A McKean-Vlasov approach to distributed electricity generation development

This paper analyses the interaction between centralised carbon emissive technologies and distributed intermittent non-emissive technologies. In our model, there is a representative consumer who can satisfy her electricity demand by investing in distributed generation (solar panels) and by buying power from a centralised firm at a price the firm sets. Distributed generation is intermittent and induces an externality cost to the consumer. The firm provides non-random electricity generation subject to a carbon tax and to transmission costs. The objective of the consumer is to satisfy her demand while mini\-mising investment costs, payments to the firm and intermittency costs. The objective of the firm is to satisfy the consumer's residual demand while minimising investment costs, demand deviation costs, and maximising the payments from the consumer. We formulate the investment decisions as McKean-Vlasov control problems with stochastic coefficients. We provide explicit, price model-free solutions to the optimal decision problems faced by each player, the solution of the Pareto optimum, and the Stackelberg equilibrium where the firm is the leader. We find that, from the social planner's point of view, the carbon tax or transmission costs are necessary to justify a positive share of distributed capacity in the long-term, whatever the respective investment costs of both technologies are. The Stackelberg equilibrium is far from the Pareto equilibrium and leads to an over-investment in distributed energy and to a much higher price for centralised energy

Algebra in teacher training in the normal school of São Paulo: the first teaching programs of this discipline

This text presents partial results of a research in development that aims to identify the purposes of Algebra in the formation of teachers in the Normal School of São Paulo, from 1880 to 1930. In this text, in particular, it is considered a specific period of time frame of research: mid-1890s; and, in order to analyse the first programs of this discipline for the training of primary teachers, documents such as legislation, a compendium of Algebra and programs of teaching of the normal and secondary course are mobilized. It was found that, in this period, the teaching of Algebra for the future teachers was more restricted than that offered in the secondary course. This reduced format of the Normal School program may be associated with the duration of normal course studies, shorter than the secondary course, but mainly, for the different purposes of the courses

Linear Quadratic Reinforcement Learning: Sublinear Regret in the Episodic Continuous-Time Framework

In this paper we study a continuous-time linear quadratic reinforcement learning problem in an episodic setting. We first show that na\"ive discretization and piecewise approximation with discrete-time RL algorithms yields a linear regret with respect to the number of learning episodes $N$. We then propose an algorithm with continuous-time controls based on a regularized least-squares estimation, and establish a sublinear regret bound in the order of $\tilde{O}(\sqrt{N})$. The analysis consists of two parts: parameter estimation error, which relies on properties of sub-exponential random variables and double stochastic integrals; and perturbation analysis, which establishes the robustness of the associated continuous-time Riccati equation by exploiting its regularity property.Comment: 25 page