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We consider a new framework where a continuous, though bounded, random variable has unobserved bounds that vary over time. In the context of univariate time series, we look at the bounds as parameters of the distribution of the bounded random variable. We introduce an extended log-likelihood estimation and design algorithms to track the bound through online maximum likelihood estimation. Since the resulting optimization problem is not convex, we make use of recent theoretical results on stochastic quasiconvex optimization, to eventually derive an Online Normalized Gradient Descent algorithm. We illustrate and discuss the workings of our approach based on both simulation studies and a real-world wind power forecasting problem.

Forecasting is of the utmost importance to the integration of renewable energy into power systems and electricity markets. Indeed, to get electricity from conventional generators such as fuel-based or nuclear power plants, one is in charge of the production, whereas renewable energy sources are fundamentally variable and weather-dependent. Full benefits from their integration can only be reaped if one is given reliable, trustworthy forecasts and therefore the opportunity to accommodate the actual renewable power generation in an optimal way. In this thesis, we focus on offshore wind power short-term forecasting, as wind power fluctuations at horizons of a few minutes ahead particularly affect the system balance and are the most significant offshore. Those very short-term lead times are not only crucial but also the most difficult to improve the forecasts for, especially compared to the simple but very effective persistence benchmark. Forecasts characterize but do not eliminate uncertainty. Therefore, they ought to be probabilistic, taking the form of distributions. Wind power generation is a stochastic process which is double-bounded by nature, by zero when there is no production and by the nominal power for high-enough wind speeds. It is non-linear and non-stationary. For short-term forecasting, statistical methods have proved to be more skilled and accurate. However, they often rely on stationary, Gaussian distributions, which cannot be appropriate for wind power generation. We start by extending previous works on generalized logit-normal distributions for wind power generation. First, we develop a rigorous statistical framework to estimate the full parameter vector of the distribution through maximum likelihood inference. Then, we derive the corresponding recursive maximum likelihood estimation and propose a recursive algorithm which can track the full parameter of the distribution in an online fashion. From the observation that bounds are alway

We consider a new framework where a continuous, though bounded, random variable has unobserved bounds that vary over time. In the context of univariate time series, we look at the bounds as parameters of the distribution of the bounded random variable. We introduce an extended log-likelihood estimation and design algorithms to track the bound through online maximum likelihood estimation. Since the resulting optimization problem is not convex, we make use of recent theoretical results on Normalized Gradient Descent (NGD) for quasiconvex optimization, to eventually derive an Online Normalized Gradient Descent algorithm. We illustrate and discuss the workings of our approach based on both simulation studies and a real-world wind power forecasting problem., Comment: 43 pages, 9 figures