  An important part of the Smart Grid literature on residential Demand Response
deals with game-theoretic consumption models. Among those papers, the hourly
billing model is of special interest as an intuitive and fair mechanism. We
focus on this model and answer to several theoretical and practical questions.
First, we prove the uniqueness of the consumption profile corresponding to the
Nash equilibrium, and we analyze its efficiency by providing a bound on the
Price of Anarchy. Next, we address the computational issue of the equilibrium
profile by providing two algorithms: the cycling best response dynamics and a
projected gradient descent method, and by giving an upper bound on their
convergence rate to the equilibrium. Last, we simulate this demand response
framework in a stochastic environment where the parameters depend on forecasts.
We show numerically the relevance of an online demand response procedure, which
reduces the impact of inaccurate forecasts.
