Applied Mathematics

We formulate an equilibrium model of intraday trading in electricity markets. Agents face balancing constraints between their customers consumption plus intraday sales and their production plus intraday purchases. They have continuously updated forecast of their customers consumption at maturity. Forecasts are prone to idiosyncratic noise as well as common noise (weather). Agents production capacities are subject to independent random outages, which are each modeled by a Markov chain. The equilibrium price is defined as the price that minimizes trading cost plus imbalance cost of each agent and satisfies the usual market clearing condition. Existence and uniqueness of the equilibrium are proved, and we show that the equilibrium price and the optimal trading strategies are martingales. The main economic insights are the following. (i) When there is no uncertainty on generation, it is shown that the market price is a convex combination of forecasted marginal cost of each agent, with deterministic weights. Furthermore, the equilibrium market price is consistent with Almgren and Chriss's model, and we identify the fundamental part as well as the permanent market impact. It turns out that heterogeneity across agents is a necessary condition for Samuelson's effect to hold. We show that when heterogeneity lies only on costs, Samuelson's effect holds true. A similar result stands when heterogeneity lies only on market access quality. (ii) When there is production uncertainty only, we provide an approximation of the equilibrium for large number of players. The resulting price exhibits increasing volatility with time. (Joint work with Andrea Cosso and René Aïd)

Electricity markets balance an increasingly intermittent supply with price-inelastic demand, while climate change and electrification of mobility are contributing to transforming diurnal and seasonal demand patterns. Electricity systems face an increasing level of stochasticity, and market participants need to inform their dispatch decisions based on 24-hour price forecasts for participation in Day-Ahead Markets, which in turn depend on supply and demand forecasts. The arrival of grid-scale electricity storage is also creating new scope for prices forecasts, while smaller scale storage systems act as price takers. In the long term, large-scale deployment of grid-scale electricity storage has the potential of significantly reducing price variation through arbitrage, which could shift the “value added” of forecasting from short-term (intra-day) to long-term predictions, and from supporting operational (dispatch) decisions to supporting capacity (investment) decisions.

The rising U.S. offshore wind sector holds great promise, both environmentally and economically, to unlock vast supplies of clean, domestic, and renewable energy. To harness this valuable resource, Gigawatt (GW)-scale offshore wind projects are already under way at several locations off of the U.S. coastline. This promising future, however, is still clouded with uncertainties on how to optimally manage those ultra-scale offshore wind assets, which would be operating under harsh environmental and operational conditions, in relatively under-explored territories, and at unprecedented scales. I will present some of our progress in formulating tailored forecasting and optimization models aimed at minimizing some of those uncertainties. Our models and analyses are largely tailored and tested using data from the U.S. Mid-Atlantic—where several GW-scale wind projects are currently under development.

Forecasts of renewable power production and electricity demand for multiple time periods and/or spatial expanses are required to operate modern power systems. Furthermore, probabilistic forecasts are necessary to facilitate economic decision-making and risk management. This gives rise to the challenge of producing forecasts that capture dependency between variables, over time, and between multiple locations. The Gaussian Copula has been widely used for multivariate energy forecasting, including for wind power, and is readily scalable given that the entire dependency structure is described by a single covariance matrix; however, estimating this covariance matrix in high dimensional problems remains a research challenge. Furthermore, it has been found empirically that this covariance matrix is often non-stationary and evolves over time. Two methods are presented for parameterising covariance matrices to enable conditioning on explanatory variables and as a step towards more robust estimation.

We consider two approaches, one based on modelling the parameters of covariance functions using additive models, and the second modelling individual elements of the modified Cholesky decomposition, again using additive models. We show how this gives rise to a wide range of possible parametric structures and discuss model selection and estimation strategies. Finally, we demonstrate though two case studies the improvement in forecast quality that these methods yield, and the importance and value of capturing the dynamics of dependency structures in wind power forecasting and net-load forecasting in the presence of embedded renewables.