Mathematical programming procedures to optimize the collector field subsystem of a power tower system

Abstract

The collector field of a solar tower system can be viewed as being composed of cells, each of which contains arrays of heliostats. Optimal design of a collector field involves determining the number, spacing and arrangement of heliostats within each cell so that the total cost per unit energy is minimized. This optimal design problem is formulated as a mathematical programming (MP) problem. This MP problem is then decomposed into two subproblems. The first subproblem is to determine the optimal spacing, arrangement, and amount of mirror surface in each cell so that the total energy collected is maximized. It is shown that the dynamic programming (DP) procedure can be employed to solve this subproblem. DP is particularly effective and efficient for this subproblem because it has the capability of determining the optimal design of one cell at a time, in a recursive fashion, until all cells are designed. This cellwise decomposition significantly simplifies the computational procedure. Furthermore, the optimal energy collected can be determined parametrically as a function of the total mirror area used in the field. This function is then utilized in the second problem: determination of the optimal total area so that the total cost over total energy, both functions of total area, is minimized. This yields a fractional programming problem. Mathematical structure of the above problems, advantages of using mathematical programming concepts, and future research needs are also discussed.