Stochastic control with state constraints via the Fokker–Planck equation. Application to renewable energy plants with batteries
Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 89-110.

Although renewable energies are beneficial to reduce carbon emissions, its intermittent characteristics may result in power-supply issues in distribution grid. Battery energy storage system is generally regarded as an effective tool to deal with them. On the other hand mathematical modelling, numerical simulation, optimization and control theory are nowadays of paramount importance to handle this kind of problems and related issues. In this paper we present a methodology for the development of bidding strategies and real-time control for electricity producers in a competitive electricity marketplace. Firstly, a stochastic model of a wind power plant with battery storage is stated in the framework of stochastic differential equations (SDE). Then, a stochastic control problem with state constraints is introduced and the corresponding optimality conditions involving the Hamilton–Jacobi–Bellman equation are deduced. For this purpose, advantage is taken from the fact that optimal control problems for stochastic ordinary differential equations (SDE) can be equivalently formulated as optimal control problems for deterministic partial differential equations (PDE), namely, the corresponding Fokker–Planck equation.

Bien que les énergies renouvelables permettent de réduire les émissions de carbone, leurs caractéristiques intermittentes peuvent entraîner des problèmes d’approvisionnement en électricité dans les réseaux de distribution. Le système de stockage d’énergie par batterie est généralement considéré comme un outil efficace pour y remédier. D’autre part, la modélisation mathématique, la simulation numérique, l’optimisation et la théorie du contrôle sont aujourd’hui d’une importance capitale pour traiter ce type de problèmes et les questions connexes. Dans cet article, nous présentons une méthodologie pour le développement de stratégies de soumission et de contrôle en temps réel pour les producteurs d’électricité sur un marché de l’électricité concurrentiel. Tout d’abord, un modèle stochastique d’une centrale éolienne avec stockage sur batterie est présenté dans le cadre des équations différentielles stochastiques (EDS). Ensuite, un problème de contrôle stochastique avec des contraintes d’état est introduit et les conditions d’optimalité correspondantes impliquant l’équation de Hamilton–Jacobi–Bellman sont déduites. À cette fin, on tire parti du fait que les problèmes de contrôle optimal pour les équations différentielles ordinaires stochastiques peuvent être formulés de manière équivalente comme des problèmes de contrôle optimal pour les équations aux dérivées partielles déterministes, à savoir l’équation de Fokker–Planck correspondante.

Accepted:
Online First:
Published online:
DOI: 10.5802/crmeca.236
Keywords: Renewable energy plant, optimal energy biddings, stochastic control, Fokker–Planck equation, Hamilton–Jacobi–Bellman equation
Mot clés : Installations d’énergie renouvelable, offres d’énergie optimales, contrôle stochastique, équation de Fokker–Planck, équation de Hamilton–Jacobi–Bellman

Alfredo Bermúdez 1; Iago Padín 2

1 CITMAga and Universidade de Santiago de Compostela, Spain
2 Department of Applied Mathematics. Universidade de Santiago de Compostela. Spain
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Alfredo Bermúdez; Iago Padín. Stochastic control with state constraints via the Fokker–Planck equation. Application to renewable energy plants with batteries. Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 89-110. doi : 10.5802/crmeca.236. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.236/

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