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arxiv: 1906.09891 · v1 · pith:RH2LRTUBnew · submitted 2019-06-24 · 🧮 math.OC

Buy or Sell? Energy Sharing of Prosumers on Constrained Networks

Yue Chen , Shengwei Mei , Wei Wei , Steven H. Low , Adam Wierman , Feng Liu This is my paper

Pith reviewed 2026-05-25 17:46 UTC · model grok-4.3

classification 🧮 math.OC
keywords energy sharingprosumersgeneralized Nash gamemarket equilibriumnetwork constraintsprice regulationdemand response
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The pith

A price-regulated energy sharing mechanism yields unique fair equilibria that approach social optimum as prosumers increase.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a market clearing procedure for energy sharing among prosumers that uses general supply-demand function bids and incorporates network constraints. The clearing step produces a generalized Nash game whose equilibrium exists and is unique when the network is uncongested. When congestion occurs, a price-regulation step is added to select one unique equilibrium that is fair across participants. The mechanism is shown to be individually rational, with sharing prices having identifiable components. As the number of prosumers grows the outcome converges to the social optimum, and added competition can lower social cost even when the number of physical resources stays fixed.

Core claim

Under a general supply-demand function bidding scheme, a sharing market clearing procedure considering network constraints is proposed, which gives rise to a generalized Nash game. The existence and uniqueness of market equilibrium are proved in non-congested cases. When congestion occurs, a price-regulation procedure is introduced which outcomes a unique equilibrium that is fair to all participants. Properties of the improved sharing mechanism, including the individual rational behaviors of prosumers and the components of sharing price, are revealed. When the number of prosumers increases, the proposed sharing mechanism approaches social optimum. Even with fixed number of resources, introdu

What carries the argument

The price-regulation procedure applied after the generalized Nash game market clearing, which selects a unique fair equilibrium from the possibly infinite set that arises under congestion.

If this is right

  • The mechanism produces individually rational outcomes for every prosumer.
  • The sharing price decomposes into explicit, identifiable components.
  • Social cost falls as the number of competing prosumers rises even when total resources stay constant.
  • The market outcome converges to the social optimum when the number of prosumers grows without bound.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same regulation step could be tested in other network resource games such as water allocation or bandwidth sharing.
  • Convergence speed to the social optimum might be measured by running the mechanism on real distribution feeder data with increasing participant counts.
  • If the bidding functions are estimated from historical data rather than assumed known, the uniqueness guarantees may need re-examination.

Load-bearing premise

Every prosumer bids according to the same general supply-demand function and the network constraints are modeled so that equilibria can be read directly from those bidding functions.

What would settle it

A simulation of a congested network in which the price-regulated equilibrium is not unique or fair, or in which social cost fails to decline when more prosumers are added while resources remain fixed.

Figures

Figures reproduced from arXiv: 1906.09891 by Adam Wierman, Feng Liu, Shengwei Mei, Steven H. Low, Wei Wei, Yue Chen.

Figure 1
Figure 1. Figure 1: Procedure of the sharing mechanism. the sharing market, which shows how the bids of prosumers in￾fluence the sharing price; λi is the clearing price for prosumer i; bi is the bid of prosumer i. The sharing market is said to be cleared effectively when the following two conditions are met: (1) The net quantity ∑i qi = 0, which means ∑ i∈I (−aλi +bi) = 0 (3) (2) The energy transaction pattern corresponding t… view at source ↗
Figure 2
Figure 2. Figure 2: Best response curves in different situations. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Topology of the test system. 5 10 15 20 25 30 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Number of prosumers (I) Relative difference 20 22 24 26 28 30 0 0.005 0.01 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Average relative difference of social total costs between SCO and [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Role of competition when different line is congested. [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Tendency of total cost when introducing competition. [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
read the original abstract

The advent of intelligent agents who produce and consume energy by themselves has led the smart grid into the era of "prosumer", offering the energy system and customers a unique opportunity to revaluate/trade their spot energy via a sharing initiative. To this end, designing an appropriate sharing mechanism is an issue with crucial importance and has captured great attention. This paper addresses the prosumers' demand response problem via energy sharing. Under a general supply-demand function bidding scheme, a sharing market clearing procedure considering network constraints is proposed, which gives rise to a generalized Nash game. The existence and uniqueness of market equilibrium are proved in non-congested cases. When congestion occurs, infinitely much equilibrium may exist because the strategy spaces of prosumers are correlated. A price-regulation procedure is introduced in the sharing mechanism, which outcomes a unique equilibrium that is fair to all participants. Properties of the improved sharing mechanism, including the individual rational behaviors of prosumers and the components of sharing price, are revealed. When the number of prosumers increases, the proposed sharing mechanism approaches social optimum. Even with fixed number of resources, introducing competition can result in a decreasing social cost. Illustrative examples validate the theoretical results and provide more insights for the energy sharing research.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes an energy sharing mechanism for prosumers on constrained networks under a general supply-demand function bidding scheme. The market clearing procedure induces a generalized Nash game; existence and uniqueness of equilibria are established for the non-congested case. A price-regulation step is added to restore uniqueness under congestion while preserving fairness. The mechanism is shown to be individually rational, and the sharing price is decomposed into components. The central result is that the regulated equilibrium converges to the social optimum as the number of prosumers grows, even when the number of network resources remains fixed; illustrative examples are provided to support the claims.

Significance. If the convergence and uniqueness results hold, the work supplies a concrete, incentive-compatible mechanism that improves efficiency in prosumer markets without requiring central coordination. The explicit treatment of network constraints and the limit property under increasing competition are useful for smart-grid design. The rigorous proofs of equilibrium existence/uniqueness and individual rationality constitute a clear technical contribution in the generalized Nash setting.

major comments (2)
  1. [§4] §4 (price-regulation procedure): the claim that the regulated equilibrium is unique and fair relies on the specific form of the price adjustment; it is not shown whether this adjustment preserves the original social-optimum limit when congestion is active, or whether the limit result in §5 continues to hold after regulation.
  2. [Theorem 3] Theorem 3 (convergence to social optimum): the statement that the mechanism approaches the social optimum as the number of prosumers N→∞ is stated for the non-congested case; the extension to the congested, price-regulated regime is asserted but the proof sketch does not explicitly address how the correlation of strategy sets under congestion affects the limit argument.
minor comments (2)
  1. Notation for the bidding functions (supply and demand curves) is introduced without an explicit table summarizing all symbols; this makes cross-referencing between the game formulation and the equilibrium conditions cumbersome.
  2. [§6] The illustrative examples in §6 are described as validating the theory, but the manuscript does not report the precise network topology, parameter values, or solver tolerances used; adding these would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the insightful comments on the price-regulation procedure and the convergence result. We address each major comment below and will revise the manuscript to provide the requested clarifications and proof details.

read point-by-point responses

  1. Referee: [§4] §4 (price-regulation procedure): the claim that the regulated equilibrium is unique and fair relies on the specific form of the price adjustment; it is not shown whether this adjustment preserves the original social-optimum limit when congestion is active, or whether the limit result in §5 continues to hold after regulation.

    Authors: We agree that the interaction between the price-regulation step and the large-N limit requires explicit treatment. The regulation selects the unique fair equilibrium by adding a uniform adjustment term that equalizes the dual variables associated with the shared network constraints. In the revised §5 we will prove that this adjustment term vanishes asymptotically as N→∞ (even with fixed network resources) because the increased competition drives the prosumers' bids toward the social marginal cost; the regulated equilibrium therefore converges to the same social optimum as the unregulated non-congested equilibrium. The argument relies on showing that the regulation-induced deviation is O(1/N) under standard regularity conditions on the supply-demand functions. revision: yes

  2. Referee: [Theorem 3] Theorem 3 (convergence to social optimum): the statement that the mechanism approaches the social optimum as the number of prosumers N→∞ is stated for the non-congested case; the extension to the congested, price-regulated regime is asserted but the proof sketch does not explicitly address how the correlation of strategy sets under congestion affects the limit argument.

    Authors: The referee correctly notes that the current proof sketch for the congested case does not detail the effect of the coupled strategy sets. In the revision we will expand the proof of Theorem 3 to handle the regulated regime explicitly. After regulation the game can be recast as a variational inequality whose feasible set is the product of individual strategy sets (the coupling is absorbed into the price adjustment). Standard arguments for aggregative games with an increasing number of players then show that any sequence of regulated equilibria converges to the unique solution of the social optimization problem; the correlation induced by congestion does not survive in the limit because the regulation term becomes vanishingly small. We will include the full variational-inequality formulation and the requisite limit argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from game model

full rationale

The paper derives existence/uniqueness of equilibria for the generalized Nash game from the bidding scheme and network constraints, then introduces price regulation for uniqueness under congestion, and proves the limit property as prosumer count grows. These steps are presented as direct consequences of the model equations without reducing to fitted parameters, self-definitional loops, or load-bearing self-citations. The social-optimum convergence and competition effects are scoped as analytical results from the improved mechanism, not inputs renamed as outputs. No quoted reductions match the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on a standard domain assumption about how prosumers bid via supply-demand functions and on the mathematical structure of generalized Nash games under network constraints; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Prosumers bid according to a general supply-demand function scheme
    Invoked as the basis for the market clearing procedure and equilibrium analysis.

pith-pipeline@v0.9.0 · 5763 in / 1239 out tokens · 27086 ms · 2026-05-25T17:46:50.395435+00:00 · methodology

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Reference graph

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