arXiv subjects

Clémence Alasseur

Publications and source records attributed to Clémence Alasseur.

13 recordsLinked to original sources

arXiv2026

A Rank-Based Reward between a Principal and a Field of Agents: Application to Energy Savings

In this paper, we consider the problem of a Principal aiming at designing a reward function for a population of heterogeneous agents. We construct an incentive based on the ranking of the agents, so that a competition among the latter is initiated. We place ourselves in the limit setting of mean-field type interactions and prove the existence and uniqueness of the equilibrium distribution for a given reward, for which we can find an explicit representation. Focusing first on the homogeneous setting, we characterize the optimal reward function using a convex reformulation of the problem and provide an interpretation of its behaviour. We then show that this characterization still holds for a sub-class of heterogeneous populations. For the general case, we propose a convergent numerical method which fully exploits the characterization of the mean-field equilibrium. We develop a case study related to the French market of Energy Saving Certificates based on the use of realistic data, which shows that the ranking system allows to achieve the sobriety target imposed by the regulation.

math.OC↗

arXiv2024

Deep learning algorithms for FBSDEs with jumps: Applications to option pricing and a MFG model for smart grids

In this paper, we introduce various machine learning solvers for (coupled) forward-backward systems of stochastic differential equations (FBSDEs) driven by a Brownian motion and a Poisson random measure. We provide a rigorous comparison of the different algorithms and demonstrate their effectiveness in various applications, such as cases derived from pricing with jumps and mean-field games. In particular, we show the efficiency of the deep-learning algorithms to solve a coupled multi-dimensional FBSDE system driven by a time-inhomogeneous jump process with stochastic intensity, which describes the Nash equilibria for a specific mean-field game (MFG) problem for which we also provide the complete theoretical resolution. More precisely, we develop an extension of the MFG model for smart grids introduced in Alasseur, Campi, Dumitrescu and Zeng (Annals of Operations Research, 2023) to the case when the random jump times correspond to the jump times of a doubly Poisson process. We first provide an existence result of an equilibria and derive its semi-explicit characterization in terms of a system of FBSDEs in the linear-quadratic setting. We then compare the MFG solution to the optimal strategy of a central planner and provide several numerical illustrations using the deep-learning solvers presented in the first part of the paper.

math.NA↗

arXiv2024

Ergodic control of a heterogeneous population and application to electricity pricing

We consider a control problem for a heterogeneous population composed of agents able to switch at any time between different options. The controller aims to maximize an average gain per time unit, supposing that the population is of infinite size. This leads to an ergodic control problem for a "mean-field" Markov Decision Process in which the state space is a product of simplices, and the population evolves according to controlled linear dynamics. By exploiting contraction properties of the dynamics in Hilbert's projective metric, we prove that the infinite-dimensional ergodic eigenproblem admits a solution and show that the latter is in general non unique. This allows us to obtain optimal strategies, and to quantify the gap between steady-state strategies and optimal ones. In particular, we prove in the one-dimensional case that there exist cyclic policies -- alternating between discount and profit taking stages -- which secure a greater gain than constant-price policies. On numerical aspects, we develop a policy iteration algorithm with "on-the-fly" generated transitions, specifically adapted to decomposable models, leading to substantial memory savings. We finally apply our results on realistic instances coming from an electricity pricing problem encountered in the retail markets, and numerically observe the emergence of cyclic promotions for sufficient inertia in the customer behavior.

math.OC↗

arXiv2023

A mean field model for the development of renewable capacities

We propose a model based on a large number of small competitive producers of renewable energies, to study the effect of subsidies on the aggregate level of capacity, taking into account a cannibalization effect. We first derive a model to explain how long-time equilibrium can be reached on the market of production of renewable electricity and compare this equilibrium to the case of monopoly. Then we consider the case in which other capacities of production adjust to the production of renewable energies. The analysis is based on a master equation and we get explicit formulae for the long-time equilibria. We also provide new numerical methods to simulate the master equation and the evolution of the capacities. Thus we find the optimal subsidies to be given by a central planner to the installation and the production in order to reach a desired equilibrium capacity.

math.OC↗

arXiv2023

Quadratic Regularization of Bilevel Pricing Problems and Application to Electricity Retail Markets

We consider the profit-maximization problem solved by an electricity retailer who aims at designing a menu of contracts. This is an extension of the unit-demand envy-free pricing problem: customers aim to choose a contract maximizing their utility based on a reservation bill and multiple price coefficients (attributes). A basic approach supposes that the customers have deterministic utilities; then, the response of each customer is highly sensitive to price since it concentrates on the best offer. A second classical approach is to consider logit model to add a probabilistic behavior in the customers' choices. To circumvent the intrinsic instability of the former and the resolution difficulties of the latter, we introduce a quadratically regularized model of customer's response, which leads to a quadratic program under complementarity constraints (QPCC). This allows to robustify the deterministic model, while keeping a strong geometrical structure. In particular, we show that the customer's response is governed by a polyhedral complex, in which every polyhedral cell determines a set of contracts which is effectively chosen. Moreover, the deterministic model is recovered as a limit case of the regularized one. We exploit these geometrical properties to develop a pivoting heuristic, which we compare with implicit or non-linear methods from bilevel programming, showing the effectiveness of the approach. Throughout the paper, the electricity retailer problem is our guideline, and we present a numerical study on this application case.

math.OC↗

arXiv2023

A Quantization Procedure for Nonlinear Pricing with an Application to Electricity Markets

We consider a revenue maximization model, in which a company aims at designing a menu of contracts, given a population of customers. A standard approach consists in constructing an incentive-compatible continuum of contracts, i.e., a menu composed of an infinite number of contracts, where each contract is especially adapted to an infinitesimal customer, taking his type into account. Nonetheless, in many applications, the company is constrained to offering a limited number of contracts. We show that this question reduces to an optimal quantization problem, similar to the pruning problem that appeared in the max-plus based numerical methods in optimal control. We develop a new quantization algorithm, which, given an initial menu of contracts, iteratively prunes the less important contracts, to construct an implementable menu of the desired cardinality, while minimizing the revenue loss. We apply this algorithm to solve a pricing problem with price-elastic demand, originating from the electricity retail market. Numerical results show an improved performance by comparison with earlier pruning algorithms.

math.OC↗

arXiv2023

Decomposition of convex high dimensional aggregative stochasticcontrol problems

We consider the framework of convex high dimensional stochastic control problems, in which the controls are aggregated in the cost function. As first contribution, we introduce a modified problem, whose optimal control is under some reasonable assumptions an $\varepsilon$-optimal solution of the original problem. As second contribution, we present a decentralized algorithm whose convergence to the solution of the modified problem is established. Finally, we study the application of the developed tools in an engineering context, studying a coordination problem for large populations of domestic thermostatically controlled loads (TCLs)

math.OC↗

arXiv2021

Computation and implementation of an optimal mean field control for smart charging

This paper addresses an optimal control problem for a large population of identical plug-in electric vehicles (PEVs). The number of PEVs being large, the mean field assumption is formulated to describe the evolution of the PEVs population and its interaction with the central planner. The resulting problem of optimal control of partial differential equations (PDEs) is discretized. Using convex analysis tools, we show the existence of an optimal solution and the convergence of the Chambolle-Pock algorithm to a solution. The implementation of this optimal control to the finite population of PEVs is detailed and we illustrate our approach with two numerical examples.

math.OC↗

arXiv2021

MFG model with a long-lived penalty at random jump times: application to demand side management for electricity contracts

We consider an energy system with $n$ consumers who are linked by a Demand Side Management (DSM) contract, i.e. they agreed to diminish, at random times, their aggregated power consumption by a predefined volume during a predefined duration. Their failure to deliver the service is penalised via the difference between the sum of the $n$ power consumptions and the contracted target. We are led to analyse a non-zero sum stochastic game with $n$ players, where the interaction takes place through a cost which involves a delay induced by the duration included in the DSM contract. When $n \to \infty$, we obtain a Mean-Field Game (MFG) with random jump time penalty and interaction on the control. We prove a stochastic maximum principle in this context, which allows to compare the MFG solution to the optimal strategy of a central planner. In a linear quadratic setting we obtain an semi-explicit solution through a system of decoupled forward-backward stochastic differential equations with jumps, involving a Riccati Backward SDE with jumps. We show that it provides an approximate Nash equilibrium for the original $n$-player game for $n$ large. Finally, we propose a numerical algorithm to compute the MFG equilibrium and present several numerical experiments.

math.OC↗

arXiv2020

Optimal contracts under adverse selection for staple goods: efficiency of in-kind insurance

An income loss can have a negative impact on households, forcing them to reduce their consumption of some staple goods. This can lead to health issues and, consequently, generate significant costs for society. We suggest that consumers can, to prevent these negative consequences, buy insurance to secure sufficient consumption of a staple good if they lose part of their income. We develop a two-period/two-good principal-agent problem with adverse selection and endogenous reservation utility to model insurance with in-kind benefits. This model allows us to obtain semi-explicit solutions for the insurance contract and is applied to the context of fuel poverty. For this application, our model allows to conclude that, even in the least efficient scenario from the households point of view, i.e., when the insurance is provided by a monopoly, this mechanism decreases significantly the risk of fuel poverty of households by ensuring them a sufficient consumption of energy. The effectiveness of in-kind insurance is highlighted through a comparison with income insurance, but our results nevertheless underline the need to regulate such insurance market.

econ.GN↗

arXiv2020

A Principal-Agent approach to Capacity Remuneration Mechanisms

We propose to study electricity capacity remuneration mechanism design through a Principal-Agent approach. The Principal represents the aggregation of electricity consumers (or a representative entity), subject to the physical risk of shortage, and the Agent represents the electricity capacity owners, who invest in capacity and produce electricity to satisfy consumers' demand, and are subject to financial risks. Following the methodology of Cvitanic et al. (2017), we propose an optimal contract, from consumers' perspective, which complements the revenue capacity owners achieved from the spot energy market, and incentivizes both parties to perform an optimal level of investments while sharing the physical and financial risks. Numerical results provide insights on the necessity of a capacity remuneration mechanism and also show how this is especially true when the level of uncertainties on demand or production side increases.

econ.GN↗

arXiv2019

An adverse selection approach to power pricing

We study the optimal design of electricity contracts among a population of consumers with different needs. This question is tackled within the framework of Principal-Agent problems in presence of adverse selection. The particular features of electricity induce an unusual structure on the production cost, with no decreasing return to scale. We are nevertheless able to provide an explicit solution for the problem at hand. The optimal contracts are either linear or polynomial with respect to the consumption. Whenever the outside options offered by competitors are not uniform among the different type of consumers, we exhibit situations where the electricity provider should contract with consumers with either low or high appetite for electricity.

math.OC↗

arXiv2018

Peer-to-Peer Electricity Market Analysis: From Variational to Generalized Nash Equilibrium

We consider a network of prosumers involved in peer-to-peer energy exchanges, with differentiation price preferences on the trades with their neighbors, and we analyze two market designs: (i) a centralized market, used as a benchmark, where a global market operator optimizes the flows (trades) between the nodes, local demand and flexibility activation to maximize the system overall social welfare; (ii) a distributed peer-to-peer market design where prosumers in local energy communities optimize selfishly their trades, demand, and flexibility activation. We first characterizethe solution of the peer-to-peer market as a Variational Equilibrium and prove that the set of Variational Equilibria coincides with the set of social welfare optimal solutions of market design (i). We give several results that help understanding the structure of the trades at an equilibriumor at the optimum. We characterize the impact of preferences on the network line congestion and renewable energy waste under both designs. We provide a reduced example for which we give the set of all possible generalized equilibria, which enables to give an approximation of the price ofanarchy. We provide a more realistic example which relies on the IEEE 14-bus network, for which we can simulate the trades under different preference prices. Our analysis shows in particular that the preferences have a large impact on the structure of the trades, but that one equilibrium(variational) is optimal.

math.OC↗