Power Magnetic Devices. Scott D. Sudhoff. Читать онлайн. Newlib. NEWLIB.NET

Автор: Scott D. Sudhoff
Издательство: John Wiley & Sons Limited
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Жанр произведения: Техническая литература
Год издания: 0
isbn: 9781119674634
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distance. The crowding distance associated with a solution xi is defined as

      Crowding tournament selection uses the concepts of front rank and crowding distance in order to decide which individuals to put into the mating pool. In this method, individuals xc1 and xc2 are randomly drawn from the current population. If one of these solutions has a better front rank than the other, it is copied into a mating pool. If the two individuals have the same front rank, then the one with the better (larger) crowding distance goes into the mating pool, as it has greater diversity in terms of objectives.

      Example 1.8A

      (1.8A-1)equation

Schematic illustration of crowding distance. Schematic illustration of elitist nondominated sorting genetic algorithm (NSGA-II).

      Using this algorithm, as the population evolves, it will come closer and closer to approaching the Pareto‐optimal set, which will lead to a family of designs. We will use multi‐objective optimization extensively in this book for the design of power magnetic devices.

      The previous sections of this chapter have focused on the design process, and some general‐purpose single‐ and multi‐objective optimization techniques. In this section, we consider methodologies to construct fitness functions.

      In constructing the fitness function, we will have a variety of metrics, such as mass and loss, and also a number of constraints related to the appropriate operation and construction of the device of interest. It will often be the case that we will have to perform multiple analyses in order to evaluate metrics, and that some of these analyses may be computationally expensive. The fact that a variety of analyses of varying computational intensity will be required will be a significant consideration in the construction of the fitness function.

      Let us begin our discussion of the construction of the fitness function with consideration of the constraints. Let us assume that we have C constraints, and use ci to denote the status of the ith constraint. If the constraint is satisfied, we will set ci = 1. If the constraint is not satisfied, we will have 0 ≤ ci < 1. It is convenient to define ci so that it approaches 1 as the constraint becomes closer to becoming satisfied.

      In order to test constraints, it is convenient to define the less‐than‐or‐equal‐to and greater‐than‐or‐equal‐to functions as

      (1.9-1)equation

      (1.9-2)equation

Schematic illustration of constraint functions.

      As an example, we may require the height of a transformer to be less than a maximum value hmx. If this is the first constraint and if we use h(x) to represent the height in terms of our design parameters, our first constraint could be calculated as

      (1.9-3)equation

      Next, let us consider our design metrics. The design metrics will also be a function of the parameter vector x. In this text, common metrics will be mass and loss. Let the number of metrics be denoted M, and the value of the ith metric be denoted mi. It will henceforth be assumed that all metric values are greater than zero.

      As we will see, metrics and objectives are closely related but not synonymous. Metrics will be based on attributes of interest. Objectives will be based on the metrics, but also be influenced by constraints.

      Let us now discuss the definition of the objective or fitness function. In keeping with the usual practice of GAs, we will assume that our fitness function (which is synonymous with the objective function) is of a form to be maximized. One approach to creating a fitness function begins with first forming a combined constraint. This can be done by averaging the constraints as