# Optimal prediction games in local electricity markets

• Randall Martyr

Student thesis: Phd

### Abstract

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.
Date of Award 1 Aug 2016 English The University of Manchester John Moriarty (Supervisor) & Goran Peskir (Supervisor)

### Keywords

• contracts for difference
• energy storage
• demand response
• balancing services
• electricity market reform
• electricity markets
• optimal control
• optimal impulse control
• optimal switching
• stochastic games
• optimal stopping
• stochastic control
• power system economics

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