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Chapter 14. Cost Minimization - 14 COST MINIMIZATION...

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543 14. COST MINIMIZATION Summary Transformer design is primarily determined by minimizing the overall cost, including the cost of materials, labor, and losses. This minimization, however, must take into account constraints which may be imposed on the transferred power, the impedance, the flux density, the overall height of the tank, etc. Since the cost and constraints are generally non-linear functions of the design variables, a non-linear constrained optimization method is required. We examine several such methods, developing in greater detail the one which appears to be best suited to our needs. It is then applied to transformer design, considering for simplicity only major cost components and constraints. 14.1 INTRODUCTION A transformer must perform certain functions such as transforming power from one voltage level to another without overheating or without damaging itself when certain abnormal events occur, such as lightning strikes or short circuits. Moreover, it must have a reasonable lifetime (> 20 years) if operated under rated conditions. Satisfying these basic requirements still leaves a wide latitude in possible designs. A transformer manufacturer will therefore find it in its best economic interest to choose, within the limitations imposed by the constraints, that combination of design parameters which results in the lowest cost unit. To the extent that the costs and constraints can be expressed analytically in terms of the design variables, the mathematical theory of optimization with constraints can be applied to this problem, Optimization is a fairly large branch of mathematics with major specialized subdivisions such as linear programming, unconstrained optimization, and linear or non-linear equality or inequality constrained optimization. Transformer design optimization falls into the most general category of such methods, namely non-linear equality and inequality constrained optimization. In this area, there are no algorithms or iteration schemes which guarantee that a global optimum © 2002 by CRC Press
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COST MINIMIZATION 544 (in our case a minimum) will be found. Most of the algorithms proposed in the literature will converge to a local optimum with varying degrees of efficiency although even this is not guaranteed if one starts the iterations too far from a local optimum. Some insight is therefore usually required to find a suitable starting point for the iterations. Often past experience can serve as a guide. There is however a branch of optimization theory called geometric programming which does guarantee convergence to a global minimum, provided the function to be minimized, called the objective or cost function, and the constraints are expressible in a certain way [Duf67]. These restricted functional forms, called posynomials, will be discussed later. This method is very powerful if the problem functions conform to this type. Approximating techniques have been developed to cast other types of functions into posynomial form. Later versions of this
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