Mathematical Problems in Engineering
Volume 8 (2002), Issue 3, Pages 181-196
A new formulation of the equivalent thermal in optimization of
University of Oviedo, Department of Mathematics, E.U.I.T.I, C./Manuel Llaneza s/n, 33208 Gijón, Asturias, Spain
Received 17 May 2001; Revised 10 October 2001
Copyright © 2002 L. Bayón et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, we revise the classical formulation of the problem depriving it of the concepts that are superfluous from the mathematical point of view. We observe that a number of power stations can be substituted by a single one that behaves equivalently to the entire set. Proceeding in this way, we obtain a variational formulation in its purest sense (without restrictions). This formulation allows us to employ the theory of calculus of variations to the highest degree. We then calculate the equivalent minimizer in the case where the cost functions are second-order polynomials. We prove that the equivalent minimizer is a second-order polynomial with piece-wise
constant coefficients. Moreover, it belongs to the class . Finally, we present various examples prompted by real systems and perform the proposed algorithms using Mathematica.