The paper describes a new and rigorous method for determining the mode of operation of an electrical power system that minimizes the operational cost. The power system consists ofMthermal generating units and one constant-head hydroplant. The operational cost includes the fixed and variable start-up and running costs of the thermal units. The method uses the maximum principle of Pontryagin to determine the generation levels of the operating units, to devise additional criteria for fathoming a vertex in the branch and bound algorithm, and to reduce the number of units considered for shut down. The choice of which units to shut down from those suggested by the maximum principle is done by branch and bound.