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Peer-to-Peer Electricity Market Analysis: From Variational to Generalized Nash Equilibrium

Hélène Le Cadre, Paulin Jacquot, Cheng Wan, Clémence Alasseur

arXiv:1812.02301v1math.OCcs.GT

TL;DR

The paper studies how prosumer trade preferences affect centralized and distributed peer-to-peer electricity markets. It characterizes variational and generalized equilibria, compares them with social-welfare optima, and evaluates congestion, waste, and efficiency in test networks. Variational equilibria coincide with social-welfare optima, while preferences substantially alter trade patterns and generalized-equilibrium analysis remains computationally limited in larger networks.

  • Problem

    The paper asks how differentiated trade preferences and decentralized decision-making affect equilibrium, social welfare, congestion, and renewable-energy waste in peer-to-peer electricity markets.

  • Method

    The authors formulate centralized and fully distributed peer-to-peer market designs, characterize variational and generalized Nash equilibria, and analyze three-node and IEEE 14-bus test cases.

  • Results

    Variational equilibria coincide with social-welfare optima, while preferences strongly affect trade structure; the study also analyzes congestion, energy waste, and generalized-equilibrium efficiency.

  • Takeaways & Limitations

    Peer-to-peer market imperfections can be studied through generalized equilibria, while the variational-equilibrium outcome remains socially optimal under the paper’s equal-valuation assumption.

  • Takeaways & Limitations

    For the IEEE 14-bus network, the authors could not compute a generalized Nash equilibrium different from the variational equilibrium because brute-force search would require a 22-dimensional space.

Abstract

from arXiv · show

We consider a network of prosumers involved in peer-to-peer energy exchanges, with differentiation price preferences on the trades with their neighbors, and we analyze two market designs: (i) a centralized market, used as a benchmark, where a global market operator optimizes the flows (trades) between the nodes, local demand and flexibility activation to maximize the system overall social welfare; (ii) a distributed peer-to-peer market design where prosumers in local energy communities optimize selfishly their trades, demand, and flexibility activation. We first characterizethe solution of the peer-to-peer market as a Variational Equilibrium and prove that the set of Variational Equilibria coincides with the set of social welfare optimal solutions of market design (i). We give several results that help understanding the structure of the trades at an equilibriumor at the optimum. We characterize the impact of preferences on the network line congestion and renewable energy waste under both designs. We provide a reduced example for which we give the set of all possible generalized equilibria, which enables to give an approximation of the price ofanarchy. We provide a more realistic example which relies on the IEEE 14-bus network, for which we can simulate the trades under different preference prices. Our analysis shows in particular that the preferences have a large impact on the structure of the trades, but that one equilibrium(variational) is optimal.

1 Introduction

The paper situates peer-to-peer electricity trading among increasingly decentralized market designs and studies how preferences, privacy, and strategic behavior affect efficiency and network operation.

  • Privacy and implementation: Privacy prevents prosumers from sharing some target-demand and renewable-generation information, requiring others to forecast it and potentially introducing bias into market outcomes.
  • Distributed optimization approaches: Distributed optimization methods in prior work coordinate local decisions toward centralized social-welfare objectives, whereas this paper assumes no central authority coordinates exchanges, prices, or information.
  • Market-design motivation: The paper compares centralized coordination with fully distributed peer-to-peer trading, where prosumers exchange energy within local communities without a central exchange authority.Distributed designs can improve resilience, but their communication mechanisms are more complex and privacy can create information asymmetry.
  • Preferences and platforms: Differentiated prices represent prosumers’ preferences for trade characteristics such as renewable generation, location, transport distance, or prosumer size.
  • Paper contributions: The paper characterizes variational equilibria, proves their equivalence to social-welfare optima, and analyzes generalized equilibria when agents do not coordinate trading-price valuations.
  • Paper contributions: It analytically studies how preferences affect network congestion and energy waste, using a three-node arbitrage example and an IEEE 14-bus network.

Notations

The paper uses bold letters to denote vectors and matrices throughout its notation.

  • Notations: Vectors and matrices are denoted by bold letters throughout the paper.

2 Prosumers and Local Communities

The model represents prosumers and local communities on a radial distribution network, with bounded demand, flexibility, bilateral trades, and local balancing constraints.

  • 2.1 Generic Framework: The network is radial, with conventional-generation root node 0 linking local energy communities to the transmission network and trading with every other node.
  • 2.1 Generic Framework: Demand and flexibility activation are bounded by node-specific capacity sets, and renewable self-generation is modeled as an exogenous random realization.
  • 2.1 Generic Framework: Each prosumer chooses demand, flexibility activation, and bilateral exchange quantities, with positive quantities denoting purchases and negative quantities denoting sales.
  • 2.1 Generic Framework: Trades satisfy reciprocity and line-capacity constraints, while each node must balance local supply and demand using net imports, flexibility, self-generation, and renewable generation.
  • 2.2 Costs and Utilities: Bilateral differentiation prices are positive and make purchases from particular neighbors more or less attractive according to trade preferences and policy taxes.
  • 2.2 Costs and Utilities: Utility equals usage benefit minus flexibility-activation and bilateral-trading costs, with quadratic activation costs and strictly concave demand benefits.
  • 2.3 Information and Market Designs: The centralized operator receives private target-demand and renewable-generation information, whereas peer-to-peer agents retain those values locally.

3 Centralized Market Design

The centralized market design has a global Market Operator maximize social welfare subject to operational, trading, and balancing constraints. Its optimality conditions relate nodal prices to preferences, congestion, and bilateral prices, and yield structural results on demand, imports, waste, and congestion.

  • The Market Operator maximizes social welfare subject to demand, flexibility, trading-flow, reciprocity, and nodal balancing constraints.
  • Nodal prices equal bilateral trade prices plus product-differentiation and congestion-price terms, linking preferences and line constraints to market outcomes.
  • At the optimum, demand, flexibility activation, and net imports at each node are linear functions of that node’s nodal price.
  • No energy waste requires at least one prosumer whose demand-capacity upper bound minus flexibility-activation lower bound is at least its renewable generation.This condition is necessary, and the paper notes that capacity sizing and renewable-generation clipping strategies are outside its scope.
  • Even without excess renewable production, sufficiently large trade-preference prices can produce energy waste.
  • Strictly positive or negative preference cycles force at least one cycle edge to full capacity, creating congestion; positive cycles also represent arbitrage opportunities that can increase social welfare.The paper suggests transaction fees as one mechanism-design response to cycling behavior.

4 Peer-to-Peer Market Design

The peer-to-peer market is modeled as a game in which prosumers independently choose demand, flexibility activation, and bilateral trades subject to individual and reciprocity constraints. Variational Equilibria coincide with centralized social-welfare optima, while generalized equilibria can be multiple and less efficient.

  • 4.1 General Nash Equilibrium and Variational Equilibrium: Each prosumer maximizes individual utility by selecting demand, flexibility activation, and bilateral trades under local, capacity, balance, and reciprocity constraints.The resulting peer-to-peer design consists of one optimization problem per agent, with shared constraints enforcing trade reciprocity.
  • 4.1 General Nash Equilibrium and Variational Equilibrium: Bilateral trade prices are equal under the complete-market scenario but may differ between prosumers when no market determines the shared-constraint price system.The latter case permits ζ_nm ≠ ζ_mn, whereas Variational Equilibrium imposes equality of shared-constraint multipliers.
  • 4.1 General Nash Equilibrium and Variational Equilibrium: Variational Equilibria coincide with the social-welfare optima of the centralized market.This equivalence holds when multipliers for shared trade constraints are equal across the participating agents.
  • 4.1 General Nash Equilibrium and Variational Equilibrium: Generalized Nash Equilibria are generally nonunique and may differ from the centralized optimum, motivating analysis of the full equilibrium set.The paper introduces SOLGNEP to denote the set of generalized Nash equilibrium solutions.
  • 4.2 Dealing with Congestion: Preferences shape congestion: with asymmetric preferences, the node with the smaller preference for the other saturates the connecting line under the stated capacity and root-preference assumptions.More generally, preference asymmetries determine which direction reaches full capacity, and directed opposite trades cannot both simultaneously saturate their capacities.
  • 4.2 Dealing with Congestion: A directed preference cycle implies that at least one trade opposing the cycle operates at full capacity.This result provides a structural congestion condition at equilibrium.
  • 4.3 Efficiency of Generalized Equilibria: Variational Equilibria have price of anarchy 1, whereas generalized equilibria can include outcomes that are not socially optimal.When bilateral prices nearly equalize, the generalized equilibrium approaches the Variational Equilibrium, but residual gaps can represent efficiency loss.

5 Test Cases

The test cases show that preference prices can redirect trades, alter congestion, and produce generalized equilibria with substantially lower welfare than the centralized optimum. In the IEEE 14-bus case, heterogeneous preferences increase trading and congestion, while the computed variational equilibrium remains the benchmark optimum.

  • 5.1 A Three Nodes Network with Arbitrage Opportunity: In the centralized solution, the preference cycle saturates the trade between nodes 1 and 2, whereas the depicted GNE congests that edge in the reverse direction.The cycle reflects that node 2 prefers buying from node 0 over node 1, so welfare increases when node 1 buys from node 2 and node 2 buys from node 0.
  • 5.1 A Three Nodes Network with Arbitrage Opportunity: All GNEs in the reduced example saturate the edge between nodes 1 and 2 in one direction or the other.The GNE set forms two connected components corresponding to the two saturation directions.
  • 5.1 A Three Nodes Network with Arbitrage Opportunity: The reduced three-node example gives a PoA bound of at least 378.3, meaning some peer-to-peer equilibria can have social welfare more than 50% below the optimum.The low-welfare GNE shown is the worst found by sampling, while evaluating the exact minimum is difficult because the problem is nonconvex.
  • 5.2 IEEE 14-bus Network: The IEEE 14-bus setup models prosumers and generators on a radial network with neighbor-limited trades, evaluated for one time period without energy waste in the illustrated solutions.The network includes consumer-only, renewable, thermal, and mixed prosumer buses, with trades limited by edge capacities.
  • 5.2 IEEE 14-bus Network: The IEEE 14-bus experiments compare four preference-price cases and compute centralized solutions, which also correspond to variational equilibria.Symmetric prices and uniform local-trade preferences produce the same VNE trade solution as uniform prices.
  • 5.2 IEEE 14-bus Network: In the IEEE 14-bus case, heterogeneous differentiation prices substantially increase traded quantities and congest ten of twenty-two edges, whereas uniform prices leave every edge uncongested.Some edges are almost unused under uniform prices, while marginal prices are heterogeneous under heterogeneous preferences.

6 Dealing with Privacy

The privacy analysis models prosumers’ forecasts of private demand and renewable generation, derives the resulting biased-forecast equilibrium, and bounds its utility impact. In a three-node example, privacy-induced social-welfare bias ranges from 1.2% to 3.6%, depending on utility-parameter symmetry.

  • 6 Dealing with Privacy: Privacy prevents prosumers from observing neighbors’ target demand and renewable generation, so they forecast these quantities and obtain a biased-forecast equilibrium.The forecasts use linear estimates with random demand and generation biases, assumed to be centered Gaussian variables.
  • 6 Dealing with Privacy: The privacy equilibrium generally differs from the full-information equilibrium, but the bias disappears when the relevant sum of forecast-error differences tends to zero.Under that condition, the estimated nodal prices converge to the true nodal prices, and utility estimation can be unbiased.
  • 6.1 Quantifying the Loss Caused by Privacy: The estimated-utility bias increases with the bilateral valuation ratio when βn > 0 and decreases when βn < 0.The valuation ratio captures the relative preference assigned to trades with the root node, linking pricing preferences to privacy-induced utility distortion.
  • 6.1 Quantifying the Loss Caused by Privacy: Each prosumer’s utility bias is bounded above by Φn over the generalized-equilibrium set characterized through parameterized variational inequalities.The equilibrium set is represented by allowing each valuation ratio rn to vary within a specified interval.
  • 6.2 Dealing with Privacy in the Three Nodes Network: 1.2%–3.6%: social-welfare bias is minimized when prosumers have identical utility parameters and maximized when their parameters are asymmetric.The three-node experiment varies ã1 and ã2 while fixing the maximum usage benefit at b̃1 = b̃2 = 60.

7 Conclusion

The paper compares centralized and distributed peer-to-peer electricity markets with differentiated trading preferences and private information. It shows that variational equilibria coincide with centralized social-welfare optima, while privacy and decentralization shape market outcomes and resilience.

  • 7 Conclusion: Variational equilibria of the distributed peer-to-peer market coincide with the social-welfare-optimal solutions of the centralized benchmark.The centralized operator optimizes trades, local demand, and flexibility activation, whereas prosumers optimize these decisions selfishly in the distributed design.
  • 7 Conclusion: The analysis characterizes how differentiated preferences affect network-line congestion and renewable-energy waste under both market designs.It also studies generalized equilibria under market imperfections and illustrates the results with numerical test cases.
  • 7 Conclusion: Distributed operation can retain functionality after a node failure or attack because decisions and information are not concentrated in one entity.The involvement of all prosumers also allows actions to adapt to the grid state.
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