In this work, we derive a probabilistic forecast of the solar irradiance during a day at a given location, using a stochastic differential equation (SDE for short) model. We propose a procedure that transforms a deterministic forecast into a probabilistic forecast: the input parameters of the SDE model are the Arome deterministic forecast computed at day D-1 for the day D. The model also accounts for the maximal irradiance from the clear sky model. The SDE model is mean-reverting towards the deterministic forecast and the instantaneous amplitude of the noise depends on the clear sky index, so that the fluctuations vanish as the index is close to 0 (cloudy) or 1 (sunny), as observed in practice. Our tests show a good adequacy of the confidence intervals of the model with the measurement.
We propose a generic multistage stochastic model for the Alternating Current Optimal Power Flow (AC OPF) problem for radial distribution networks, to account for the random electricity production of renewable energy sources and dynamic constraints of storage systems. We consider single-phase radial networks. Radial three-phase balanced networks (medium-voltage distribution networks typically have this structure) reduce to the former case. This induces a large scale optimization problem, which, given the non-convex nature of the AC OPF, is generally challenging to solve to global optimality.We derive a priori conditions guaranteeing a vanishing relaxation gap for the multi-stage AC OPF problem, which can thus be solved using convex optimization algorithms. We also give an a posteriori upper bound on the relaxation gap. In particular, we show that a null or low relaxation gap may be expected for applications with light reverse power flows or if sufficient storage capacities with low cost are available. Then, we discuss the validity of our results when incorporating voltage regulation devices. Finally, we illustrate our results on problems of planning of a realistic distribution feeder with distributed solar production and storage systems. Scenario trees for solar production
We study the mathematical modeling of the energy management system of a smart grid, related to a aggregated consumer equipped with renewable energy production (PV panels e.g.), storage facilities (batteries), and connected to the electrical public grid. The consumer pilots the use of the storage facilities in order to diminish the random fluctuations of his residual load on the public grid, so that intermittent renewable energy is better used leading globally to a much greener carbon footprint. The optimization problem is described in terms of an extended McKean-Vlasov stochastic control problem. Using the Pontryagin principle, we characterize the optimal storage control as solution of a certain McKean-Vlasov Forward Backward Stochastic Differential Equation (possibly with jumps), for which we prove existence and uniqueness. Quasi-explicit solutions are derived when the cost functions may not be linear-quadratic, using a perturbation approach. Numerical experiments support the study.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.