# 5ème Journée COSMOS

## Onglets principaux

**Lieu : **

La journée aura lieu à INRIA Paris, Salle Jacques-Louis Lions 1 (rez-de-chaussée).

Adresse : 2 rue Simone Iff à Paris dans le douzième arrondissement, proche de la gare de Lyon.

Les infos pour venir sont données là.

**Thématiques**

- optimisation stochastique
- théorie de jeux
- tarification
- revenue management
- applications (réseaux, energie, ... )

**Inscription**

La journée est gratuite mais pour des raisons d'organisations (notamment pour rentrer dans les locaux), nous vous demandons de vous inscrire. Pour s'inscrire.

**Programme**

09:30 - 10:00 Accueil

10:00 - 10:45 Clémence Alasseur (EDF) - **An adverse selection approach to power pricing**

10:45 - 11:00 Pause

11:00 - 11:45 René Aïd (U. Paris Dauphine) - **A Principal-Agent model for pricing electricity demand volatility **

12:00 - 13:30 Déjeuner

13:30 - 14:15 Benjamin Heymann (ENSTA ParisTech) - **Mechanism Design and Auctions for Electricity Network. **

14:15 - 15:00 Bruno Gaujal (Inria Grenoble Rhône-Alpes) - **A Distributed Algorithm to Compute Nash Equilibria in Potential Games**

15:00 - 15:30 Pause

15:30 - 16:15 Vincent Leclère (CERMICS, ENPC) - **Introduction to Stochastic Programming**

16:15 - 17:00 Dominique Quadri (LRI, Université Paris-Sud) - **Mathematical Programming with Constraints Equilibrium: Last-mile delivery services design under stochastic user equilibrium**

**Résumés **

**An adverse selection approach to power pricing**

*Clémence Alasseur (EDF) *

In this work, we study an optimal tarification problem for an electricity provider facing a distribution of clients having different appetences for electricity consumption. We formulate the problem in a non-standard problem of calculus of variations, which we solve explicitly in several cases. This is a joint work with Ivar Ekeland (U. Paris Dauphine), Romuald Elie (U. Marne-la-Vallée), Nicolas Hernandez (U. Paris Dauphine, U. De Chile) and Dylan Possamaï (U. Paris Dauphine).

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**A Principal-Agent model for pricing electricity demand volatility**

*René Aïd (U. Paris Dauphine)*

The development of renewable energy sources for electricity generation in the electric systems are renewed the interest for demand response programs. Indeed, the volatility of renewable energies compels systems operator and electric utilities to increase their storage capacity to be able to cope with these important variations over small time steps. Instead of using a physical storage solution, we propose a model of demand pricing that allow a producer to incite a consumer to smooth her consumption over time. We use a Principal-Agent framework where the agent’s consumption volatility is controlled, find the optimal contract and show with numerical illustrations how the agent’s consumption volatility is reduced.

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**Mechanism Design and Auctions for Electricity Network**

*Benjamin Heymann (ENSTA ParisTech)*

We present some key aspects of wholesale electricity markets modeling and more specifically focus our attention on auctions and mechanism design. Some of the results arising from those models are the computation of an optimal allocation for the Independent System Operator, the study of the equilibria (existence and unicity in particular) and the design of mechanisms to increase the social surplus.

From a more general perspective, this field of research provides clues to discuss how wholesale electricity market should be regulated. We start with a general introduction and then present some results the authors obtained recently. We also briefly expose some undergoing related work. As an illustrative example, a section is devoted to the computation of the Independent System Operator response function for a symmetric binodal setting with piece-wise linear production cost functions.

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**A Distributed Algorithm to Compute Nash Equilibria in Potential Games**

*Bruno Gaujal (Inria Grenoble Rhône-Alpes)*

We consider a finite potential game and our goal is to design a distributed algorithm (executed by each user) that learns the Nash equilibria. Each user computes its strategy according to a procedure that should satisfy:

- it only uses in-game information (stateless);

- it does not require time-coordination between the users (asynchronous);

- it tolerates outdated measurements (delay-oblivious);

- it tolerates random perturbations on the measures (robust),

- and it converges fast even if the number of players is large (scalable).

The construction of this algorithm is based on a novel penalty-based continuous dynamics and its stochastic approximation.

This is a joined work with Pierre Coucheney and Panayotis Mertikopoulos.

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**Introduction to Stochastic Programming**

*Vincent Leclère (CERMICS, ENPC) *

Stochastic programming is an approach for modeling optimization problems that involve uncertainty. Whereas deterministic optimization problems are formulated with known parameters, real world problems almost invariably include parameters which are unknown at the time a decision should be made. When the parameters are uncertain, but assumed to lie in some given set of possible values, one might seek a solution that is feasible for all possible parameter choices and optimizes a given objective function. Stochastic programming models are similar in style but try to take advantage of the fact that probability distributions governing the data are known or can be estimated. Often these models apply to settings in which decisions are made repeatedly in essentially the same circumstances, and the objective is to come up with a decision that will perform well on average.

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**Mathematical Programming with Constraints Equilibrium: Last-mile delivery services design under stochastic user equilibrium**

*Dominique Quadri (LRI, Université Paris-Sud) *

We consider a bi-level formulation for e-commerce delivery services pricing. At the upper level, the provider controls services’ tariffs. At the lower level, customers react by choosing their delivery service according to a utility function. In addition to tariffs, the utility function includes a congestion measure depending on the service. Due to correlation between services of the same family we use a nested Logit model and compute the resulting stochastic user equilibrium. A sensitivity analysis of the SUE is then conducted, it gives explicit expression of the derivatives of customers’ decisions with respect to services’ tariffs. Based on a local search that exploits the derivatives information, a new heuristic algorithm for the bi-level services design problem is developed and compared to others existing approaches.

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