The 11th of December, 2020, Mathieu Besançon succesfully defended his thesis titled
Bilevel models for Demand Response in Smart Grids.
The jury was composed by:
- Miguel F. Anjos, Professor, University of Edinburgh
- Luce Brotcorne, Directrice de Recherche, INRIA
- Ivana Ljubic, Professor, ESSEC
- Michel Gendreau, Professor, Polytechnique Montréal
- Roland Malhamé, Professor, Polytechnique Montréal
- Gilles Savard, Professor, Polytechnique Montréal
- Martin Schmidt, Professor, Universität Trier
- Frédéric Semet, Professor, Centrale Lille
- Wolfram Wiesemann, Professor, Imperial College Business School
Thesis abstract: This thesis investigates mathematical optimization models with a bilevel structure and their application to price-based Demand Response in smart grids.
The increasing penetration of renewable power generation has put power systems under higher tension. The stochastic and distributed nature of wind and solar generation increases the need for adjustment of the conventional production to the net demand, which corresponds to the demand minus the renewable generation.
Demand Response as a means to this adjustment of demand and supply is receiving growing attention. Instead of achieving the adjustment thanks to generation units, it consists in leveraging the flexibility of a part of the demand, thus changing the aggregated demand curve in time.
The first part of this thesis focuses on a Time-and-Level-of-Use Demand Response system based on a price of energy that depends on the time of consumption, but also on a capacity that is self-determined by each user of the program. This capacity is booked by the user for a specific time frame, and determines a limit for energy consumption. Several key properties of the pricing system are studied, focusing on the perspective of the supplier setting the pricing components. The supplier anticipates the decision of the customers to the prices they set, the sequential decision created by this situation is modelled
as a Stackelberg or Leader-Follower game formulated as a bilevel optimization problem.
Bilevel optimization problems embed the optimality condition of other optimization problems in their constraints. Their range of applications includes optimization for engineering, economics, power systems, or security games. The inherent computational difficulty of bilevel problems has motivated the development of customized algorithms for their resolution.
In the second part of the contributions, a variant of the bilevel optimization problem is developed, where the upper level protects its feasibility against deviations of the lower-level solution from optimality. More specifically, this near-optimal robust model maintains the upper-level feasibility for any lower-level solution that is feasible and almost optimal for the lower-level. This model introduces a robustness notion that is specific to multilevel optimization.
We derive a single-level closed-form reformulation when the lower level is a convex optimization problem and an extended formulation when it is linear.
The near-optimal robust bilevel problem is a generalization of the optimistic bilevel problem and is in general harder to solve. Nonetheless, we obtain
complexity results for the near-optimal robust bilevel problem, showing it belongs to the same complexity class as the optimistic problem under mild assumptions. Finally, we design exact and heuristic solution methods that significantly improve the solution time of the extended formulation.