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Price discrimination for energy trading in smart grid: A game theoretic approach

Wayes Tushar, Chau Yuen, David Smith, H. Vincent Poor

arXiv:1512.03485v1eess.SYcs.GTcs.SY

TL;DR

Smart-grid pricing schemes commonly use a shared unit price, while this paper examines fair price discrimination for energy users selling surplus to a shared facility controller. It models pricing as a cake-cutting game and proposes a distributed algorithm; the paper reports socially optimal, Pareto-optimal and envy-free properties, with a case study showing more energy purchased from users than from the grid at the same budget.

  • Problem

    The paper studies fair price selection among energy users because maintaining fairness in discriminatory pricing is a challenge for sustaining energy trading.

    Source: §I · pp. 1–2

  • Method

    A cake-cutting game uses a benefit function based on users’ surplus energy and price sensitivity, with a distributed algorithm for users to determine prices through interaction with the SFC.

    Source: §I · p. 2, §II · p. 4

  • Results

    The game’s solution is socially optimal and Pareto optimal, and the scheme is described as envy-free; with a 1000-cent budget, the SFC can buy 58.3 kWh more from users than exclusively from the grid.

    Source: C. Properties of CCS · p. 6, §V · p. 10

  • Takeaways & Limitations

    The paper presents discriminatory pricing as a viable way for users to sell surplus energy to an SFC under a constrained budget.

    Source: §V · p. 10

    The proposed scheme is studied for a setting that is not time-varying; extending it to a time-varying environment is left for future work.

    Source: D. Algorithm · p. 7

Abstract

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Pricing schemes are an important smart grid feature to affect typical energy usage behavior of energy users (EUs). However, most existing schemes use the assumption that a buyer pays the same price per unit of energy to all suppliers at any particular time when energy is bought. By contrast, here a discriminate pricing technique using game theory is studied. A cake cutting game is investigated, in which participating EUs in a smart community decide on the price per unit of energy to charge a shared facility controller (SFC) in order to sell surplus energy. The focus is to study fairness criteria to maximize sum benefits to EUs and ensure an envy-free energy trading market. A benefit function is designed that leverages generation of discriminate pricing by each EU, according to the amount of surplus energy that an EU trades with the SFC and the EU's sensitivity to price. It is shown that the game possesses a socially optimal, and hence also Pareto optimal, solution. Further, an algorithm that can be implemented by each EU in a distributed manner to reach the optimal solution is proposed. Numerical case studies are given that demonstrate beneficial properties of the scheme.

I. INTRODUCTION

Smart-grid pricing schemes can shape energy use and help shared facility controllers obtain energy from users, but most schemes assign all sellers the same unit price. This paper studies whether price discrimination can be fair while maximizing user benefits and sustaining energy trading.

I. INTRODUCTION

  • Most existing smart-grid pricing schemes give all energy users the same selling price per unit at a given time.The paper identifies fair price distribution among users as a challenge for sustaining discriminatory trading.Source: §I · pp. 1–2
  • In the example, discriminatory prices raise EU2’s revenue from 100 to 176 cents while reducing EU1’s from 700 to 576 cents and the SFC’s cost from 800 to 752 cents.The example compares equal prices of 20 cents/kWh with prices of 18 and 22 cents/kWh, respectively; the SFC needs 40 kWh.Source: §I · p. 2, Table I · p. 2
  • The proposed cake-cutting game lets users with less available energy choose a higher unit price, with price also adapted to users’ sensitivity.Its benefit function is designed to support discriminatory pricing and a socially optimal solution, which the paper says implies Pareto optimality and an envy-free market.Source: §I · p. 2
  • An algorithm lets each energy user interact with the SFC in a distributed manner to determine the price vector, and its convergence to the optimal solution is demonstrated.The paper also reports numerical case studies of the scheme’s properties.Source: §I · p. 2
  • The authors envision discriminatory pricing as a future addition to real-time pricing, including when an SFC seeks to reduce dependence on a dominant supplier.Their example describes the SFC buying less from a large-surplus user after offering a smaller supplier a higher price.Source: §I · p. 2

II. SYSTEM DESCRIPTION

The system models energy users selling surplus generation to a shared facility controller or the main grid, while the controller buys energy for shared facilities. User prices depend on available surplus and price sensitivity, and the proposed benefit function supports discriminatory pricing under a controller budget.

II. SYSTEM DESCRIPTION

  • Each energy user has distributed generation but no storage, and may sell surplus to the SFC or the main grid; the SFC relies on these sources and the grid for shared-facility demand.The SFC has no generation capacity and is interested in buying from users because the main-grid unit price is significantly higher.Source: §II · p. 3
  • The SFC’s budget C constrains total payments to users; the authors require it to remain below the cost of buying the same energy from the grid.They note that selecting a suitable budget may depend on factors such as users’ willingness to sell, which would require behavioral models to estimate reasonably.Source: §II · p. 3
  • Each user’s price depends on its surplus e_n and sensitivity α_n, with larger-surplus users modeled as having lower marginal benefit for the same price choice.The paper describes α_n as capturing how a user’s benefit responds to price changes; users with less surplus may require a higher unit price to find trading worthwhile.Source: §II · pp. 3–4
  • The proposed quadratic benefit function X_n models discriminatory pricing and is concave in the unit price p_n, so an excessively high price can reduce a user’s benefit.The paper also states that X_n increases with P_n and decreases with α_n.Source: §II · p. 4
  • The objective is to choose prices within the SFC budget that maximize the sum of users’ benefits.The paper motivates this system objective by noting that unprofitable trading may discourage users from participating.Source: §II · pp. 3–4

A. Brief Background

The paper frames price selection as dividing the SFC’s finite budget among energy users through a cake-cutting game. It motivates this approach as a way to preserve privacy while pursuing fair allocations.

A. Brief Background

  • A cake-cutting game divides a finite resource pool according to players’ valuation criteria; here, the SFC’s total budget is the cake and each EU receives a budget allocation.Each EU chooses a price within the budget constraint to pursue its objective.Source: A. Brief Background · p. 5
  • Higher price sensitivity shifts an EU’s maximum net benefit to a lower price, while lower surplus corresponds to a relatively higher maximizing price than other EUs.These relationships are illustrated in Fig. 2.Source: Figure 2 · p. 5
  • The authors choose a distributed cake-cutting approach over centralized techniques partly to keep EU price sensitivities and available energy private.They note that centralized access to this information could raise privacy concerns or allow the SFC to favor some EUs.Source: A. Brief Background · p. 5
  • A complete socially optimal allocation in a cake-cutting game is also Pareto optimal.The paper uses this property to connect maximizing total EU benefit with an allocation in which no player can improve without harming another.Source: A. Brief Background · p. 5

B. Proposed Game

The proposed game has EUs choose selling prices subject to the SFC’s shared budget constraint. Its cake-cutting solution is a unique global maximizer of total benefit, and the paper links complete allocation to Pareto optimality.

B. Proposed Game

  • Each EU chooses a selling price and offers it to the SFC, which checks whether the resulting total expense satisfies the shared budget constraint.The game includes EUs with surplus energy who participate when the offered price exceeds the grid’s buying price.Source: B. Proposed Game · p. 5
  • Because the budget constraint couples EUs’ price strategies, the game is treated as a variational inequality problem whose variational equilibrium is called the cake-cutting solution.The cake-cutting solution is defined by the game’s strategy-set inequalities.Source: B. Proposed Game · p. 6
  • The game has a unique cake-cutting solution that is also the unique global maximizer of the EUs’ total benefit and thus socially optimal.The proof establishes strict monotonicity over the feasible price set.Source: B. Proposed Game · p. 6
  • The authors argue that a complete allocation is needed to establish Pareto optimality and connect a socially and Pareto optimal allocation with fair, envy-free prices.They state that no EU can change strategy in a Pareto-efficient allocation without hurting another EU.Source: B. Proposed Game · p. 6

C. Properties of CCS

The paper establishes a socially optimal cake-cutting solution and shows that its budget allocation is complete and Pareto optimal. It associates this outcome with fair, envy-free prices and relates price choices to surplus and price sensitivity.

C. Properties of CCS

  • The proposed game has a socially optimal cake-cutting solution, which maximizes total benefit to the EUs.The proof uses a variational-inequality formulation and the positive price sensitivities in the Jacobian to establish strict monotonicity.Source: C. Properties of CCS · p. 6
  • Because the KKT multipliers are positive, the total budget allocation equals the SFC’s budget, making the solution complete and Pareto optimal.The authors identify each multiplier as belonging to an EU’s price-choice constraint.Source: C. Properties of CCS · p. 6
  • The authors characterize the resulting price allocation as fair and envy-free, allowing EUs to trade without envying others’ payments and supporting market transparency.They state that this property enables the discriminatory-pricing market to be sustained.Source: C. Properties of CCS · p. 6
  • EUs with greater surplus and/or greater price sensitivity choose relatively lower unit prices than EUs with less surplus and/or lower sensitivity at the optimal solution.The paper relates this price-choice property to the socially and Pareto optimal cake-cutting solution.Source: C. Properties of CCS · pp. 6–7

D. Algorithm

The paper uses the S-S hyperplane projection method for the strictly monotone game, with two-way communication between EUs and the SFC. It proves the algorithm reaches the cake-cutting solution, while noting that its analysis is static and single-time-slot.

D. Algorithm

  • The S-S hyperplane projection method uses two-way EU–SFC communication and two projections per iteration to solve the game’s variational inequality.It projects first onto the feasible set, then onto the intersection of that set with a separating half-space.Source: D. Algorithm · p. 7
  • The procedure starts with an SFC-announced budget; each EU then determines its sale quantity, and the EUs use the game to reach the optimal price vector.The SFC may set its budget using a statistical technique such as a Markov chain model based on historical budgets.Source: D. Algorithm · p. 7
  • The algorithm is guaranteed to reach the cake-cutting solution because the S-S method converges for strictly monotone variational inequalities, and the proposed game has that property.The proof states that the resulting solution set is non-empty.Source: D. Algorithm · p. 7
  • The analysis considers a static game at one time instant and does not model changing parameters across time slots.The authors identify extending the scheme to a time-varying environment as future work, with possible incorporation into real-time pricing.Source: D. Algorithm · p. 7

IV. CASE STUDY

The case study examines convergence to the game’s solution and how SFC budget, EU participation, and price sensitivity relate to trading outcomes. Its reported results show that group size and sensitivity shape per-EU benefits, while the proposed scheme benefits both the SFC and participating EUs in the tested cases.

IV. CASE STUDY

  • After 8 iterations, the price vector for 10 EUs reaches the game’s CCS, with different EUs choosing different prices.The study attributes the differing prices to differences in available energy surplus and price sensitivity.Source: §IV · p. 7, Figure 3 · p. 8
  • 1250 to 390: average net benefit per EU falls as the group grows from 10 to 40 while sharing a 1000-cent budget.This is an approximately 68% decrease.Source: §IV · p. 8, Figure 4 · p. 8
  • 15.1%: average net benefit per EU decreases as sensitivity rises from 1 to 3 with a 1000-cent budget.For a 2000-cent budget, the decrease is 9.8%; the study reports that higher sensitivity restricts price choice.Source: §IV · p. 9, Table III · p. 9
  • With a 1000-cent budget, EU participation falls from 100% for 10 EUs to 0% for 40 EUs.At group sizes of 20 and 30, participation is 40% and 20%; an EU does not trade if its payment falls below 8.00 cents/kWh.Source: §IV · p. 9, Table II · p. 9
  • The case study reports that the proposed scheme benefits both the SFC and participating EUs for the system parameters considered.The example assumes surplus energy uniformly distributed from 3.6 to 12.25 kWh, a 1000-cent SFC budget, and sensitivity values from 1 to 3; the authors note that parameters may vary with SFC needs, weather, time, and national energy policy.Source: §IV · pp. 7, 10

V. CONCLUSION

The paper concludes that its cake cutting game supports discriminate pricing for surplus energy sold to an SFC, with a socially optimal, Pareto-optimal solution and a distributed algorithm that converges to it. Numerical comparisons report benefits for both EUs and the SFC relative to trading with the main grid.

V. CONCLUSION

  • The cake cutting game has a socially optimal solution that is also Pareto optimal, and each EU’s distributed algorithm converges to the optimal solution.The game is modeled as a variational inequality problem to establish the cake cutting solution.Source: §V · p. 10
  • For 10 EUs, total revenue is 1000 cents through the proposed scheme versus 648 cents from grid sales, a 352-cent difference.The passage says the proposed scheme’s revenue improvement is greater with more EUs.Source: Table V · p. 10, §V · p. 10
  • 58.3 kWh more energy can be bought through the proposed scheme than exclusively from the grid for a 1000-cent SFC budget.The passage reports this comparison and says the pattern holds across all considered SFC budgets.Source: §V · p. 10
  • Future work includes relating the SFC budget to the number of participating EUs and studying real-time pricing and time-varying environments.The proposed budget-to-participant relationship could be studied with an interactive Stackelberg game under imperfect budget information.Source: §V · p. 10
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