems, the choice of the best procedure, namely the Genetic algorithm, will be performed in Chapter 5. Then in the first part of Chapter 5, the didactic network is optimized according to two objectives: the fuel consumption in compression stations and the mass load of gas delivery. In the second part, this didactic network is considered again for hydrogen transportation, and three objectives are taken into account: the fuel consumption in compression stations, the mass load of gas delivery and the percentage of injected hydrogen into the network.

Multiobjective optimization

Chapter

3

Contents

1 Introduction . . . 43

2 General properties of a multiobjective constrained opti- mization problem . . . 44

3 General Multiobjective Optimization methods . . . 48

4 Solution procedures . . . 51

5 Mathematical examples . . . 66

1 Introduction 43

Chapter

3

1

Introduction

As shown in Chapter 2, the natural gas network system can be formulated as a multiobjective optimization problem. In many other engineering fields, most of process optimization problems became multiobjective optimization ones. When dealing with process optimization, the current trend is to consider other ob- jectives besides the traditional economic criterion, related to sustainability, en- vironment and safety. So, this chapter deals with the most commonly used multiobjective methods in chemical engineering. Two mathematical examples are presented as comparison purposes. Then, from the basis of well-known chemical engineering problems, the choice of the multiobjective optimization algorithm is performed in Chapter 4.

Among the diversity of multiobjective optimization methods, two important classes have to be distinguished: first scalarization approaches, second genetic and evolutionary methods. Complete reviews are proposed in literature for both

classes [Hao et al.,1999;Grossmann,2002; Biegler and Grossmann,2004]. A

thorough analysis of both classes was previously studied byPonsich[2005] with

the support of batch plant design problems.

The first class, namely deterministic methods, assumes the verification of mathematical properties of the objective function and constraints, such as con- tinuity, differentiability and convexity. In practice, these assumptions (parti- cularly convexity) do not always hold, and the convergence towards a global optimum is no longer guaranteed. This working mode enables only to ensure to get a local optimum, what is a great advantage versus stochastic methods.

The second class, namely stochastic methods, is based on the evaluation of the objective function at different points of the search space. These points are chosen through a set of heuristics, combined with generations of random numbers. Thus, stochastic procedures cannot guarantee to obtain an optimum. However by allowing occasional objective function increases (for minimization problems) they may go out of local optimum gaps. Even if stochastic methods do not require any mathematical property for the objective function and cons- traints, they may be difficult to implement for problems involving a significant number of equality constraints.

Besides, the efficiency of a given method for a particular example is hardly predictable, and the only certainty we have is expressed by the No Free Lunch

Chapter

3

(NFL)Theory[Wolpert and Macready,1997]: there is no method that outdoes

all the other ones for any considered problem. This feature generates a common lack of explanation concerning the use of a method for the solution of a parti-

cular example. Several works were carried out on the NFL:Droste et al.[2002]

show that each heuristic which is able to optimize some functions efficiently follows some ideas about the structure of considered functions in black-box op-

timization; Griffiths and Orponen [2005] study the NFL in the framework of

Boolean functions; Service [2010] generalizes the NFL theorem to non totally

ordered objectives spaces. However, for any particular application, the resolu- tion strategy has to be selected in one of the two classes of methods.

This chapter recalls three classical types of procedures used in multiobjective optimization. The choice of the most adequate method will be performed in the next chapter, where several chemical process optimization problems are studied. The present chapter is organized as follows. First, the general properties of a multiobjective problem are presented. Then, three classical solution procedures

(Weighted-sum, "-constraint and Evolutionary procedures) are recalled. More

precisely, three algorithms (Adaptive Weighted-Sum, Augmented "-Constraint

and NSGA-IIb) are described. Finally, two mathematical problems are solved for performing a preliminary comparison of the three algorithms.

2

General properties of a multiobjective constrained

optimization problem