An aggregation method for improving Lagrangian relaxation-based auction implementation and generation scheduling

Lagrangian relaxation has been widely used in hydrothermal scheduling. A well recognized deficiency of Lagrangian relaxation is solution oscillation caused by the existence of integer decision variables and linear or piece-wise linear subproblem cost functions, resulting in slow convergence. When there are identical or similar subproblems, this deficiency becomes serious, intensifying the slow convergence and creating an arbitrary nature of primal solutions. The latter may cause difficulty in using LR to solve the Independent System Operator's auction problems because of the unfair treatment of identical or similar bids submitted. In this paper, an aggregation method is developed to reduce the oscillation of subproblem solutions in the primal space and to improve the convergence of the dual problem. For convex hydro subproblems, aggregation is performed as a convex combination of subproblem solutions across iterations. For mixed-integer thermal subproblems, the aggregation is applied to each operating region, and dynamic programming is then used to optimize the states across the time horizon. The multipliers are updated at the high level based on the aggregate solutions after solving a portion of subproblems rather than solving all the subproblems. The convergence is established. Compared with the solution obtained without aggregation, the new solution is less sensitive to the perturbation of multipliers, is more feasible, and generally leads to better feasible schedules. Identical or similar units could be differentiated as they are solved with different multipliers, and more commitments with the similar costs can be obtained. Alternative solutions are thus fairly selected in a probabilistic sense. Consequently the unfairness difficulty is alleviated, making LR practical in solving ISO's auction problem. Extensive numerical testing demonstrates the above properties of the method, and better schedules are obtained without increasing computational requirements.