The control of reactive generation in a hydroelectric system so as to minimize the transmission losses for specified real-power production and subject to inequality constraints of voltage and reactive production is discussed. The resulting optimization problem is one of nonlinear programming. Because the Bonneville Power Administration has efficient computer programs for solving the power flow equations, the most expedient approach to obtaining the optimum values for reactive generation consists of adjusting the inputs to the power flow program iteratively by means of an algorithm based on the dual (Lagrange) variables resulting from the Kuhn and Tucker theorem of nonlinear programming. A simple derivation of this theorem is given and the significance of the dual variables is discussed. These variables, which are easily computed once the solution of the power flow equations is available, serve a variety of additional purposes related to tariffication, short-term planning of equipment additions, optimization of real production and sales, and accuracy requirements of the measurement and telemetry subsystems for real-time control. The resulting optimization procedure leads to an acceptable scheme for the on-line control of reactive power; the transmission losses are reduced by 3 to 4%.