Prices stabilization for inexact unit-commitment problems

A widespread and successful approach to tackle unit-commitment problems is constraint decomposition: by dualizing the linking constraints, the large-scale nonconvex problem decomposes into smaller independent subproblems. The dual problem consists then in finding the best Lagrangian multiplier (the optimal “price”); it is solved by a convex nonsmooth optimization method. Realistic modeling of technical production constraints makes the subproblems themselves difficult to solve exactly. Nonsmooth optimization algorithms can cope with inexact solutions of the subproblems. In this case however, we observe that the computed dual solutions show a noisy and unstable behaviour, that could prevent their use as price indicators. In this paper, we present a simple and easy-to-implement way to stabilize dual optimal solutions, by penalizing the noisy behaviour of the prices in the dual objective. After studying the impact of a general stabilization term on the model and the resolution scheme, we focus on the penalization by discrete total variation, showing the consistency of the approach. We illustrate our stabilization on a synthetic example, and real-life problems from EDF (the French Electricity Board).