**5.1. Optimization of the pumps' schedule in the Fátima system**

The feasibility of the developed HGA model in the search of the best operational strategy for a lowest energy cost in the real Fátima system was analysed. Two algorithms were devel‐ oped and linked to the GA. The first one, a repair algorithm from an analysis done in the unfeasible solutions generated by the GA, alters the decision variables in the attempt of making these solutions feasible. After finishing the generations of the GA, the second algo‐ rithm acts in these solutions, making a local search in the attempt of finding optimal locals.

The efficiency of the algorithm developed HGA in the search of the solution with lowest op‐ erational cost is confirmed, whereas the convergence occurred six times faster. One of the biggest limitations of the GA is the treatment of problems with high quantity of constraints. The application of penalties only allows the identification of unfeasible solutions. In prob‐ lems of this kind it is probable that along the candidates' generation, the quantity of unfeasi‐ ble candidates does not decrease, making the search of good solutions very difficult. With the implementation of repair algorithms, the appearance of super-candidates occurs in less time, since the alterations in the individuals are done directly in its problematic genes.

An evaluation analysis about the necessity of use genetic operators, when these algorithms are applied directly in a large set of solutions generated randomly, also shows final good results. To determine the best strategy among thousands of possible solutions it must also be taken into consideration the hydraulic reliability of the system.

The HGA model presented can be implemented in any network. Furthermore, its applica‐ tion is practical and useful, being able to be used by water supply companies, making easier the best decision aiming at the energy efficiency in pumping systems.
