A Mathematical Model for the Efficiency Curves of Hydroelectric units

Hydropower generation is a function of the discharge of the generating units and the difference between the forebay and tailrace levels of the reservoir, and is subject to penstock head losses and to the generation unit efficiency factor. This factor is in turn a function of the turbined flow and the head in the reservoir, and is usually obtained from the hill curves - or hill diagrams - of a unit. In the mid / short-term operation planning of hydrothermal systems, the unit efficiency factors are usually considered constant. However, when a hydro producer aims at determining an hourly dispatch according to a given daily or weekly target set by the system operator, it is essential to consider variations in each unit efficiency factor with respect to operating conditions, in order to distribute the generation of the plant among units in a more economic way. This paper studies two mathematical models for the turbine- generator efficiency. Model coefficients are computed by a linear regression technique using points taken from the level sets of a typical hill curve. For each model, different variants are considered in order to satisfy some specific conditions. By analyzing each model's results, the variants that are most suitable for the dispatch problem of a hydro utility are chosen. For two important hydro plants in the Brazilian system, we give a comparative assessment between the generation obtained when considering an average unit efficiency factor and the generation resulting from considering the variation of efficiency with respect to the operating conditions of the unit.