An extended dynamic programming (EDP) approach is developed to optimize the ac steady-state operation of radial electrical distribution systems (EDS). Based on the optimality principle of the recursive Hamilton-Jacobi-Bellman equations, the proposed EDP approach determines the optimal operation of the EDS by setting the values of the controllable variables at each time period. A suitable definition for the stages of the problem makes it possible to represent the optimal ac power flow of radial EDS as a dynamic programming problem, wherein the 'curse of dimensionality' is a minor concern, since the number of state and control variables at each stage is low and the time complexity of the algorithm grows linearly with the number of nodes of the EDS. The proposed EDP is applied to solve the economic dispatch of the DG units installed in a radial EDS. The effectiveness and the scalability of the EDP approach is illustrated using real-scale systems and comparisons with commercial programming solvers. Finally, generalizations to consider other EDS operation problems are also discussed.