Mathematical Models Classifications

To the best of our knowledge, a vast number of works have been done on the optimization approaches and developing suitable algorithms to find the optimal solutions for the natural gas distribution and transmission networks. In this part, we classify the earliest papers that focused on the mathematical modeling in Table 19.5. Some general findings are obviously seen in this table. For example, the optimization has a more effective role in transmission network in comparison to distribution network because the natural gas spends more time under high pressures in transmission networks because of the long distances among produc­ers and city gate stations and because more instruments are used in the transmis­sion network. Therefore, more problems have been defined in this segment. Moreover, as it would appear from the table, although some data in the natural gas industry are not deterministic, to simplify the problem researchers have not considered uncertainties in the form of fuzzy or statistical data such as what Carvalho and Ferreira [15] and Davidon et al. [10] did. Other issues such as type of problems, number of objectives, and more useful solution methods regarding the optimization of mathematical models developed for natural gas net­work planning are presented in the following subsections.

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Types of Problem

In general, optimization problems can be classified based on the type of variables (continuous, integer, or mixed) and nature of functions used as the objective func­tion and constraints. Considering these two factors, six types of problems are defined as presented in Table 19.6. Because of the nonlinearity behavior of the com­pressor station units and other factors and existence of mixed continuous and integer variables, most of the optimization problems of the natural gas network planning are categorized in nonlinear problems. Dependent on the defined variables, they can be Non-Linear Programming (NLP), Integer Non-Linear Programming (INLP), and Mixed Integer Non-Linear Programming(MINLP).

Number of Objectives

In practice to plan for a natural gas network optimally, more than one objective, generally conflict objectives, should be considered. For example, minimizing net­work flows or investment cost versus the maximum satisfaction of the customers involve two conflict objectives, which should be achieved simultaneously.

In theory, if all objectives are seen in the solution methodology, problems become more difficult, especially when a large number of objectives are consid­ered. In a few cases, all objectives are transformed into a single objective, but in most situations, it is not possible. Therefore, a number of researchers focus only on one objective and do not pay attention to others or relax them, and another group of researchers keeps the nature of problems and applies a multiobjective optimiza­tion method based on an approach. For example, if goal values of objective func­tions are known, the goal programming approach can be a suitable option.

Solution Methods

The nature of natural gas network problems that are categorized in the group of NP-hard problems is nonconvex and nonlinear. Therefore, the design and selection of proper solution methods are very critical in this field. Mallinson et al. [26] described two general methods to optimize natural gas network problems. In the first method, numbers of optimization problem variables are reduced by eliminating the flow variables; in the second one, to achieve a better behavior, the optimization problem is solved without removing any variable. By reduction techniques, the solution of the problem becomes easier, but finding the suitable algorithm has

Table 19.6 Classification of Optimization Problems in Mathematical Models

Integer

Mixed