A new formulation of the equivalent thermal in optimization of hydrothermal systems

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 C-1. Finally, we present various examples prompted by real systems and perform the proposed algorithms using Mathematica.