ALASSEUR Clemence

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.

This thesis is composed of two independent parts. The first part focuses on the application of the Principal-Agent problem (c.f. Cvitanic & Zhang (2013) and Cvitanic. et al. (2018)) for solving modeling problems in energy markets. The second one deals with the joint laws of a martingale and its current maximum.We first focus on the electricity capacity market, and in particular capacity remuneration mechanisms. Given the increasing share of renewable energies in the electricity production, "classical" power plants (e.g. gas or coal) are less and less used, which makes them unprofitable and not economically viable. However, their closure would expose consumers to the risk of a blackout in the event of a peak in electricity demand, since electricity cannot be stored. Thus, generation capacity must always be maintained above demand, which requires a "capacity payment mechanism" to remunerate seldom used power plants, which can be understood as an insurance to be paid against electricity blackouts.We then address the issue of incentives for decarbonization. The objective is to propose a model of an instrument that can be used by a public agent (the state) to encourage the different sectors to reduce their carbon emissions in a context of moral hazard (where the state does not observe the effort of the actors and therefore cannot know whether a decrease in emissions comes from a decrease in production and consumption or from a management effort. The second part (independent) is motivated by model calibration and arbitrage on a financial market with barrier options. It presents a result on the joint laws of a martingale and its current maximum. We consider a family of probabilities in dimension 2, and we give necessary and sufficient conditions ensuring the existence of a martingale such that its marginal laws coupled with those of its current maximum coincide with the given probabilities.We follow the methodology of Hirsch and Roynette (2012) based on a martingale construction by DHS associated with a well-posed Fokker-Planck PDE verified by the given marginal laws under regularity assumptions, then in a general framework with regularization and boundary crossing.

In this thesis, we are mainly interested in three research topics, relatively independent, but nevertheless related through the thread of interactions and incentives, as highlighted in the introduction constituting the first chapter.In the first part, we present extensions of contract theory, allowing in particular to consider a multitude of players in principal-agent models, with drift and volatility control, in the presence of moral hazard. In particular, Chapter 2 presents a continuous-time optimal incentive problem within a hierarchy, inspired by the one-period model of Sung (2015) and enlightening in two respects: on the one hand, it presents a framework where volatility control occurs in a perfectly natural way, and, on the other hand, it highlights the importance of considering continuous-time models. In this sense, this example motivates the comprehensive and general study of hierarchical models carried out in the third chapter, which goes hand in hand with the recent theory of second-order stochastic differential equations (2EDSR). Finally, in Chapter 4, we propose an extension of the principal-agent model developed by Aïd, Possamaï, and Touzi (2019) to a continuum of agents, whose performances are in particular impacted by a common hazard. In particular, these studies guide us towards a generalization of the so-called revealing contracts, initially proposed by Cvitanić, Possamaï and Touzi (2018) in a single-agent model.In the second part, we present two applications of principal-agent problems to the energy domain. The first one, developed in Chapter 5, uses the formalism and theoretical results introduced in the previous chapter to improve electricity demand response programs, already considered by Aïd, Possamaï and Touzi (2019). Indeed, by taking into account the infinite number of consumers that a producer has to supply with electricity, it is possible to use this additional information to build the optimal incentives, in particular to better manage the residual risk implied by weather hazards. In a second step, chapter 6 proposes, through a principal-agent model with adverse selection, an insurance that could prevent some forms of precariousness, in particular fuel precariousness.Finally, we end this thesis by studying in the last part a second field of application, that of epidemiology, and more precisely the control of the diffusion of a contagious disease within a population. In chapter 7, we first consider the point of view of individuals, through a mean-field game: each individual can choose his rate of interaction with others, reconciling on the one hand his need for social interactions and on the other hand his fear of being contaminated in turn, and of contributing to the wider diffusion of the disease. We prove the existence of a Nash equilibrium between individuals, and exhibit it numerically. In the last chapter, we take the point of view of the government, wishing to incite the population, now represented as a whole, to decrease its interactions in order to contain the epidemic. We show that the implementation of sanctions in case of non-compliance with containment can be effective, but that, for a total control of the epidemic, it is necessary to develop a conscientious screening policy, accompanied by a scrupulous isolation of the individuals tested positive.

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 exibility 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 exibility activation. We first characterize the 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 equilibrium or at the optimum. We characterize the impact of preferences on the network line congestion and renewable energy surplus 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 of anarchy. We provide a more realistic example which relies on the IEEE 14-bus network, for which we can simulate the trades under dierent 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. Finally, the learning mechanism needed to reach an equilibrium state in the peer-to-peer market design is discussed together with privacy issues.

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We consider a stylized model for a power network with distributed local power generation and storage. This system is modeled as network connection a large number of nodes, where each node is characterized by a local electricity consumption, has a local electricity production (e.g. photovoltaic panels), and manages a local storage device. Depending on its instantaneous consumption and production rates as well as its storage management decision, each node may either buy or sell electricity, impacting the electricity spot price. The objective at each node is to minimize energy and storage costs by optimally controlling the storage device. In a non-cooperative game setting, we are led to the analysis of a non-zero sum stochastic game with $N$ players where the interaction takes place through the spot price mechanism. For an infinite number of agents, our model corresponds to an Extended Mean-Field Game (EMFG). In a linear quadratic setting, we obtain and explicit solution to the EMFG, we show that it provides an approximate Nash-equilibrium for $N$-player game, and we compare this solution to the optimal strategy of a central planner.

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We consider the framework of high dimensional stochastic control problem, 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 ε-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 to a problem of coordination of energy consumption and production of domestic appliances.

Power producers are interested in valuing their power plant production. By trading into forward contracts, we propose to reduce the contingency of the associated income considering the fixed costs and using an asymmetric risk criterion. In an asymptotic framework, we provide an optimal hedging strategy through a solution of a nonlinear partial differential equation. As a numerical experiment, we analyze the impact of the fixed costs structure on the hedging policy and the value of the assets.

No summary available.

We consider a stylized model for a power network with distributed local power generation and storage. This system is modeled as network connection a large number of nodes, where each node is characterized by a local electricity consumption, has a local electricity production (e.g. photovoltaic panels), and manages a local storage device. Depending on its instantaneous consumption and production rates as well as its storage management decision, each node may either buy or sell electricity, impacting the electricity spot price. The objective at each node is to minimize energy and storage costs by optimally controlling the storage device. In a non-cooperative game setting, we are led to the analysis of a non-zero sum stochastic game with $N$ players where the interaction takes place through the spot price mechanism. For an infinite number of agents, our model corresponds to an Extended Mean-Field Game (EMFG). In a linear quadratic setting, we obtain and explicit solution to the EMFG, we show that it provides an approximate Nash-equilibrium for $N$-player game, and we compare this solution to the optimal strategy of a central planner.

The growing demand for broadband satellite services and the congestion of systems operating at traditional frequencies are driving the development of new systems at transmission bands above 10 GHz. However, the behavior of the LMSC (Land Mobile Satellite Channel) is not well known and needs to take into account atmospheric disturbances such as rain. The mobility of antennas for transmission and reception is also part of the problems of satellite services. To design new systems and methods of compensating for losses by dynamic adaptation, models of the channel and rainfall are then necessary. Our work brings first an analysis of the satellite propagation channel at Ku and Ka frequencies by studying on the one hand the normalized received power and on the other hand the precipitation rate time series. We then propose models as well as methods to extract their parameters for these two types of signals. Two approaches, based on MCMC (Monte Carlo Markov Chain) tools, allow a segmentation of the normalized power of the channel as well as the extraction of the parameters of the underlying hidden Markov model. A procedure for evaluating a two-level Markov chain to model the precipitation rate signal is also described. Finally the developed methods are applied to experimental data and provide Markov models of the satellite channel power signal and precipitation rates. The comparison of first and second order statistics between the models and the measurements attests to their quality.