[Paper Review] Analysis and Implementation of a Hourly Billing Mechanism for Demand Response Management

Paulin Jacquot, Olivier Beaude+2 more|arXiv (Cornell University)|Dec 22, 2017

Smart Grid Energy ManagementEngineering46 references59 citations

TL;DR

The paper analyzes the hourly billing demand response model, proves NE uniqueness under convex price functions, bounds Price of Anarchy, proposes two convergence algorithms, and validates an online DR procedure with forecasts.

ABSTRACT

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.

Motivation & Objective

• Motivate residential demand response through an hourly billing mechanism with fairness properties.
• Establish theoretical guarantees for equilibrium existence, uniqueness, and efficiency under convex price functions.
• Develop and analyze decentralized algorithms to compute Nash equilibria efficiently.
• Propose and test an online DR procedure that adapts to forecast updates in a stochastic environment.

Proposed method

• Model the DR problem as a N-player minimalization game with an hourly proportional billing (b_n).
• Assume convex and increasing price functions and derive a unique Nash equilibrium (Theorem 1).
• Provide a price of anarchy bound (Theorem 2) and analyze convergence rates for two algorithms (Algorithm 1 CB RD and Algorithm 2 SIRD) (Theorems 3 and 4).
• Introduce an online receding-horizon DR procedure (Algorithm 3) to handle forecast updates.
• Validate with simulations using real consumption data and stochastic forecasts.
• Discuss two affine-price special cases and extensions to online implementations.

Experimental results

Research questions

• RQ1 Does the hourly billing mechanism yield a unique Nash equilibrium under convex, increasing price functions?
• RQ2 What is the efficiency of the equilibrium as measured by the Price of Anarchy, and how tight are the bounds?
• RQ3 Can decentralized algorithms (best-response and projected gradient) reliably converge to the NE, and at what rates?
• RQ4 How does an online, forecast-updating DR procedure perform compared to offline planning in a stochastic environment?

Key findings

• The model guarantees a unique Nash equilibrium under Assumption 1 (convex, increasing prices).
• The Price of Anarchy is bounded and near one under the proposed conditions, with a concrete bound given in Theorem 2.
• Two decentralized NE computation methods converge: Cycling Best Response Dynamics (CBRD) with a proven rate for affine prices (Theorem 3) and Simultaneous Improving Response Dynamics (SIRD) with a geometric convergence bound (Theorem 4).
• The online DR procedure (Algorithm 3) is consistent: with perfect forecasts, the NE matches the forecast-driven NE in subsequent periods (Theorem 5).
• simulations with real EV charging data show the online approach can reduce sensitivity to forecast errors compared to offline planning.