Our work is driven by a class of practical problems of sequential decision making in the context of electric power generation under uncertainties. These problems are usually treated as receding horizon deterministic optimization problems, and/or as scenario-based stochastic programs. Stochastic programming allows to compute a first stage decision that is hedged against the possible futures and -- if a possibility of recourse exists -- this decision can then be particularized to possible future scenarios thanks to the information gathered until the recourse opportunity.
Although many decomposition techniques exist, stochastic programming is currently not tractable in the context of day-ahead electric power generation and furthermore does not provide an explicit recourse strategy. The latter observation also makes this approach cumbersome when one wants to evaluate its value on independent scenarios.
We propose a supervised learning methodology to learn an explicit recourse strategy for a given generation schedule, from optimal adjustments of the system under simulated perturbed conditions. This methodology may thus be complementary to a stochastic programming based approach. With respect to a receding horizon optimization, it has the advantages of transferring the heavy computation offline, while providing the ability to quickly infer decisions during online exploitation of the generation system. Furthermore the learned strategy can be validated offline on an independent set of scenarios.
On a realistic instance of the intra-day electricity generation rescheduling problem, we explain how to generate disturbance scenarios, how to compute adjusted schedules, how to formulate the supervised learning problem to obtain a recourse strategy, how to restore feasibility of the predicted adjustments and how to evaluate the recourse strategy on independent scenarios. We analyze different settings, namely either to predict the detailed adjustment of all the generation units, or to predict more qualitative variables that allow to speed up the adjustment computation procedure by facilitating the ``classical' optimization problem. Our approach is intrinsically scalable to large-scale generation management problems, and may in principle handle all kinds of uncertainties and practical constraints. Our results show the feasibility of the approach and are also promising in terms of economic efficiency of the resulting strategies.
The solutions of the optimization problem of generation (re)scheduling must satisfy many constraints. However, a classical learning algorithm that is (by nature) unaware of the constraints the data is subject to may indeed successfully capture the sensitivity of the solution to the model parameters. This has nevertheless raised our attention on one particular aspect of the relation between machine learning algorithms and optimization algorithms. When we apply a supervised learning algorithm to search in a hypothesis space based on data that satisfies a known set of constraints, can we guarantee that the hypothesis that we select will make predictions that satisfy the constraints? Can we at least benefit from our knowledge of the constraints to eliminate some hypotheses while learning and thus hope that the selected hypothesis has a better generalization error?
In the second part of this thesis, where we try to answer these questions, we propose a generic extension of tree-based ensemble methods that allows incorporating incomplete data but also prior knowledge about the problem. The framework is based on a convex optimization problem allowing to regularize a tree-based ensemble model by adjusting either (or both) the labels attached to the leaves of an ensemble of regression trees or the outputs of the observations of the training sample. It allows to incorporate weak additional information in the form of partial information about output labels (like in censored data or semi-supervised learning) or -- more generally -- to cope with observations of varying degree of precision, or strong priors in the form of structural knowledge about the sought model.
In addition to enhancing the precision by exploiting information that cannot be used by classical supervised learning algorithms, the proposed approach may be used to produce models which naturally comply with feasibility constraints that must be satisfied in many practical decision making problems, especially in contexts where the output space is of high-dimension and/or structured by invariances, symmetries and other kinds of constraints.