Source-linked AI summary
A Game-Theoretic Approach to Energy Trading in the Smart Grid
Yunpeng Wang, Walid Saad, Zhu Han, H. Vincent Poor, Tamer Başar
TL;DR
The paper studies how geographically distributed storage units should choose surplus-energy sale quantities amid complex smart-grid trading interactions. It combines a noncooperative game with a double-auction market, proves equilibrium existence, and provides a guaranteed-convergent algorithm. Simulations report higher average utility per seller than a greedy scheme, with improvements varying across market settings.
Problem
The paper addresses complex energy-trading decisions among storage units and the challenge of establishing equilibrium when auction pricing creates discontinuous utilities.
Method
The paper combines a noncooperative game in which sellers choose sale quantities with a double auction that determines traded prices and quantities.
Results
72.3% to 234.4% performance improvement in average utility per seller relative to the greedy scheme is reported as the number of buyers increases.
Takeaways & Limitations
The proposed framework provides a strategic storage-selling model with an auction-based market, Nash-equilibrium existence, and an algorithm guaranteed to converge to an equilibrium.
Abstract
from arXiv · showhide
Electric storage units constitute a key element in the emerging smart grid system. In this paper, the interactions and energy trading decisions of a number of geographically distributed storage units are studied using a novel framework based on game theory. In particular, a noncooperative game is formulated between storage units, such as PHEVs, or an array of batteries that are trading their stored energy. Here, each storage unit's owner can decide on the maximum amount of energy to sell in a local market so as to maximize a utility that reflects the tradeoff between the revenues from energy trading and the accompanying costs. Then in this energy exchange market between the storage units and the smart grid elements, the price at which energy is traded is determined via an auction mechanism. The game is shown to admit at least one Nash equilibrium and a novel proposed algorithm that is guaranteed to reach such an equilibrium point is proposed. Simulation results show that the proposed approach yields significant performance improvements, in terms of the average utility per storage unit, reaching up to 130.2% compared to a conventional greedy approach.
I. INTRODUCTION
The paper addresses complex energy-trading decisions among storage units in smart grids by combining strategic seller choices with auction-based market pricing. It contributes a double-auction game framework, equilibrium analysis, and a guaranteed learning algorithm.
- Energy storage deployment creates challenges involving interactions between storage units, their owners, and other smart-grid elements.
- The proposed market uses a double auction for multiple buyers and sellers, with seller quantities influencing the trading price dynamically.
- The paper establishes Nash-equilibrium existence for the resulting discontinuous-utility game and proposes a learning algorithm guaranteed to reach an equilibrium.
- The framework lets storage units strategically choose how much stored energy to sell while accounting for trading benefits and associated costs.Costs may include battery life reduction and storage-unit efficiency.
- The paper analyzes both strategic energy-selling decisions and the mechanism determining trading prices in the storage-unit market.
II. SYSTEM MODEL
The system model contains energy-deficient smart-grid elements that buy surplus energy from distributed storage units. Buyers have specified demands, while sellers choose sale quantities subject to capacity, retention, and reservation-price constraints.
- K energy-deficient grid elements acquire energy from N customer-owned storage units holding surplus energy.
- Buyers seek fixed energy amounts xk over a specified period, representing practical requirements or anticipated additional demand.
- Each seller chooses a sale amount ai while respecting its maximum market quantity, storage capacity, and retained energy.Bi is the maximum amount offered, Ci,max is storage capacity, and Di is energy retained by seller i.
- Sellers set reservation prices si below which they will not trade energy.
- The market determines exchanged quantities and prices, while sellers balance revenue from surplus energy against the consequences of choosing ai.
III. A GAME-THEORETIC APPROACH TO ENERGY TRADING
This section introduces the noncooperative seller game and the double-auction mechanism used to determine energy-trading outcomes. The surrounding notation organizes the market’s sellers, buyers, and auction inputs.
- The paper formulates a noncooperative game among storage-unit sellers before analyzing the energy-trading mechanism.
- The section introduces notation for the total number of sellers and other elements used in the trading model.
- The market analysis uses a double auction to study energy trading among multiple sellers and buyers.
A. Noncooperative Game Model
The noncooperative game models each storage unit as a player choosing a sale quantity to maximize utility. Utility depends on market prices, exchanged quantities, buyer demands and bids, and increasing selling costs.
- Each seller i chooses a strategy ai from Ai := [0, Bi], where ai is the energy amount offered for sale.
- Each seller maximizes a utility that captures revenues from traded energy and the cost of selling it.
- The utility depends on opponents’ strategies through the strategy vector a and the opponents’ action vector a−i.
- Prices pik(a) and exchanged quantities qik determine seller i’s market revenue for trades with buyer k.
- The selling-cost function f(·) increases with the amount of energy sold.
B. Double Auction Mechanism for Market Analysis
The paper uses a double auction to determine trading prices, traded quantities, and participating sellers and buyers from strategic offers, while preserving truthful reporting. Supply and demand are sorted and intersected to identify participants; an allocation rule then handles over-demand and oversupply.
- Auction inputs and sorting: The double auction derives trading prices, traded quantities, and participating sellers and buyers from sellers’ offered quantities, reservation prices, buyers’ requested quantities, and bids.The mechanism is applied to the strategy vector a and the market’s reservation prices, quantities, and bids.
- Auction inputs and sorting: Sellers are sorted by increasing reservation price and buyers by decreasing reservation bid before generating supply and demand curves.Equal prices or bids may be grouped into a single virtual seller or buyer.
- Price and participation: At the supply-demand intersection, sellers i < L and buyers k < M trade, while the trading price is selected within [sL, bM].The intersection and participating indices can vary with the sellers’ maximum-energy strategy vector a.
- Seller utilities: After the unified trading price is found, each participating seller’s utility depends on the quantity sold rather than the identity of the buyer, under the stated quadratic cost example.The analysis can accommodate other cost functions f as well.
- Energy allocation: The allocation rule divides the traded volume to preserve auction truthfulness, with sellers selling all offered energy under excess demand and sharing oversupply burdens when supply exceeds demand.The mechanism iteratively reallocates oversupply when a seller’s calculated quantity would otherwise become negative.
- Truthfulness: No seller or buyer benefits from misreporting its reservation price or bid, so the resulting double auction is strategy-proof and truthful.Truthful reporting is the stated property of the proposed allocation mechanism.
IV. PROPOSED SOLUTION AND ALGORITHM
The paper establishes that the proposed storage-unit game has a pure-strategy Nash equilibrium despite auction-induced utility discontinuities, then develops an iterative best-response algorithm guaranteed to converge to one. The solution uses utility regularity, auction-aware analysis, and seller updates toward best responses.
- Nash-equilibrium existence: The game possesses at least one pure-strategy Nash equilibrium despite discontinuities introduced by the double-auction trading price.The proof relies on compact action sets and utility properties including graph continuity, upper semi-continuity, and quasi-concavity.
- Nash-equilibrium existence: The utility is upper semi-continuous and quasi-concave in each seller’s offered quantity, supporting equilibrium existence.The analysis treats jump points and shows that the utility first increases and then may decrease around an inflection point.
- Iterative algorithm: A best response is any strategy that gives a seller utility at least as high as every other strategy in its action set, given the other sellers’ strategies.This definition provides the decision rule used by the iterative energy-trading algorithm.
- Iterative algorithm: The proposed iterative algorithm approaches sellers’ best responses and is guaranteed to converge to a Nash equilibrium for a suitable inertia weight.The weight addresses price-changing loops that can arise because the auction price is piecewise continuous.
- Scope boundary: The convergence analysis assumes that the sellers can sustain and share the oversupply burden; cases where oversupply cannot be split among all sellers are omitted.The paper states that analogous results can be generated for those cases but does not analyze them because of space limitations.
- Practical operation: The algorithm can be implemented sequentially or in parallel, after which equilibrium bids are submitted to the practical double auction.Sequential updates observe prior players’ actions, whereas parallel updates respond to the previous iteration’s actions.
V. SIMULATION RESULTS AND ANALYSIS
Simulations evaluate the proposed game and auction approaches across market sizes, update schemes, time-dependent trading, and penalty factors. The results show increased utility and trading activity relative to greedy behavior, while convergence and equilibrium actions depend on the algorithm and setting.
- 75–220 MWh is the simulated surplus range per storage unit, with seller reservation prices of $10–$50/MWh and buyer bids of $15–$60/MWh.
- Equilibrium actions: At equilibrium, competition makes seller actions generally decrease, so not all players sell their maximum available energy.The sequential algorithm can initially favor the first player because that player has a higher opportunity to sell energy.
- Utility performance: 130.2% is the maximum average-utility improvement reported for the proposed approach relative to the conventional greedy approach.This maximum occurs at K = 5 buyers and N = 4 sellers.
- Convergence and complexity: The parallel algorithm requires many more iterations than the sequential algorithm, although the sequential algorithm has lower computational complexity O(L + M).The sequential algorithm’s iteration count is similar to classical best-response dynamics when those dynamics converge.
- Algorithm comparison: The proposed sequential and parallel algorithms produce average actions greater than the greedy strategy, with nearly identical equilibrium actions.The higher actions are associated with enhanced utility in the corresponding comparison.
- Penalty sensitivity: Increasing the penalty factor τ decreases utility because the penalty term grows while auction revenue remains unchanged.At τ = 1, the penalty strongly influences each player’s utility, although total utility remains positive.
- Utility performance: 72.3%–234.4% is the reported utility improvement range as the number of buyers increases for N = 6 sellers, relative to the greedy scheme.The range corresponds to K = 4 and K = 10, respectively.
- Time-dependent trading: The time-dependent game lets all players act as buyers or sellers each hour while respecting battery-capacity limits.Player 2 reaches maximum capacity twice and sells at the second and fourth hours.
VI. CONCLUSIONS
The paper models storage-unit energy selling as a noncooperative game combined with double-auction pricing, establishes equilibrium existence, and provides an algorithm to reach equilibrium while improving average utility.
- Each storage unit strategically chooses its maximum energy surplus offered for sale to optimize benefits against associated costs.
- A double-auction mechanism determines market trading prices and produces a strategy-proof outcome.
- The formulated game admits a Nash equilibrium, whose properties are analyzed.
- A novel algorithm enables storage units to reach a Nash equilibrium in the model.
- Simulation results show improved average utility while storage units act strategically.