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Prioritizing Consumers in Smart Grid: A Game Theoretic Approach

Wayes Tushar, Jian A. Zhang, David Smith, H. Vincent Poor, Sylvie Thiebaux

arXiv:1312.0659v1eess.SYcs.GTcs.SY

TL;DR

Consumer participation is a central challenge in smart-grid energy management, so this paper proposes a consumer-centric energy-trading scheme with adaptive pricing and a Stackelberg game. The game has a socially optimal equilibrium, and simulations report noticeable performance improvements over a conventional feed-in-tariff scheme.

  • Problem

    Smart-grid energy management needs to motivate consumers to participate actively and voluntarily so its benefits can be realized.

    Source: §I · p. 1

  • Method

    A consumer-centric scheme uses adaptive per-EC pricing and a Stackelberg game to model CPS and EC decisions and derive a decentralized solution.

    Source: §I · p. 1, A. Unit Price Model · p. 2

  • Results

    The game has a socially optimal variational equilibrium, and simulations show noticeable performance improvements over a conventional feed-in-tariff scheme.

    Source: A. Existence of Equilibrium · p. 5, §VII · p. 9

  • Takeaways & Limitations

    The proposed scheme pairs limited-communication decentralized implementation with consumer-benefit and CPS-cost objectives.

    Source: §VII · p. 9

    The per-time-slot approach ignores inter-temporal elasticity of substitution in demand-side flexibility.

    Source: §VII · p. 9

Abstract

from arXiv · show

This paper proposes an energy management technique for a consumer-to-grid system in smart grid. The benefit to consumers is made the primary concern to encourage consumers to participate voluntarily in energy trading with the central power station (CPS) in situations of energy deficiency. A novel system model motivating energy trading under the goal of social optimality is proposed. A single-leader multiple-follower Stackelberg game is then studied to model the interactions between the CPS and a number of energy consumers (ECs), and to find optimal distributed solutions for the optimization problem based on the system model. The CPS is considered as a leader seeking to minimize its total cost of buying energy from the ECs, and the ECs are the followers who decide on how much energy they will sell to the CPS for maximizing their utilities. It is shown that the game, which can be implemented distributedly, possesses a socially optimal solution, in which the benefits-sum to all consumers is maximized, as the total cost to the CPS is minimized. Numerical analysis confirms the effectiveness of the game.

I. INTRODUCTION

The paper addresses the challenge of motivating voluntary consumer participation in smart-grid energy management with a consumer-centric trading scheme. It models energy-deficient trading between the CPS and ECs, aiming to balance consumer benefits and purchasing costs.

I. INTRODUCTION

  • Consumer participation is a key challenge: the scheme should make consumers primary beneficiaries so they actively and voluntarily provide grid services.The system model frames trading as a way to encourage surplus-energy consumers to participate while controlling purchasing costs.Source: §I · p. 1
  • The proposed scheme uses adaptive, per-consumer pricing to encourage participation, particularly among ECs with smaller energy surpluses, while minimizing CPS purchasing cost.The paper extends prior work with a generalized model, performance analysis, and broader simulations.Source: §I · p. 1
  • A single-leader, multiple-follower Stackelberg game supports decentralized decisions with limited CPS–EC interaction, and the paper proves algorithm optimality and convergence.The contributions also include parameter insights from analytical and numerical results.Source: §I · p. 1

II. SYSTEM MODEL

  • The system comprises a CPS and ECs such as electric vehicles, renewable farms, smart homes, and biogas plants; during peak demand, the CPS may buy their surplus energy.The modeled trading can also extend to outages and emergencies when the CPS cannot meet demand.Source: §II · p. 2
  • Energy management is modeled per time slot: the CPS sets a fixed energy-deficiency target, and participating ECs have available sale energy that can vary by source and conditions.The scheme can be reapplied across successive slots with updated parameters and participants.Source: §II · p. 2
  • Social optimality is defined as maximizing total welfare while motivating surplus-energy ECs to participate and controlling overall purchasing cost.The paper emphasizes balancing EC expectations and returns rather than maximizing an individual EC’s benefit in isolation.Source: §II · p. 2

A. Unit Price Model

The unit-price model assigns adaptive prices across ECs to encourage energy trading while keeping CPS purchasing costs under control. Price bounds constrain incentives and spending.

A. Unit Price Model

  • The CPS can set different, adaptive unit prices based on each EC’s offered energy, with incentives aimed especially at ECs holding smaller surpluses.The paper motivates differentiated incentives by noting that smaller-surplus ECs may prefer storing energy under a feed-in tariff.Source: A. Unit Price Model · p. 2
  • The total unit energy price P scales normalized EC prices and makes the direct purchasing cost proportional to P.P is fixed per time slot and may be set using a real-time price estimator.Source: A. Unit Price Model · pp. 2–3
  • Each EC’s unit price is bounded by pmin and pmax to discourage trading deterrence at low prices and excessive CPS costs at high prices.Prices outside the specified range are rounded to the relevant bound.Source: A. Unit Price Model · p. 3

B. Utility Model and ECs’ Objectives

The utility model sums EC benefits while accounting for payment, offered energy, available surplus, and individual preferences. The CPS cost model incorporates purchasing and associated costs into its minimization objective.

B. Utility Model and ECs’ Objectives

  • The EC objective maximizes the sum of individual utilities, with each utility depending on offered energy, available energy, and unit price.The paper motivates a summed objective as a way to pursue EC benefits within social optimality rather than optimize one participant alone.Source: B. Utility Model and ECs’ Objectives · p. 3
  • The proposed utility increases with price and offered energy but is concave in offered energy, reflecting possible saturation or declining benefit from excessive supply.The paper links this declining benefit to limited battery capacity and the risk of calendar ageing.Source: B. Utility Model and ECs’ Objectives · p. 3
  • Each EC’s utility combines direct income from selling energy with a possible loss term that reflects its preference for how much energy to offer.The parameter cn can be chosen by the EC; a larger value corresponds to preferring to sell less.Source: B. Utility Model and ECs’ Objectives · p. 3

C. Cost Model and the CPS’s Objective

  • The CPS cost model accounts for direct energy-purchase costs, transmission and storage costs, and costs associated with insufficient purchasing.The purchasing-cost term is weighted to generate differentiated prices; the shortfall cost can represent shed load.Source: C. Cost Model and the CPS’s Objective · p. 3
  • For simplicity, the model sets βn = α and discards the αEdef term before formulating the CPS’s cost-minimization objective.The paper states that this simplification has little influence on the scheme.Source: C. Cost Model and the CPS’s Objective · pp. 3–4

D. Optimization Problem

The CPS cannot directly control EC decisions, and parameters such as available energy and preferences may be unknown to it. The paper therefore motivates a decentralized game-theoretic mechanism to coordinate the trading decisions.

D. Optimization Problem

  • The CPS lacks direct control over EC decisions, and may not know parameters such as available energy or preferences.These decisions are made by individual customers.Source: D. Optimization Problem · p. 4
  • A decentralized mechanism is needed so ECs can choose the energy they sell while pursuing the joint optimization problem.The mechanism must also capture how the CPS makes decisions for the prescribed trading.Source: D. Optimization Problem · p. 4
  • The paper proposes game theory to represent interactions between EC selling decisions and CPS decisions in energy trading.The stated goal is to realize the optimization without direct CPS control of ECs.Source: D. Optimization Problem · p. 4

III. NON-COOPERATIVE GAME FORMULATION

The paper models CPS–consumer energy trading as a single-leader, multiple-follower Stackelberg game. CPS pricing and consumer supply decisions reach an energy management equilibrium solution (EMES) when the consumers respond with a generalized Nash equilibrium.

III. NON-COOPERATIVE GAME FORMULATION

  • The CPS sets unit prices as leader, while each energy consumer (EC) responds as a follower by choosing its energy offer through a generalized Nash game.The ECs jointly maximize social welfare under a shared supply constraint.Source: §III · p. 4
  • ECs retain privacy and do not communicate with one another; they may interact with the CPS through controlled signaling such as smart-meter messages.The consumers’ decisions remain coupled through the shared constraint on total energy supplied.Source: §III · p. 4
  • The CPS first announces a common price so EC offers reveal supply capacity, then optimizes its price vector and ECs select their equilibrium offers.The resulting price-and-energy pair is called an energy management equilibrium solution (EMES).Source: §III · p. 4
  • At EMES, no EC can improve its utility by unilaterally deviating, and changing the CPS price cannot lower its total cost.The game’s strategic form identifies the CPS and EC strategies, objectives, and strategy sets.Source: §III · p. 4

A. Existence of Equilibrium

The paper establishes that the consumer game has a socially optimal variational equilibrium (VE), and that this supports an equilibrium solution for the Stackelberg game. Its existence depends on the EC game admitting a generalized Nash equilibrium.

A. Existence of Equilibrium

  • A solution to the Stackelberg game exists if the EC game has a generalized Nash equilibrium, and it is socially optimal when that equilibrium is socially optimal.The CPS optimization has a unique solution because its cost function is strictly convex over a convex set.Source: A. Existence of Equilibrium · p. 5
  • The VE is the relevant equilibrium because the EC game’s shared constraint makes it jointly convex, and the VE is socially optimal among its generalized Nash equilibria.The paper describes the solution as efficient and proves VE existence and uniqueness through the utility function’s pseudo-gradient.Source: A. Existence of Equilibrium · p. 5
  • The EC game possesses a socially optimal variational equilibrium (VE), which is socially optimal among generalized Nash equilibria in the jointly convex game.The paper formulates the EC generalized Nash equilibrium problem as a variational inequality to study existence and efficiency.Source: A. Existence of Equilibrium · p. 5
  • The paper states that consumer-centric energy management possesses a socially optimal solution.The result is linked to the VE of the consumers’ game in response to the CPS price vector.Source: A. Existence of Equilibrium · p. 5

B. Decision Making Process

The decision-making analysis uses Karush-Kuhn-Tucker conditions for the ECs’ and CPS’s optimization problems. At equilibrium, an EC’s energy offer depends on its unit price, available energy, and preference parameter.

B. Decision Making Process

  • The paper formulates Karush-Kuhn-Tucker conditions for both EC and CPS optimization problems using Lagrange multipliers.The following EC decision analysis derives the equilibrium condition for an EC’s generalized Nash equilibrium problem.Source: B. Decision Making Process · p. 5, 1) ECs’ decisions · p. 5
  • The EC-side KKT solution supplies the equilibrium decision condition for how much energy an EC offers to the CPS.The paper presents the KKT formulation as a way to clarify the players’ decision-making process.Source: 1) ECs’ decisions · p. 5, B. Decision Making Process · p. 5
  • At equilibrium, an EC’s offered energy is proportional to its unit price and available energy, scaled by its constant c_n.An EC that prefers to sell more energy can choose a smaller c_n.Source: 1) ECs’ decisions · p. 5

2) CPS’s decision:

The CPS sets unit prices inversely to the energy offered, subject to minimum and maximum price bounds. The pricing parameter r controls price differentiation, and the simplified cost model does not change the scheme’s efficiency.

2) CPS’s decision:

  • The CPS unit price is inversely proportional to the energy an EC offers, subject to p_min ≤ p_n ≤ p_max.Prices outside the allowed range are rounded to the nearest bound.Source: 2) CPS’s decision · p. 5
  • The unit energy price is linearly proportional to P, the CPS’s direct purchasing cost.The passage relates that cost to the total purchase amount.Source: 2) CPS’s decision · p. 5
  • A smaller r creates a larger difference between EC unit prices, giving consumers with less energy to sell greater incentives to trade.The pricing mechanism can be adjusted by choosing different r values for different motivating strategies.Source: 2) CPS’s decision · p. 5
  • Using the more general CPS cost function yields a similar pricing rule and does not affect the proposed scheme’s efficiency.The result follows from replacing the simplified cost function with the more general one in the CPS decision process.Source: 2) CPS’s decision · pp. 5–6

V. ALGORITHM

The algorithm alternates CPS price optimization with EC energy decisions, using limited communication to reach the energy management equilibrium solution (EMES). It checks EC convergence through shared slack-variable values and is guaranteed to converge to an optimal solution.

V. ALGORITHM

  • ECs solve a strongly monotone variational inequality with SSHPM, while the CPS can use a standard convex optimization technique to set prices.SSHPM uses two projections per iteration, constructing a separating hyperplane and updating within the feasible set.Source: §V · p. 6
  • The ECs reach a variational equilibrium when their slack variables are equal, and the Stackelberg game reaches EMES when that equilibrium holds for the optimized price vector.The CPS checks the slack-variable values and either ends the iterations or requests another EC energy update.Source: §V · p. 6
  • The CPS first announces the energy deficit and total unit price; ECs use an equal initial price to choose offered energy, which informs CPS price optimization.The CPS then sets optimized EC-specific prices, and the ECs revise their supply decisions at those prices.Source: §V · p. 6
  • The proposed algorithm is guaranteed to converge to an optimal solution because the EC problem is strongly monotone and the CPS problem is strictly convex.The guarantee applies under constraints (1) and (2).Source: §V · p. 6

VI. NUMERICAL RESULTS

The simulations examine convergence, network size, the price upper bound, and comparison with a Feed-in-Tariff scheme. The proposed scheme yields higher average utility per EC and lower CPS cost than FIT in the reported comparisons, while network size and price bounds affect outcomes.

VI. NUMERICAL RESULTS

  • Results average 1,000 independent simulation runs over random EC capacities, with no observed failure to produce a solution in any iteration.The reported setup excludes other purchasing costs, including costs associated with insufficient energy purchases.Source: §VI · p. 7
  • Approximately 6 iterations bring EC utility and offered energy to EMES; lower-capacity ECs receive higher unit prices, while CPS prices fluctuate before settling.The simulation uses a 700 kWh energy deficit and 5 ECs.Source: Figure 1 · p. 7, §VI · p. 7
  • As EC count rises from 5 to 15, CPS cost falls, then rises from 20 to 25 because minimum payments apply to more ECs; per-EC utility falls as EC count increases.Per-EC utility increases with greater energy deficiency in the reported cases.Source: §VI · p. 7, Figure 2 · p. 8
  • As pmax increases, average CPS cost decreases and then stabilizes; the CPS can use the wider price bound to lower payments to other ECs after paying a higher price to a lower-capacity EC.The figure considers N = 5; cost is also reported as proportional to P.Source: §VI · pp. 7–8, Figure 3 · p. 8
  • 1.5 times better average utility per EC and lower average total CPS cost are reported for the proposed scheme than FIT.The utility ratio averages results across networks of 5, 10, 15, 20, and 25 ECs; the reported CPS cost advantage increases with total unit price P.Source: §VI · pp. 8–9, Figure 4 · p. 9

VII. CONCLUSION

The scheme combines end-user benefit maximization with a minimum CPS cost, and its Stackelberg equilibrium is socially optimal. Simulations report improvement over conventional feed-in tariffs, while the per-time-slot model leaves inter-temporal flexibility unmodeled.

VII. CONCLUSION

  • The Stackelberg game reaches an equilibrium comprising socially optimal energy and price vectors for the ECs and the CPS.Source: §VII · p. 9
  • Simulation showed noticeable performance improvements over a conventional feed-in-tariff scheme.Source: §VII · p. 9
  • The per-time-slot approach ignores inter-temporal demand flexibility, which the authors propose to address with learning curves and state equations.Source: §VII · p. 9
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