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.

In this paper, the optimal schedule of dispatchable distributed generation (DG) units connected to radial electrical distribution systems (EDS) is solved using an extended dynamic programming approach. The objective of the optimal DG scheduling problem is to determine the hour-by-hour active generation output of each dispatchable DG unit, in order to minimize the total active power losses of the EDS and the generation costs. The proposed extended dynamic programming (EDP) is an advantageous approach because convexity is not required to obtain a global optimal solution, and the 'curse of dimensionality' is not a concern since the computational complexity of the algorithm grows linearly with the size of the network. Besides, the state variables have only two dimensions, one to represent the active power flows and the other to represent the nodal voltages. A 56-nodes MV distribution system with two dispatchable DG units is used to evaluate the performance of the proposed EDP approach, considering a deterministic and a stochastic case. A set of Monte Carlo simulations is used to analyze the influence of uncertainties. Results confirm that the proposed methodology is a suitable approach to unveil the best operation schedule for dispatchable DG units.