Cette thèse est constituée de deux parties pouvant être lues indépendamment. Dans la première partie on s'intéresse à la modélisation mathématique du risque de liquidité. L'aspect étudié ici est la contrainte sur les dates des transactions, c'est-à-dire que contrairement aux modèles classiques où les investisseurs peuvent échanger les actifs en continu, on suppose que les transactions sont uniquement possibles à des dates aléatoires discrètes. On utilise alors des techniques de contrôle optimal (programmation dynamique, équations d'Hamilton-Jacobi-Bellman) pour identifier les fonctions valeur et les stratégies d'investissement optimales sous ces contraintes. Le premier chapitre étudie un problème de maximisation d'utilité en horizon fini, dans un cadre inspiré des marchés de l'énergie. Dans le deuxième chapitre on considère un marché illiquide à changements de régime, et enfin dans le troisième chapitre on étudie un marché où l'agent a la possibilité d'investir à la fois dans un actif liquide et un actif illiquide, ces derniers étant corrélés. Dans la deuxième partie on présente des méthodes probabilistes de quantification pour résoudre numériquement un problème de switching optimal. On considère d'abord une approximation en temps discret du problème et on prouve un taux de convergence. Ensuite on propose deux méthodes numériques de quantification : une approche markovienne où on quantifie la loi normale dans le schéma d'Euler, et dans le cas où la diffusion n'est pas contrôlée, une approche de quantification marginale inspirée de méthodes numériques pour le problème d'arrêt optimal.

Optimal transmission switching (OTS) enables us to remove selected transmission lines from service as a cost reduction method. A mixed integer programming (MIP) model has been proposed to solve the OTS problem based on the direct current optimal power flow (DCOPF) approximation. Previous studies indicated computational issues regarding the OTS problem and the need for a more accurate model. In order to resolve computational issues, especially in large real systems, the MIP model has been followed by some heuristics to find good, near optimal, solutions in a reasonable time. The line removal recommendations based on DCOPF approximations may result in poor choices to remove from service. We assess the quality of line removal recommendations that rely on DCOPF-based heuristics, by estimating actual cost reduction with the exact alternating current optimal power flow (ACOPF) model, using the IEEE 118-bus test system. We also define an ACOPF-based line-ranking procedure and compare the quality of its recommendations to those of a previously published DCOPF-based procedure. For the 118-bus system, the DCOPF-based line ranking produces poor quality results, especially when demand and congestion are very high, while the ACOPF-based heuristic produces very good quality recommendations for line removals, at the expense of much longer computation times. There is a need for approximations to the ACOPF that are accurate enough to produce good results for OTS heuristics, but fast enough for practical use for OTS decisions.

Optimal transmission switching (OTS) enables us to remove selected transmission lines from service as a cost reduction method. A mixed integer programming (MIP) model has been proposed to solve the OTS problem based on the direct current optimal power flow (DCOPF) approximation. Previous studies indicated computational issues regarding the OTS problem and the need for a more accurate model. In order to resolve computational issues, especially in large real systems, the MIP model has been followed by some heuristics to find good, near optimal, solutions in a reasonable time. The line removal recommendations based on DCOPF approximations may result in poor choices to remove from service. We assess the quality of line removal recommendations that rely on DCOPF-based heuristics, by estimating actual cost reduction with the exact alternating current optimal power flow (ACOPF) model, using the IEEE 118-bus test system. We also define an ACOPF-based line-ranking procedure and compare the quality of its recommendations to those of a previously published DCOPF-based procedure. For the 118-bus system, the DCOPF-based line ranking produces poor quality results, especially when demand and congestion are very high, while the ACOPF-based heuristic produces very good quality recommendations for line removals, at the expense of much longer computation times. There is a need for approximations to the ACOPF that are accurate enough to produce good results for OTS heuristics, but fast enough for practical use for OTS decisions.

Nous étudions le lien entre EDS rétrogrades et certains problèmes d'optimisation stochas- tique ainsi que leurs applications en finance. Dans la première partie, nous nous intéressons à la représentation par EDSR de problème d'optimisation stochastique séquentielle : le contrôle impul- sionnel et le switching optimal. Nous introduisons la notion d'EDSR contrainte à sauts et montrons qu'elle donne une représentation des solutions de problème de contrôle impulsionnel markovien. Nous lions ensuite cette classe d'EDSR aux EDSRs à réflexions obliques et aux processus valeurs de problèmes de switching optimal. Dans la seconde partie nous étudions la discrétisation des EDSRs intervenant plus haut. Nous introduisons une discrétisation des EDSRs contraintes à sauts utilisant l'approximation par EDSRs pénalisées pour laquelle nous obtenons la convergence. Nous étudions ensuite la discrétisation des EDSRs à réflexions obliques. Nous obtenons pour le schéma proposé une vitesse de convergence vers la solution continument réfléchie. Enfin dans la troisième partie, nous étudions un problème de liquidation optimale de portefeuille avec risque et coût d'exécution. Nous considérons un marché financier sur lequel un agent doit liquider une position en un actif risqué. L'intervention de cet agent influe sur le prix de marché de cet actif et conduit à un coût d'exécution modélisant le risque de liquidité. Nous caractérisons la fonction valeur de notre problème comme solution minimale d'une inéquation quasi-variationnelle au sens de la viscosité contrainte.

Local electricity markets can be defined broadly as 'future electricity market designs involving domestic customers, demand-side response and energy storage'. Like current deregulated electricity markets, these localised derivations present specific stochastic optimisation problems in which the dynamic and random nature of the market is intertwined with the physical needs of its participants. Moreover, the types of contracts and constraints in this setting are such that 'games' naturally emerge between the agents. Advanced modelling techniques beyond classical mathematical finance are therefore key to their analysis. This thesis aims to study contracts in these local electricity markets using the mathematical theories of stochastic optimal control and games. Chapter 1 motivates the research, provides an overview of the electricity market in Great Britain, and summarises the content of this thesis. It introduces three problems which are studied later in the thesis: a simple control problem involving demand-side management for domestic customers, and two examples of games within local electricity markets, one of them involving energy storage. Chapter 2 then reviews the literature most relevant to the topics discussed in this work. Chapter 3 investigates how electric space heating loads can be made responsive to time varying prices in an electricity spot market. The problem is formulated mathematically within the framework of deterministic optimal control, and is analysed using methods such as Pontryagin's Maximum Principle and Dynamic Programming. Numerical simulations are provided to illustrate how the control strategies perform on real market data. The problem of Chapter 3 is reformulated in Chapter 4 as one of optimal switching in discrete-time. A martingale approach is used to establish the existence of an optimal strategy in a very general setup, and also provides an algorithm for computing the value function and the optimal strategy. The theory is exemplified by a numerical example for the motivating problem. Chapter 5 then continues the study of finite horizon optimal switching problems, but in continuous time. It also uses martingale methods to prove the existence of an optimal strategy in a fairly general model. Chapter 6 introduces a mathematical model for a game contingent claim between an electricity supplier and generator described in the introduction. A theory for using optimal switching to solve such games is developed and subsequently evidenced by a numerical example. An optimal switching formulation of the aforementioned game contingent claim is provided for an abstract Markovian model of the electricity market. The final chapter studies a balancing services contract between an electricity transmission system operator (SO) and the owner of an electric energy storage device (battery operator or BO). The objectives of the SO and BO are combined in a non-zero sum stochastic differential game where one player (BO) uses a classic control with continuous effects, whereas the other player (SO) uses an impulse control (discontinuous effects). A verification theorem proving the existence of Nash equilibria in this game is obtained by recursion on the solutions to Hamilton-Jacobi-Bellman variational PDEs associated with non-zero sum controller-stopper games.

The Autonomic Power System (APS) grand challenge was a multi-disciplinary EPSRC-funded research project that examined novel techniques that would enable the transition between today's and 2050's highly uncertain and complex energy network. Being part of the APS, this thesis reports on the sub-project 'RR2: Avoiding High-Impact Low Probability events'. The goal of RR2 is to develop new algorithms for controlling risk exposure to high-impact low probability (Hi-Lo) events through the provision of appropriate risk-sensitive control strategies. Additionally, RR2 is concerned with new techniques for identifying and modelling risk in future energy networks, in particular, the risk of Hi-Lo events. In this context, this thesis investigates two distinct problems arising from energy risk management. On the one hand, we examine the problem of finding managerial strategies for exercising the operational flexibility of energy assets. We look at this problem from a risk perspective taking into account non-linear risk preferences of energy asset managers. Our main contribution is the development of a risk-sensitive approach to the class of optimal switching problems. By recasting the problem as an iterative optimal stopping problem, we are able to characterise the optimal risk-sensitive switching strategies. As byproduct, we obtain a multiplicative dynamic programming equation for the value function, upon which we propose a numerical algorithm based on least squares Monte Carlo regression. On the other hand, we develop tools to identify and model the risk factors faced by energy asset managers. For this, we consider a class of models consisting of superposition of Gaussian and non-Gaussian Ornstein-Uhlenbeck processes. Our main contribution is the development of a Bayesian methodology based on Markov chain Monte Carlo (MCMC) algorithms to make inference into this class of models. On extensive simulations, we demonstrate the robustness and efficiency of the algorithms to different data features. Furthermore, we construct a diagnostic tool based on Bayesian p-values to check goodness-of-fit of the models on a Bayesian framework. We apply this tool to MCMC results from fitting historical electricity and gas spot price data- sets corresponding to the UK and German energy markets. Our analysis demonstrates that the MCMC-estimated models are able to capture not only long- and short-lived positive price spikes, but also short-lived negative price spikes which are typical of UK gas prices and German electricity prices. Combining together the solutions to the two problems above, we strive to capture the interplay between risk, uncertainty, flexibility and performance in various applications to energy systems. In these applications, which include power stations, energy storage and district energy systems, we consistently show that our risk management methodology offers a tradeoff between maximising average performance and minimising risk, while accounting for the jump dynamics of energy prices. Moreover, the tradeoff is achieved in such way that the benefits in terms of risk reduction outweigh the loss in average performance.

Cette thèse est composée de trois parties. Dans la première nous montrons l'existence et l'unicité de la solution continue et à croissance polynomiale, au sensviscosité, du système non linéaire de m équations variationnelles de type intégro-différentiel à obstacles unilatéraux interconnectés. Ce système est lié au problème du switching optimal stochastique lorsque le bruit est dirigé par un processus de Lévy. Un cas particulier du système correspond en effet à l’équation d’Hamilton-Jacobi-Bellman associé au problème du switching et la solution de ce système n’est rien d’autre que la fonction valeur du problème. Ensuite, nous étudions un système d’équations intégro-différentielles à obstacles bilatéraux interconnectés. Nous montrons l’existence et l’unicité des solutions continus à croissance polynomiale, au sens viscosité, des systèmes min-max et max-min. La démarche conjugue les systèmes d’EDSR réfléchies ainsi que la méthode de Perron. Dans la dernière partie nous montrons l’égalité des solutions des systèmes max-min et min-max d’EDP lorsque le bruit est uniquement de type diffusion. Nous montrons que si les coûts de switching sont assez réguliers alors ces solutions coïncident. De plus elles sont caractérisées comme fonction valeur du jeu de switching de somme nulle. / There are three main results in this thesis. The first is existence and uniqueness of the solution in viscosity sense for a system of nonlinear m variational integral-partial differential equations with interconnected obstacles. From the probabilistic point of view, this system is related to optimal stochastic switching problem when the noise is driven by a Lévy process. As a by-product we obtain that the value function of the switching problem is continuous and unique solution of its associated Hamilton-Jacobi-Bellman system of equations. Next, we study a general class of min-max and max-min nonlinear second-order integral-partial variational inequalities with interconnected bilateralobstacles, related to a multiple modes zero-sum switching game with jumps. Using Perron’s method and by the help of systems of penalized unilateral reflected backward SDEs with jumps, we construct a continuous with polynomial growth viscosity solution, and a comparison result yields the uniqueness of the solution. At last, we deal with the solutions of systems of PDEs with bilateral inter-connected obstacles of min-max and max-min types in the Brownian framework. These systems arise naturally in stochastic switching zero-sum game problems. We show that when the switching costs of one side are smooth, the solutions of the min-max and max-min systems coincide. Furthermore, this solution is identified as the value function of the zero-sum switching game.