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Autonomous Demand Side Management Based on Energy Consumption Scheduling and Instantaneous Load Billing: An Aggregative Game Approach

He Chen, Yonghui Li, Raymond H. Y. Louie, Branka Vucetic

arXiv:1306.3383v2math.OC

TL;DR

The paper studies selfish consumers scheduling future energy use in DSM, where billing must encourage peak shifting while remaining fair. It formulates an aggregative game, characterizes its Nash equilibrium, and develops distributed algorithms for centralized and peer-to-peer settings. Simulations show rapid equilibrium convergence and a PAR reduction from 2.3189 to 1.6161, or 30.31% less, after DSM.

  • Problem

    DSM needs a billing mechanism that motivates consumers to participate while fairly accounting for peak-time and off-peak energy use.

  • Method

    The paper formulates an aggregative game and develops distributed proximal-point, agreement-based, and asynchronous gossip-based algorithms for computing its Nash equilibrium.

  • Results

    PAR decreases from 2.3189 to 1.6161, or 30.31% less, after the DSM program, while the algorithms quickly converge to the Nash equilibrium.

  • Takeaways & Limitations

    Instantaneous-load billing shifts on-peak consumption and can benefit both consumers and overall grid efficiency, while distributed algorithms support equilibrium computation with or without aggregate-load access.

Abstract

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In this paper, we investigate a practical demand side management scenario where the selfish consumers compete to minimize their individual energy cost through scheduling their future energy consumption profiles. We propose an instantaneous load billing scheme to effectively convince the consumers to shift their peak-time consumption and to fairly charge the consumers for their energy consumption. For the considered DSM scenario, an aggregative game is first formulated to model the strategic behaviors of the selfish consumers. By resorting to the variational inequality theory, we analyze the conditions for the existence and uniqueness of the Nash equilibrium (NE) of the formulated game. Subsequently, for the scenario where there is a central unit calculating and sending the real-time aggregated load to all consumers, we develop a one timescale distributed iterative proximal-point algorithm with provable convergence to achieve the NE of the formulated game. Finally, considering the alternative situation where the central unit does not exist, but the consumers are connected and they would like to share their estimated information with others, we present a distributed agreement-based algorithm, by which the consumers can achieve the NE of the formulated game through exchanging information with their immediate neighbors.

I. INTRODUCTION

The paper frames DSM as a smart-grid mechanism for shifting peak consumption, then addresses fairness and decentralized computation through an aggregative-game formulation and distributed algorithms.

  • I. INTRODUCTION: Real-time pricing links energy prices to aggregate load, encouraging consumers to shift consumption from peak to non-peak periods.Flattened demand can improve whole-grid operating efficiency.
  • I. INTRODUCTION: Existing total-load billing charges equal total consumption equally across time, motivating instantaneous-load billing to charge more for peak-time use.The paper identifies billing design as important for consumer participation and fairness.
  • I. INTRODUCTION: The paper models selfish consumers’ scheduling decisions as an aggregative game and analyzes Nash-equilibrium existence and uniqueness using variational inequality theory.Each payoff depends on the consumer’s own action and the aggregate of all consumers’ actions.
  • I. INTRODUCTION: With a central unit broadcasting aggregate consumption, the paper develops a parallel one-timescale distributed iterative proximal-point algorithm for computing the Nash equilibrium.Strict monotonicity supports avoiding the two-timescale algorithms used in related settings.
  • I. INTRODUCTION: Without a central unit, connected consumers exchange estimated information with neighbors through agreement-based and asynchronous gossip-based algorithms.The gossip approach removes synchronization requirements and permits uncoordinated update step sizes.

II. SYSTEM MODEL

The system schedules consumers’ energy profiles over future time slots under individual constraints, using an increasing aggregate-load price and instantaneous-load billing.

  • II. SYSTEM MODEL: The network contains N consumers served by one provider, with each controller responsible for scheduling consumption over H future time slots.Each consumer’s total energy requirement is fixed in advance.
  • A. Energy Consumption Model: Each consumer’s profile is constrained by minimum and maximum per-slot energy levels and a total energy requirement.These constraints define each consumer’s feasible consumption set and the Cartesian-product feasible set for all consumers.
  • B. Instantaneous Load Billing: Instantaneous-load billing sets each slot’s price as an increasing smooth function of total demand and charges consumers according to slot price and their slot consumption.The scheme is intended to shift peak-time use and improve charging fairness.
  • B. Instantaneous Load Billing: The price model uses time-slot parameters ah > 0, bh ≥ 1, and ch ≥ 0, with Lh denoting total energy consumed in slot h.Its increasing convex form makes prices grow more rapidly as aggregate load increases.
  • B. Instantaneous Load Billing: The adopted billing method is reported as fairer than total-load billing because it accounts for when consumers use energy, not only how much they use.The paper states that simulations validate this fairness comparison.
  • B. Instantaneous Load Billing: A consumer’s total cost can be computed from the aggregate profile qΣ, without requiring the individual profiles of other consumers.This aggregative dependence supports the game formulation and distributed information exchange.

III. GAME FORMULATION AND ANALYSIS

The paper models selfish consumers’ energy-scheduling decisions as an aggregative game and uses variational inequality theory to characterize its Nash equilibrium. Under a condition linking the polynomial price exponent and consumer population, the game has a unique equilibrium.

  • A. Aggregative Game Formulation: Each consumer’s optimization problem is coupled with the aggregated energy consumption of all consumers.This coupling motivates the aggregative-game representation.
  • A. Aggregative Game Formulation: The section formulates DSM scheduling as an aggregative game in which N consumers choose feasible energy profiles to minimize individual total energy costs.Each payoff depends on the consumer’s own profile and the aggregated consumption profile.
  • B. NE Analysis: The game’s action sets and payoff functions are examined before applying variational inequality theory to the equilibrium problem.The supplied lemma states that each payoff is continuously differentiable and convex in the consumer’s own action.
  • B. NE Analysis: The Nash equilibrium is equivalent to a solution of the variational inequality VI(Q, F), with Q formed as the Cartesian product of consumers’ feasible sets.The mapping is defined from the gradients of the consumers’ total-cost functions.
  • B. NE Analysis: The mapping F is investigated for monotonicity to establish equilibrium properties.The supplied passage introduces this monotonicity analysis but does not state its full proposition.
  • B. NE Analysis: If bh < 3 + 4/(N −1) for every time slot h, the formulated aggregative game admits a unique Nash equilibrium.The condition imposes an upper bound on the polynomial price function’s exponential factor that decreases with N.
  • B. NE Analysis: The uniqueness guarantee depends specifically on the relationship between the polynomial price exponent and the number of consumers.The paper notes that this relationship is sufficient for uniqueness under Proposition 1’s condition.

IV. DISTRIBUTED ITERATIVE PROXIMAL-POINT ALGORITHM WITH A CENTRAL UNIT

With a central unit broadcasting the latest aggregate load, the paper develops a distributed iterative proximal-point algorithm for computing the game’s Nash equilibrium. The method uses a single timescale and converges to the unique equilibrium under the stated condition and step-size requirements.

  • Central-unit setting: A central unit calculates and broadcasts the aggregated energy-consumption profile after consumers update their individual profiles.Consumers receive the aggregate, update locally, and send their new profiles back to the central unit.
  • Central-unit setting: The proposed distributed iterative proximal-point algorithm computes the Nash equilibrium in the central-unit setting.The algorithm is designed for the aggregative game formulated earlier.
  • Algorithm design: The variational inequality framework defines the equilibrium condition through a vector x* satisfying (y − x*)^T F(x*) ≥ 0 for all y in K.This supplies the equilibrium formulation underlying the algorithmic analysis.
  • Algorithm design: Unlike the earlier proximal-decomposition methods, the proposed method is single-timescale and requires only one projection step per iteration.The paper motivates this design using the strictly monotone mapping of the formulated problem.
  • Convergence: Under Proposition 1’s condition, Algorithm 1 converges to the unique Nash equilibrium when its step-size sequence satisfies the stated requirements.The convergence claim applies to the generated sequence of energy-consumption profiles.
  • Scope: The central-unit algorithm cannot be directly implemented when no central unit provides the aggregated consumption profile.The paper therefore introduces agreement- and gossip-based alternatives for the no-central-unit setting.

V. DISTRIBUTED SYNCHRONOUS AGREEMENT-BASED ALGORITHM WITHOUT A CENTRAL UNIT

For networks without a central unit, consumers estimate the aggregate load through local communication and update their schedules using agreement-based methods. The synchronous algorithm converges under connectivity, step-size, weight, and equilibrium-uniqueness conditions, while remaining compatible with privacy-preserving information exchange.

  • Algorithm setting: Without a central unit, connected consumers exchange estimated aggregate-load information with immediate neighbors to compute the game’s Nash equilibrium.The connection topology is modeled as an undirected static graph.
  • Privacy: The agreement-based method requires consumers to share estimates of average consumption rather than their exact energy-consumption profiles.The paper presents this as avoiding consumers’ security and privacy concerns.
  • Information estimation: Each consumer estimates the average consumption through a weighted combination of its own and neighboring estimates, then forms an estimated aggregate load.The estimated aggregate is N times the estimated average consumption.
  • Synchronous algorithm: Consumers update their energy profiles by applying a Euclidean projection based on the estimated aggregate load.The synchronous algorithm performs this update for every consumer at each iteration.
  • Weight design: The agreement-based algorithm uses nonnegative neighbor weights, with zero weight assigned to non-neighbors and self-excluded entries as specified.The weights satisfy the stated row and column summation conditions; τ controls the relative neighbor contribution.
  • Convergence: If the consumer graph is connected, the step size decreases appropriately, the weights satisfy the required conditions, and Proposition 1 holds, Algorithm 2 converges to the unique Nash equilibrium.These are the assumptions stated in Proposition 3.
  • Asynchronous extension: The asynchronous gossip algorithm lets consumers use update-frequency-based step sizes without synchronization between consumers.The method is designed for computing the same game equilibrium without a central unit.
  • Asynchronous extension: The synchronous no-central-unit algorithm still requires synchronization and coordinated step sizes, which are challenging in very large networks.This limitation motivates the subsequent asynchronous gossip-based algorithm.

VI. DISTRIBUTED ASYNCHRONOUS GOSSIP-BASED ALGORITHM WITHOUT A CENTRAL UNIT

The paper develops an asynchronous gossip-based algorithm that computes the game’s Nash equilibrium without a central unit. Randomly activated neighboring consumers exchange estimates and update asynchronously, with convergence under connectivity and suitable conditions.

  • Algorithm design: The algorithm computes the unique Nash equilibrium without requiring a central unit.It is designed for connected consumers that exchange estimated information with neighbors.
  • Asynchronous gossip protocol: Consumer clocks follow independent Poisson processes, so one consumer is randomly activated and contacts a randomly selected neighbor.Each neighbor has an equal chance of being selected.
  • Asynchronous gossip protocol: At each iteration, only two randomly selected neighboring consumers exchange average-consumption estimates and update their energy profiles.All other consumers remain unchanged during that iteration.
  • Convergence: The energy-consumption profile converges almost surely to the unique Nash equilibrium when Proposition 1’s condition holds and the undirected consumer graph is connected.The convergence claim is stated for the sequence generated by Algorithm 3.
  • Implementation properties: Consumers need not synchronize and may use uncoordinated step sizes based on their individual update frequencies.The algorithm also requires no private information to be exchanged between consumers.
  • Scope boundary: The paper adopts pairwise gossip for simplicity and leaves extensions to random subsets of consumers for future work.The stated extension would allow more than one pair to exchange estimates and update at each iteration.

VII. NUMERICAL RESULTS

Numerical experiments with 50 residential consumers validate algorithm convergence and show that the DSM program shifts consumption away from peak hours, lowers PAR, and supports differentiated billing.

  • Simulation setup: The simulations model 50 consumers scheduling a full day across 24 one-hour time slots starting at 8 AM.Initial consumer energy requirements are representative of residential usage, ranging from 10 kWh to 30 kWh.
  • Simulation setup: The connection structure defines immediate neighbors for information exchange in Algorithms 2 and 3.Directly linked consumers exchange information during algorithm iterations.
  • Algorithm convergence: Both Algorithms 1 and 2 stabilize each consumer’s energy cost after approximately 10 iterations, while Algorithm 3 approaches the same outcome after 200 iterations.These observations support the stated convergence propositions for the developed algorithms.
  • DSM effects: 30.31% less PAR: the aggregated load’s peak-to-average ratio decreases from 2.3189 before DSM to 1.6161 afterward.The authors report that this produces a generally flattened demand profile, reducing consumers’ energy cost and benefiting grid efficiency.
  • Billing fairness: The proposed billing method charges consumer 43 less than consumer 50 despite consumer 43 having greater total daily energy consumption.The reported charges are B43 = 10.56 and B50 = 10.66; the method considers both total consumption and usage timing.
  • Overall performance: The proposed DSM program achieves a total energy cost almost the same as the optimal social-welfare solution.The authors note that a theoretical price-of-anarchy analysis is outside this paper’s scope.

VIII. CONCLUSIONS

The paper formulates DSM as an aggregative game and develops distributed algorithms for computing its Nash equilibrium. Numerical results indicate convergence to the equilibrium, shifted on-peak consumption, and near-optimal total energy cost.

  • VIII. CONCLUSIONS: The DSM problem models selfish consumers competing to minimize individual energy costs through consumption scheduling and instantaneous load billing.The formulation includes sufficient conditions for Nash-equilibrium existence and uniqueness.
  • VIII. CONCLUSIONS: The proposed DSM program significantly reduces total energy cost compared with operation before DSM.Figure 6 compares the costs before DSM, after DSM, and under social-welfare optimization.
  • VIII. CONCLUSIONS: Three distributed algorithms address scenarios with and without access to real-time aggregated-load information.The algorithms require no private information exchange between consumers.
  • VIII. CONCLUSIONS: The algorithms quickly converge to the Nash equilibrium and efficiently shift on-peak consumption, benefiting consumers and the grid.These outcomes are reported from numerical results.

APPENDIX

The appendix establishes convexity of the billing-related function in each consumer’s decision profile by showing that its Hessian is positive semidefinite.

  • APPENDIX: For fixed q−n, the proof reduces convexity of Bn in qn to positive semidefiniteness of its Hessian.This uses the convexity criterion for twice-differentiable functions.
  • APPENDIX: The argument applies the convexity check while holding the other consumers’ profiles fixed.The fixed profile is denoted q−n.
  • APPENDIX: The Hessian is positive semidefinite because the relevant matrix is diagonal with strictly positive diagonal elements.This completes the convexity proof.

B. Proof of Proposition 1

The proof establishes uniqueness of the variational-inequality solution by proving strict monotonicity of the game mapping over a compact, convex feasible set.

  • B. Proof of Proposition 1: A variational inequality over a compact, convex feasible set has a unique solution when its mapping is strictly monotone.This criterion is used to establish uniqueness of the Nash equilibrium.
  • B. Proof of Proposition 1: Strict monotonicity is reduced to proving positive definiteness of a Jacobian-related matrix.The proof uses the fact that transposition preserves definiteness.
  • B. Proof of Proposition 1: The matrix argument examines its symmetric form and shows positivity through the smallest eigenvalue.The nonsymmetric matrix is handled by analyzing an associated symmetric matrix.
  • B. Proof of Proposition 1: The eigenvalue calculation accounts for two non-zero eigenvalues and N −2 zero eigenvalues in the auxiliary matrix.The proof explicitly assumes N ≥2.
  • B. Proof of Proposition 1: A sufficient positivity condition is obtained by requiring κh > 0, equivalently (N + 1+bh)^2 > N.The proof derives this condition from σh > 0.

C. Proof of Proposition 2

The proof verifies the regularity conditions needed for the iterative proximal-point algorithm by establishing Lipschitz continuity of the game mapping on the feasible set.

  • C. Proof of Proposition 2: The proximal-point convergence proof requires the formulated NEP Gλ to satisfy the listed regularity conditions.The proof invokes the general convergence result for iterative proximal-point algorithms.
  • C. Proof of Proposition 2: The proof establishes Lipschitz continuity of each component function f h in q, which yields Lipschitz continuity of F(q).The argument applies the Euclidean norm and componentwise bounds.
  • C. Proof of Proposition 2: The componentwise Lipschitz bound follows from algebraic substitution and the triangle inequality.The proof derives the bound through intermediate expressions before applying the inequality.
  • C. Proof of Proposition 2: For fixed qh_n, the function f h_n is Lipschitz continuous with a positive finite constant.The proof introduces constants clip_n,h,2 and clip_n,h to satisfy the required bounds.
  • C. Proof of Proposition 2: These bounds establish the required condition for all q and s in Q.The proof concludes after substituting the derived component bounds.

D. Proof of Proposition 3

The proof of Proposition 3 invokes established convergence conditions for distributed agreement-based algorithms and verifies that the aggregative game satisfies the required assumptions. It uses Lipschitz-continuity arguments for the relevant function components and concludes the proof.

  • Proposition 3 follows from established sufficient conditions and convergence proofs for a general distributed agreement-based algorithm.The game must satisfy all conditions listed in Assumptions 8–13 of [13, Ch. 4].
  • The adopted weight structure and step-size assumption, together with Lemma 1 and Proposition 1, support the required conditions.
  • F_n(q_n, qΣ) is Lipschitz continuous in qΣ when each function element f_n^h(q_n, qΣ) is Lipschitz continuous in qΣ.
  • The componentwise Lipschitz-continuity verification completes the proof.The argument refers to Appendix C for the validity of the required continuity properties.
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