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Preliminaries on electricity markets
In this paper we consider a short-term electricity market, populated with small agents with identical characteristics. These agents face uncertain demand or supply at some future time, and use the electricity market to manage the associated risk. While our primary interest is to study the impact of increasing renewable penetration on intraday market prices, the market participants may in principle represent both renewable producers with uncertain generation forecasts and industrial consumers / utilities with uncertain demand. To simplify the language and notation, in the sequel, unless speci ed otherwise, we will refer to forecasts of all agents as demand forecasts (if the agent is a producer, its demand forecast will therefore be negative). In most countries, the short-term electricity markets have the following structure (the speci c times correspond to the EPEX Spot/Intraday market):
• The day-ahead market is a one-o trading venue, where the agents may make bids until 12PM (noon) on the day preceding the delivery day. At 12:55 the price is xed using the merit order mechanism and the market clears. A major part of the electricity production is sold on the day-ahead market.
• At 3PM on the day preceding the delivery day, the intraday market opens, allowing con-tinuous trading for each quarter-hour of the delivery day. The intraday market has higher trading cost than the day-ahead market, and is mainly used by market participants to adjust their day-ahead positions following forecast updates.
• 15 minutes before delivery the intraday trading for the given delivery period closes. At this point, negative production imbalances must be compensated to the market operator at the ’high imbalance settlement price’, which is higher than the last intraday price, and for pos-itive production imbalances, the producer is compensated at the ’low imbalance settlement price’, which is lower than the intraday market price1. In addition, high imbalances carry a reputational cost for the market participant. To avoid paying the imbalance penalties, the aggregate position (day-ahead plus intraday) held by the agent at the delivery time must therefore be equal to the realized demand.
To represent this market structure in a simpli ed way, we consider a xed delivery period, starting at time T , and assume that the day-ahead market allows agents to trade instantaneously, without transaction costs, at time t = 0, at price denoted by S0. Then, between t = 0 and t = T , the agents may trade in the intraday market, at price (Pt)0 t T , which contains a market impact component, and subject to transaction costs. Finally, at time T , if there is an imbalance, the agents must purchase the missing amount /sell the extra amount of electricity at price ST without transaction cost, and in addition, pay a penalty depending on the absolute value of the imbalance. In the following section we provide details of the model and compute explicitly the optimal strategies of the agents and the equilibrium intraday market price.
Optimal trading strategies and equilibrium price
In this section we introduce our model of electricity market and derive explicit expressions for the equilibrium price and optimal equilibrium trading strategies of the agents. We consider both the N-agent setting (Section 3.3.1) and the mean- eld game setting (Section 3.3.2). Section 3.3.3 clari es the relationship between the equilibrium strategies and prices in the N-agent market and those of the mean eld game limit.
N-player setting
In this section we assume that in the market there are N identical agents, and we denote by it the position of i-th agent at time t. As is common in optimal execution literature, we assume that the position of i-th agent is an absolutely continuous process, and we de ne the rate of trading _it. We introduce a ltered probability space ( ; F; F := (Ft)t2[0;T ]; P) to which all processes are adapted, and which models the information available in the market to all the agents. The position of the i-th agent at time t is given by it = i0 + R0t _isds with i0 2 F0 denoting the position of the agent in the day-ahead market. The fundamental electricity price process is denoted by (St)t2[0;T ], where S0 corresponds to the day-ahead market price and St for 0 < t < T denotes the intraday market price net of the price impact component. The intraday market price with the price impact component is denoted by (PtN )t2[0;T ]. The strategies of the becoming-a-balance-responsible-party/Imbalance-settlement-price.html for details agents impact the market price PtN as follows: where tN = Pi=1 ti is the average position of the agents and a is a constant. The parameter N N describes the size of the market (number of agents), it is therefore natural that the trading strategy of each agent has an e ect of order of 1=N on the market price. The permanent component of the price impact of trades in our model is thus linear, which is the only shape compatible with the absence of arbitrage, see Huberman and Stanzl (2004); Gatheral (2010). On the other hand, the transient component of market impact is not modelled directly. Literature on market microstructure mostly shows that metaorders have a concave transient impact on prices (see Bershova and Rakhlin (2013), Bacry et al. (2015), Bucci et al. (2020) and Bouchaud (2010)). However, for the sake of simplicity and in order to obtain an analytical solution for our model, we choose a linear impact function as in the seminal papers by Almgren and Chriss (1999, 2001) and many other more recent papers, including Aid et al. (2016) in the context of electricity markets. The transient component of the market impact is taken into account indirectly, via a trading cost penalty.
Relationship between N-player setting and MFG setting
In this section, we study the relationship between the equilibrium strategies and prices in the N-agent market and those of the mean eld game limit, and prove the following results.
• The market price and the agent’s strategy in the N-agent model converge to their respective mean eld values as N ! 1. This shows that to understand the behavior of agents and prices in the realistic N-agent market, one can use the mean- eld game model, which does not require the knowledge of individual forecasts, but only that of the common information ltration.
• An approximate equilibrium (« -Nash equilibrium) in the N-player setting may be con-structed from the MFG solution. In other words, an agent trading in the N-agent market may construct a strategy whose gain is su ciently close to the optimal equilibrium gain using the mean- eld game solution, which does not require the knowledge of the private forecasts of the other agents. To address these questions, we need to make more precise assumptions on the probabilistic setup of the problem. In particular, since we would like to study the convergence of the N-agent problem as N ! 1, we consider an in nity of agents. In addition, all N-agent problems and the mean eld problem must be de ned on the same probability space.
• The process S adapted to the ltration F0 and satis es (3.2).
• The processes (Xi)1i=1 are square integrable F-martingales.
• There exists a square intergrable F-martingale X, such that for all i 1, and all t 2 [0; T ], almost surely, E[XtijFt0] = Xt.
Table of contents :
1 Introduction
1.1 General context and motivation
1.1.1 Energy and technology transition
1.1.2 Successive short term electricity markets: a brief overview
1.1.3 Main practical challenges of the thesis
1.2 Theoretical background
1.2.1 Stochastic games
1.2.2 Probabilistic forecasts
1.2.3 Neural networks and PAC-Bayesian inequalities
1.3 Main contributions of the thesis
1.3.1 Optimal strategies for renewable producers
1.3.2 Price formation and main characteristics of the intraday market
1.3.3 Probabilistic forecasts in sequential decision making problems
1.3.4 Learning by aggregated shallow neural networks
1.4 Outline of the thesis
2 Introduction (Francais)
2.1 Context general et motivation
2.1.1 Une transition energetique et technologique
2.1.2 Les marches successifs court terme de l’electricite: un bref apercu
2.1.3 Les principaux enjeux pratiques de la these
2.2 Cadre theorique
2.2.1 Jeux stochastiques
2.2.2 Previsions probabilistes
2.2.3 Reseaux de neurones et inegalites PAC-Bayesiennes
2.3 Principales contributions de la these
2.3.1 Strategies optimales pour des producteurs renouvelables
2.3.2 Formation des prix et principales caracteristiques du marche infrajournalier
2.3.3 Previsions probabilistes dans des problemes de decision sequentiels
2.3.4 Apprentissage par reseaux de neurones agreges a une couche cachee
2.4 Structure de la these
3 Price formation and optimal trading in intraday electricity markets
3.1 Introduction
3.2 Preliminaries on electricity markets
3.3 Optimal trading strategies and equilibrium price
3.3.1 N-player setting
3.3.2 Mean-eld game setting
3.3.3 Relationship between N-player setting and MFG setting
3.4 Intraday electricity prices: theoretical insights
3.4.1 Price impact of forecast adjustments
3.4.2 Volatility and Samuelson’s eect
3.4.3 Price-forecast covariance
3.4.4 Trading costs
3.5 Empirical results and numerical illustrations
3.5.1 Stylized features of intraday electricity market prices
3.5.2 Numerical illustration of our model
3.6 Appendix
3.6.1 Proof of Theorem
3.6.2 Proofs of Propositions 19 and
4 Price formation and optimal trading in intraday electricity markets with a major player
4.1 Introduction
4.2 Preliminaries
4.3 A game of a major and minor agents
4.4 Approximate Nash equilibrium in the N-player Stackelberg game
4.5 Numerical illustration
4.6 Conclusion
4.7 Appendi
4.7.1 Proofs of Lemma 29 and Proposition
4.7.2 Proof of Theorem
4.7.3 Proofs of Corollary 34 and Corollary
4.7.4 Proof of Proposition
5 Decision making with dynamic probabilistic forecasts
5.1 Introduction
5.2 Modeling probabilistic forecasts
5.2.1 Forecast of a real-valued quantity
5.2.2 Forecast of a positive quantity
5.3 Fitting forecast models to data
5.3.1 Presentation of the dataset
5.3.2 Model calibration
5.3.3 Numerical illustrations
5.4 Application to wind power trading
5.4.1 Description of the problem
5.4.2 Numerical resolution
5.5 Appendix
5.5.1 Talagrand diagrams and PIT histograms for lead times 36h and 48h
6 Risk bounds for aggregated shallow neural networks using Gaussian priors
6.1 Introduction
6.2 Preliminaries and notations
6.2.1 General framework
6.2.2 PAC-Bayesian type upper bounds
6.2.3 Shallow neural networks
6.2.4 Spherical Gaussian prior distribution
6.3 Examples of application
6.3.1 Fixed design regression
6.3.2 Random design regression
6.3.3 Density estimation
6.3.4 Classication for -risk
6.4 Risk bounds
6.4.1 Risk bound for an arbitrary centered Gaussian prior
6.4.2 Oracle inequality for optimized Gaussian prior
6.5 Approximation bounds
6.5.1 Bounds for sigmoidal activation functions
6.5.2 Bounds for the ReLU activation function
6.6 Worst-case risk bounds over smoothness classes
6.6.1 Sigmoid activation functions
6.6.2 ReLU activation function
6.6.3 Related works for risk bound of neural networks
6.7 Conclusion and outlook
6.8 Appendix
6.8.1 Some useful lemmas
6.8.2 Proof of Proposition 47
6.8.3 Proof of Corollary 48
6.8.4 Proof of Lemma 51
6.8.5 Proof of Proposition 55 .
