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Game Design and Analysis for Price based Demand Response: An Aggregate Game Approach

Maojiao Ye, Guoqiang Hu

arXiv:1508.02636v3econ.GNeess.SYmath.OC

TL;DR

The paper addresses energy-consumption control when users’ costs depend on an aggregate load they do not know. It combines aggregate-game modeling, load estimation, and neighbor communication to seek Nash equilibria, with convergence results for general, unique-equilibrium, and stubborn-player settings.

  • Problem

    Users’ costs depend on aggregate energy consumption that is unknown during Nash seeking, creating a distributed energy-consumption game-design problem.

  • Method

    The paper uses an aggregative game on an undirected connected graph, average consensus for aggregate-load estimation, and stability analysis for Nash-seeking strategies.

  • Results

    The proposed strategies provide local convergence for games with multiple equilibria, non-local convergence for unique equilibria, and convergence of rational players toward best-response neighborhoods with stubborn players.

  • Takeaways & Limitations

    Neighbor-based aggregate-load estimation enables distributed Nash-equilibrium seeking without requiring users to share their own energy consumptions with opponents.

Abstract

from arXiv · show

In this paper, an aggregate game approach is proposed for the modeling and analysis of energy consumption control in smart grid. Since the electricity user's cost function depends on the aggregate load, which is unknown to the end users, an aggregate load estimator is employed to estimate it. Based on the communication among the users about their estimations on the aggregate load, Nash equilibrium seeking strategies are proposed for the electricity users. By using singular perturbation analysis and Lyapunov stability analysis, a local convergence result to the Nash equilibrium is presented for the energy consumption game that may have multiple Nash equilibria. For the energy consumption game with a unique Nash equilibrium, it is shown that the players' strategies converge to the Nash equilibrium non-locally. More specially, if the unique Nash equilibrium is an inner Nash equilibrium, then the convergence rate can be quantified. Energy consumption game with stubborn players is also investigated. Convergence to the best response strategies for the rational players is ensured. Numerical examples are provided to verify the effectiveness of the proposed methods.

I. INTRODUCTION

The paper models smart-grid energy consumption control as an aggregate game for price-anticipating users. It develops neighbor-based Nash-seeking strategies that reduce centralized communication and address games with multiple or unique equilibria.

  • Contributions: The paper adopts an aggregate game to model energy consumption control in a smart grid.Users interact through the sum of their actions, namely aggregate energy consumption.
  • Contributions: Users update actions through neighbor communication, reducing communication with a centralized agent and relieving single-node congestion.The proposed graph-based scheme uses communication with neighboring users rather than relying exclusively on a centralized agent.
  • Contributions: The study considers aggregate games with multiple isolated Nash equilibria and proposes an average-consensus-based Nash-seeking strategy.It also studies HVAC energy consumption games with unique equilibria using primal-dual dynamics.
  • Contributions: If the unique HVAC Nash equilibrium is inner, the proposed strategy yields exponential convergence.With stubborn players, rational players’ actions converge to a neighborhood of their best-response strategies.
  • Contributions: Users communicate estimates of aggregate energy consumption with neighbors without sharing their own consumption with opponents.The paper presents this as protecting users’ energy-consumption privacy.
  • Preliminaries: An aggregate game is a normal-form game whose costs depend on each player’s action and a linear aggregate of the full action profile.The paper also defines Nash equilibrium as an action profile where no player can reduce cost by unilaterally changing its action.

B. Graph Theory

The paper formulates energy consumption control for price-anticipating users whose costs depend on aggregate consumption. Users communicate over an undirected connected graph while estimating the unknown aggregate load.

  • Graph model: Users communicate through an undirected and connected graph, with neighboring sets defined from graph edges.The graph Laplacian is defined from the degree matrix and adjacency matrix.
  • Market model: The simplified electricity buying and selling model is illustrated in Fig. 1.The supplied caption identifies the figure as a simplified illustration of the electricity buying and selling model.
  • User model: Each user schedules energy consumption by minimizing its own cost using an energy-management controller and advanced metering infrastructure.The infrastructure enables bidirectional communication among users and with a centralized agent.
  • Cost model: User cost combines load-curtailment cost with billing payment P(l̄)l_i, where price P(l̄) depends on aggregate consumption l̄.Aggregate consumption is the sum of all users’ energy consumptions.
  • Constraints: Each user’s consumption must remain within its acceptable minimum and maximum range.The strategy-design problem treats users as players and consumption as their actions.
  • Information assumptions: The aggregate l̄ is unknown to players during Nash seeking, while users may communicate with neighbors about aggregate-consumption estimates.The formulation assumes an isolated pure-strategy Nash equilibrium exists and user costs are smooth.

IV. ENERGY CONSUMPTION GAME DESIGN AND ANALYSIS

The general energy-consumption game omits consumption constraints for analysis and assumes an isolated, stable Nash equilibrium. Players use estimates of aggregate consumption to formulate a consensus-based search.

  • Problem setting: The section studies the general energy-consumption game without specifying the pricing function.For simplicity, the individual consumption constraints are omitted in this section.
  • Assumptions: The analysis assumes an isolated, stable Nash equilibrium.The equilibrium is denoted by l* in the stated assumptions.
  • Assumptions: The paper imposes a strict diagonal-dominance condition on the relevant matrix at the Nash equilibrium.This is stated as an additional assumption for the general-game analysis.
  • Estimation and seeking: Each player uses D_i, its estimate of aggregate energy consumption, to rewrite its objective.A consensus-based method is then proposed to search for the Nash equilibrium without the consumption constraints.

B. Nash Equilibrium Seeking for the Aggregate Energy Consumption Game

A consensus-based strategy estimates aggregate consumption while users seek a Nash equilibrium. Under the stated conditions and sufficiently small singular-perturbation parameter, the closed-loop states converge exponentially near the equilibrium.

  • Strategy design: Each player updates its strategy using neighbor communication, with δ a small positive parameter and κ_i intermediate variables.The gain is scaled as k̄_i = δk_i.
  • System representation: The concatenated closed-loop system combines the users’ consumption, aggregate estimates, and intermediate variables.The analysis rewrites these states in vector form using graph-related coordinates.
  • Singular perturbation analysis: For fixed l, the quasi-steady states D^e(l) and κ^e(l) are unique.Their uniqueness follows from the stated Hurwitz property.
  • Convergence result: Theorem 2 establishes exponential convergence to (l*, 1/N Σ_i l*_i, κ^e(l*)) when 0 < δ < δ* and initial errors are sufficiently small.The result is local because the initial deviations must satisfy smallness conditions.
  • Stability analysis: The boundary-layer equilibrium is exponentially stable, while the reduced system’s equilibrium l* is locally exponentially stable.The reduced-system stability uses the Hurwitz property and the Gershgorin Circle Theorem.
  • Conclusion: All states remain bounded, and the consumption trajectory produced by the strategy converges to the Nash equilibrium under the given conditions.The section concludes by contrasting this result with the subsequent unique-equilibrium HVAC analysis.

V. ENERGY CONSUMPTION GAME AMONG A NETWORK OF HVAC SYSTEMS

The HVAC energy consumption game models load-curtailment costs and assumes a parameter condition that ensures a unique Nash equilibrium. A Nash seeking strategy is then developed for this unique equilibrium.

  • The HVAC load-curtailment cost is modeled as a function of users’ energy consumption and indoor-temperature maintenance needs.The model includes the energy required to maintain each HVAC system’s indoor temperature.
  • For N > 3, the condition a < min_i∈N … N−3 ensures uniqueness of the Nash equilibrium.The paper assumes this condition for the remainder of the analysis.
  • The section proposes a Nash seeking strategy for players to search for the unique Nash equilibrium under the given HVAC model.

A. Nash Equilibrium Seeking for Energy Consumption Game of HVAC Systems

This section establishes a potential-game formulation for the HVAC energy consumption game and designs primal-dual dynamics for Nash equilibrium seeking. Under stated initialization and regularity conditions, the strategy converges asymptotically to the equilibrium, with singular perturbation analysis providing the broader convergence result.

  • The HVAC energy consumption game is shown to be a potential game, allowing Nash seeking to be formulated through optimization of a potential function.
  • The proposed strategy uses primal-dual dynamics with positive dual-variable initialization to handle the game’s constrained optimization formulation.The dynamics are parameterized by scaled positive constants, and η_ij(0) > 0 is required.
  • For sufficiently small δ, the full system’s state satisfies a local convergence result obtained using singular perturbation analysis.The state includes deviations in actions, aggregate-load estimates, auxiliary variables, and dual variables.
  • The potential function is strictly convex because its Hessian is positive definite under the stated condition, while linear constraints yield strong duality.The optimal solution is therefore characterized by a saddle point of the Lagrangian.
  • The saddle point is globally asymptotically stable under the proposed dynamics when η_ij(0) > 0, so players’ strategies converge asymptotically to the Nash equilibrium.

B. Nash Equilibrium Seeking for Energy Consumption Game of HVAC Systems with A Unique Inner Nash Equilibrium

For a unique inner Nash equilibrium, the paper simplifies the seeking dynamics when constraints do not alter the equilibrium and proves exponential convergence under a sufficiently small perturbation parameter.

  • When constraints do not affect the equilibrium value, the proposed updating strategy becomes a special case of the general strategy.
  • For 0 < δ < δ*, the actions, aggregate-load estimates, and auxiliary variables converge exponentially to their equilibrium values.
  • An inner Nash equilibrium is one whose action satisfies the interior first-order condition ∂C_i/∂l_i = 0 for every player.
  • Theorem 4 gives a stronger result than Theorem 2 under the specified HVAC model when the equilibrium is inner and constraints do not affect its value.

C. Energy Consumption Game of HVAC Systems with Stubborn Players

The paper extends HVAC energy-consumption-game analysis to settings with stubborn players, whose fixed consumption is accommodated while rational players seek best responses. The section also notes neighbor-only communication and presents simulation parameters.

  • Energy Consumption Game of HVAC Systems with Stubborn Players: A stubborn player keeps its energy consumption constant while the remaining rational players continue the coordination process.
  • Energy Consumption Game of HVAC Systems with Stubborn Players: Rational players use different seeking dynamics depending on whether constraints affect the values of their best response strategies.They adopt (35) when constraints do not affect best responses and (27) otherwise.
  • Energy Consumption Game of HVAC Systems with Stubborn Players: Under the stated condition and sufficiently small δ, rational players’ deviations from their best responses and aggregate-estimation errors converge exponentially to zero.
  • Energy Consumption Game of HVAC Systems with Stubborn Players: The stability argument uses a reduced system whose matrix −kH1 is Hurwitz by the Gershgorin Circle Theorem.
  • Energy Consumption Game of HVAC Systems with Stubborn Players: The analysis states that analogous results can be derived when multiple stubborn players exist.
  • Energy Consumption Game of HVAC Systems with Stubborn Players: The proposed strategy communicates only aggregate-load estimates and auxiliary variables with neighbors, not users’ own energy consumptions.The section’s simulation considers a communication graph for electricity users; Table I lists simulation parameters.

A. Simulation Setup

The simulation considers five commercial/industrial electricity users equipped with HVAC systems and connected through an undirected, connected communication graph.

  • The setup models a network of 5 commercial/industrial users equipped with HVAC systems.
  • Users communicate with one another through an undirected and connected graph.
  • The users’ cost functions and simulation parameters are specified for the energy-consumption control problem.

B. Energy Consumption Control of HVAC Systems

The simulations evaluate aggregate-game Nash seeking for HVAC energy consumption, including boundary and inner equilibria and a stubborn-player setting. The proposed strategies converge to the relevant equilibrium or best-response outcomes while estimating aggregate consumption.

  • B. Energy Consumption Control of HVAC Systems: The boundary-case Nash equilibrium is (45, 46.4, 51.3, 56.2, 61.1) kWh, with aggregate consumption 259.9 kWh.This equilibrium is not an inner Nash equilibrium.
  • B. Energy Consumption Control of HVAC Systems: The proposed strategy in (27) makes users’ energy consumptions converge to the Nash equilibrium.Users’ aggregate-consumption estimates also converge to the actual aggregate energy consumption.
  • C. Energy Consumption Control of HVAC Systems with A Unique inner Nash equilibrium: The inner-equilibrium case has equilibrium profile (41.5, 46.4, 51.3, 56.2, 61.1) kWh and aggregate consumption 256.7 kWh.The Nash equilibrium is unique and inner, and strategy (35) is used in simulation.
  • C. Energy Consumption Control of HVAC Systems with A Unique inner Nash equilibrium: Under strategy (35), users’ energy consumptions converge to the unique Nash equilibrium.
  • D. Energy Consumption Control with Stubborn Players: With player 5 fixed at 100 kWh, players 1–4 have best responses of 40.8, 45.7, 50.6, and 55.5 kWh, respectively.The resulting aggregate energy consumption is 292.7 kWh.
  • D. Energy Consumption Control with Stubborn Players: With a stubborn player, the other players’ actions converge to best-response strategies relative to the stubborn action.Rational players use (35), while the stubborn player uses (41).
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