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arxiv: 2407.07068 · v1 · submitted 2024-07-09 · 📡 eess.SY · cs.SY· math.OC

Chance-Constrained Energy Storage Pricing for Social Welfare Maximization

Ning Qi , Ningkun Zheng , Bolun Xu This is my paper

Pith reviewed 2026-05-23 22:46 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords chance-constrained optimizationenergy storagesocial welfare maximizationeconomic dispatchopportunity costpower system pricinguncertainty managementmarket design
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The pith

Chance-constrained pricing for energy storage maximizes social welfare by bounding opportunity costs and coupling them to future prices.

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

The paper proposes a framework for pricing energy storage within economic dispatch to maximize social welfare instead of private profit. It relies on a two-stage chance-constrained model that accounts for system balance and uncertainty through tractable reformulations of joint chance constraints. Analytical results establish that storage opportunity cost is convex and grows with net load uncertainty, while prices remain bounded and couple linearly to future energy and reserve prices. Simulations demonstrate that this design lowers electricity payments by 17.4 percent on average and system costs by 3.9 percent relative to a price-taker profit-maximizing approach, with larger gains as renewable and storage capacities expand.

Core claim

The central claim is that a two-stage chance-constrained formulation provides a theoretical framework for pricing energy storage in social welfare maximization. Tractable reformulations are given for the joint chance constraints. The storage opportunity cost is convex and increases with greater net load uncertainty. The storage opportunity prices are bounded and are linearly coupled with future energy and reserve prices. The approach reduces electricity payments and system costs while reducing storage profit margins.

What carries the argument

two-stage chance-constrained formulation with tractable reformulations for joint chance constraints

If this is right

  • Storage opportunity cost is convex and increases with greater net load uncertainty.
  • Storage opportunity prices are bounded and linearly coupled with future energy and reserve prices.
  • The market design reduces electricity payments by an average of 17.4%.
  • System costs are reduced by 3.9% on average.
  • These reductions scale up with the renewable and storage capacity.

Where Pith is reading between the lines

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

  • System operators could apply the framework to generate default bids for storage or to benchmark bids for market power.
  • The convexity of opportunity costs may allow for more efficient computational methods in larger dispatch problems.
  • Linear coupling suggests that storage pricing can be integrated into existing multi-period market clearing without major changes.
  • Testing on systems with higher uncertainty levels could confirm the scaling of cost reductions.

Load-bearing premise

The tractable reformulations for the joint chance constraints accurately capture the original probabilistic constraints without significant approximation errors.

What would settle it

Observing cases where the chance-constrained prices lead to actual system imbalance probabilities exceeding the target or fail to achieve the reported payment reductions in out-of-sample tests.

Figures

Figures reproduced from arXiv: 2407.07068 by Bolun Xu, Ningkun Zheng, Ning Qi.

Figure 1
Figure 1. Figure 1: Prices and costs under different risk aversions and different [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Cumulative cost curve of 76 conventional generators in the ISO-NE [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Opportunity prices under different initial SoC and uncertainty levels. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Simulated and theoretical opportunity prices over four simulation days. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: Expected opportunity prices under different cost functions and [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Prices and costs under different market participations of energy [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Economic performance under different pricing mechanisms and [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
read the original abstract

This paper proposes a novel framework to price energy storage in economic dispatch with a social welfare maximization objective. This framework can be utilized by power system operators to generate default bids for storage or to benchmark market power in bids submitted by storage participants. We derive a theoretical framework based on a two-stage chance-constrained formulation which systematically incorporates system balance constraints and uncertainty considerations. We present tractable reformulations for the joint chance constraints. Analytical results show that the storage opportunity cost is convex and increases with greater net load uncertainty. We also show that the storage opportunity prices are bounded and are linearly coupled with future energy and reserve prices. We demonstrate the effectiveness of the proposed approach on an ISO-NE test system and compare it with a price-taker storage profit-maximizing bidding model. Simulation results show that the proposed market design reduces electricity payments by an average of 17.4% and system costs by 3.9% while reducing storage's profit margins, and these reductions scale up with the renewable and storage capacity.

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 a two-stage chance-constrained formulation for pricing energy storage in economic dispatch to maximize social welfare. It derives tractable reformulations of the joint chance constraints, proves that the resulting storage opportunity cost is convex and increases with net-load uncertainty, shows that opportunity prices are bounded and linearly coupled to future energy and reserve prices, and reports ISO-NE simulations in which the design reduces electricity payments by 17.4 % and system costs by 3.9 % relative to a price-taker profit-maximizing benchmark.

Significance. If the convexity, boundedness, and linear-coupling results survive the passage from the original joint chance-constrained program to the tractable surrogate, the framework supplies a principled, uncertainty-aware default-bid or market-power benchmark that could improve efficiency in systems with growing storage and renewables. The reported cost and payment reductions, if robust, would be practically relevant for ISO market design.

major comments (2)
  1. [§3] §3 (Tractable Reformulations): the analytical claims of convexity of the opportunity cost, boundedness, and linear coupling with future prices are derived on the reformulated problem. The manuscript must explicitly state which conservative approximation (Bonferroni, CVaR, scenario, etc.) is used and either prove that the stated properties transfer to the original non-convex joint chance-constrained program or quantify the gap that would invalidate the claims.
  2. [§4] §4 (Analytical Results): the proofs that opportunity cost increases with uncertainty and that prices are linearly coupled rely on the reformulated model. If the reformulation is only an inner approximation, the coupling and monotonicity statements do not necessarily hold for the original probabilistic constraints; explicit conditions or counter-example checks are required.
minor comments (2)
  1. [Abstract / §5] The abstract states average reductions of 17.4 % and 3.9 %; the main text should report the number of Monte-Carlo scenarios, the exact uncertainty model, and confidence intervals so that the simulation claims can be reproduced.
  2. [§2] Notation for the two-stage variables and the joint chance-constraint index sets should be introduced once and used consistently; several symbols appear without prior definition in the early sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment below and will revise the manuscript to improve clarity on the relationship between the original joint chance-constrained program and the tractable reformulations.

read point-by-point responses

  1. Referee: [§3] §3 (Tractable Reformulations): the analytical claims of convexity of the opportunity cost, boundedness, and linear coupling with future prices are derived on the reformulated problem. The manuscript must explicitly state which conservative approximation (Bonferroni, CVaR, scenario, etc.) is used and either prove that the stated properties transfer to the original non-convex joint chance-constrained program or quantify the gap that would invalidate the claims.

    Authors: We agree that the manuscript should explicitly name the conservative approximation and discuss its implications. In the revision we will state the specific reformulation technique used for the joint chance constraints and add a dedicated paragraph clarifying that convexity, boundedness, and linear coupling are established for the reformulated (inner-approximation) problem. We will also include a short discussion of the approximation gap, supported by references to existing bounds on Bonferroni-type and CVaR-type relaxations of joint chance constraints, together with numerical evidence from the ISO-NE study showing that the reported cost and payment reductions remain qualitatively unchanged when the original probabilistic constraints are evaluated ex post. revision: yes

  2. Referee: [§4] §4 (Analytical Results): the proofs that opportunity cost increases with uncertainty and that prices are linearly coupled rely on the reformulated model. If the reformulation is only an inner approximation, the coupling and monotonicity statements do not necessarily hold for the original probabilistic constraints; explicit conditions or counter-example checks are required.

    Authors: We acknowledge that the monotonicity and linear-coupling results are formally proven only on the reformulated model. In the revised manuscript we will add an explicit statement of the conditions (e.g., when the individual probability thresholds are sufficiently small) under which the qualitative properties carry over approximately to the original joint chance-constrained program. We will also include a brief appendix subsection that provides both a simple analytical counter-example illustrating a case where exact transfer fails and additional simulation checks confirming that the monotonicity and coupling behavior observed in the ISO-NE instances is preserved when the original chance constraints are enforced. revision: yes

Circularity Check

0 steps flagged

No significant circularity; analytical derivations are independent of inputs

full rationale

The paper derives convexity of storage opportunity cost, boundedness, and linear coupling from a two-stage chance-constrained formulation after tractable reformulations of joint chance constraints. These are presented as mathematical results on the reformulated model without any reduction to fitted parameters, self-definitions, or self-citation chains. No equations or steps in the abstract or description equate outputs to inputs by construction. The simulation results on payments and costs are separate empirical checks. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based solely on abstract; no explicit free parameters, axioms, or invented entities are described in sufficient detail to populate the ledger.

axioms (1)
  • domain assumption Uncertainty in net load admits a known or approximable distribution suitable for joint chance constraints.
    Invoked by the two-stage chance-constrained formulation and tractable reformulations.

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Forward citations

Cited by 1 Pith paper

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    A stopping-time reward and chance-constrained SoC penalty embedded in an end-to-end learning framework improves battery reachability of target ranges, raises arbitrage profit, and lowers profit variance under volatile prices.

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages · cited by 1 Pith paper

  1. [1]

    Key statistics in april,

    CAISO, “Key statistics in april,” 2024. [Online]. Available: https://www.caiso.com/documents/key-statistics-apr-2024.pdf

    work page 2024

  2. [2]

    Energy storage state-of-charge market model,

    N. Zheng, X. Qin, D. Wu et al., “Energy storage state-of-charge market model,” IEEE Trans. on Energy Markets, Policy and Regulation , vol. 1, no. 1, pp. 11–22, 2023

    work page 2023

  3. [3]

    Electricity storage and market power,

    O. Williams and R. Green, “Electricity storage and market power,” Energy policy, vol. 164, p. 112872, 2022

    work page 2022

  4. [4]

    D. S. Kirschen and G. Strbac, Fundamentals of power system economics. John Wiley & Sons, 2018

    work page 2018

  5. [5]

    Transferable energy storage bidder,

    Y . Baker, N. Zheng, and B. Xu, “Transferable energy storage bidder,” IEEE Trans. on Power Systems , 2023. IEEE TRANSACTIONS ON ENERGY MARKETS, POLICY AND REGULATION, VOL. X, NO. X, XX JULY 2024 9

    work page 2023

  6. [6]

    The price prediction for the energy market based on a new method,

    H. Ebrahimian, S. Barmayoon, M. Mohammadi et al. , “The price prediction for the energy market based on a new method,” Economic research-Ekonomska istraˇzivanja, vol. 31, no. 1, pp. 313–337, 2018

    work page 2018

  7. [7]

    Real-time electricity price forecasting of wind farms with deep neural network transfer learning and hybrid datasets,

    H. Yang and K. R. Schell, “Real-time electricity price forecasting of wind farms with deep neural network transfer learning and hybrid datasets,” Applied Energy, vol. 299, p. 117242, 2021

    work page 2021

  8. [8]

    Energy storage arbitrage under day-ahead and real-time price uncertainty,

    D. Krishnamurthy, C. Uckun, Z. Zhou et al., “Energy storage arbitrage under day-ahead and real-time price uncertainty,” IEEE Trans. on Power Systems, vol. 33, no. 1, pp. 84–93, 2017

    work page 2017

  9. [9]

    Economic capacity withholding bounds of competitive energy storage bidders,

    X. Qin, I. Lestas, and B. Xu, “Economic capacity withholding bounds of competitive energy storage bidders,” arXiv preprint arXiv:2403.05705 , 2024

    work page arXiv 2024

  10. [10]

    On truthful pricing of battery energy storage resources in electricity spot market,

    B. Xu and B. F. Hobbs, “On truthful pricing of battery energy storage resources in electricity spot market,” Oxford Energy Forum , no. 140, pp. 34–38, 2024

    work page 2024

  11. [11]

    Pricing energy storage in real-time market,

    C. Chen, L. Tong, and Y . Guo, “Pricing energy storage in real-time market,” in 2021 IEEE Power & Energy Society General Meeting (PESGM). IEEE, 2021, pp. 1–5

    work page 2021

  12. [12]

    On the efficiency of energy markets with non-merchant storage,

    L. Fr ¨olke, E. Prat, P. Pinson et al., “On the efficiency of energy markets with non-merchant storage,” Energy Systems, pp. 1–32, 2024

    work page 2024

  13. [13]

    An efficient and incentive-compatible market design for energy storage participation,

    X. Fang, H. Guo, X. Zhang et al., “An efficient and incentive-compatible market design for energy storage participation,” Applied Energy , vol. 311, p. 118731, 2022

    work page 2022

  14. [14]

    Operational valuation of energy storage under multi-stage price uncertainties,

    B. Xu, M. Korp ˚as, and A. Botterud, “Operational valuation of energy storage under multi-stage price uncertainties,” in 2020 59th IEEE Conference on Decision and Control (CDC) . IEEE, 2020, pp. 55–60

    work page 2020

  15. [15]

    Convexifying market clearing of soc-dependent bids from merchant storage participants,

    C. Chen and L. Tong, “Convexifying market clearing of soc-dependent bids from merchant storage participants,” IEEE Trans. on Power Systems, 2023

    work page 2023

  16. [16]

    Pricing energy and reserves using stochastic optimization in an alternative electricity market,

    S. Wong and J. D. Fuller, “Pricing energy and reserves using stochastic optimization in an alternative electricity market,” IEEE Trans. on Power Systems, vol. 22, no. 2, pp. 631–638, 2007

    work page 2007

  17. [17]

    Reserve requirements for wind power integration: A scenario-based stochastic programming framework,

    A. Papavasiliou, S. S. Oren, and R. P. O’Neill, “Reserve requirements for wind power integration: A scenario-based stochastic programming framework,” IEEE Trans. on Power Systems , vol. 26, no. 4, pp. 2197–2206, 2011

    work page 2011

  18. [18]

    Pricing electricity in pools with wind producers,

    J. M. Morales, A. J. Conejo, K. Liu et al., “Pricing electricity in pools with wind producers,” IEEE Trans. on Power Systems , vol. 27, no. 3, pp. 1366–1376, 2012

    work page 2012

  19. [19]

    A stochastic market design with revenue adequacy and cost recovery by scenario: Benefits and costs,

    J. Kazempour, P. Pinson, and B. F. Hobbs, “A stochastic market design with revenue adequacy and cost recovery by scenario: Benefits and costs,” IEEE Trans. on Power Systems , vol. 33, no. 4, pp. 3531–3545, 2018

    work page 2018

  20. [20]

    Pricing chance constraints in electricity markets,

    X. Kuang, Y . Dvorkin, A. J. Lamadrid et al., “Pricing chance constraints in electricity markets,” IEEE Trans. on Power Systems , vol. 33, no. 4, pp. 4634–4636, 2018

    work page 2018

  21. [21]

    Chance-constrained ac optimal power flow: Reformulations and efficient algorithms,

    L. Roald and G. Andersson, “Chance-constrained ac optimal power flow: Reformulations and efficient algorithms,” IEEE Trans. on Power Systems, vol. 33, no. 3, pp. 2906–2918, 2017

    work page 2017

  22. [22]

    A chance-constrained stochastic electricity market,

    Y . Dvorkin, “A chance-constrained stochastic electricity market,” IEEE Trans. on Power Systems , vol. 35, no. 4, pp. 2993–3003, 2019

    work page 2019

  23. [23]

    Chance-constrained optimization of demand response to price signals,

    G. Dorini, P. Pinson, and H. Madsen, “Chance-constrained optimization of demand response to price signals,” IEEE Trans. on Smart Grid , vol. 4, no. 4, pp. 2072–2080, 2013

    work page 2072

  24. [24]

    Distribution locational marginal pricing for optimal electric vehicle charging through chance constrained mixed-integer programming,

    Z. Liu, Q. Wu, S. S. Oren et al. , “Distribution locational marginal pricing for optimal electric vehicle charging through chance constrained mixed-integer programming,” IEEE Trans. on Smart Grid , vol. 9, no. 2, pp. 644–654, 2016

    work page 2016

  25. [25]

    Convex approximations of chance constrained programs,

    A. Nemirovski and A. Shapiro, “Convex approximations of chance constrained programs,” SIAM Journal on Optimization , vol. 17, no. 4, pp. 969–996, 2007

    work page 2007

  26. [26]

    Chance-constrained day-ahead scheduling in stochastic power system operation,

    H. Wu, M. Shahidehpour, Z. Li et al. , “Chance-constrained day-ahead scheduling in stochastic power system operation,” IEEE Trans. on Power Systems, vol. 29, no. 4, pp. 1583–1591, 2014

    work page 2014

  27. [27]

    Chance-constrained generic energy storage operations under decision-dependent uncertainty,

    N. Qi, P. Pinson, M. R. Almassalkhi et al., “Chance-constrained generic energy storage operations under decision-dependent uncertainty,” IEEE Trans. on Sustainable Energy , vol. 14, no. 4, pp. 2234–2248, 2023

    work page 2023

  28. [28]

    A versatile probability distribution model for wind power forecast errors and its application in economic dispatch,

    Z.-S. Zhang, Y .-Z. Sun, D. W. Gao et al. , “A versatile probability distribution model for wind power forecast errors and its application in economic dispatch,” IEEE Trans. on Power Systems , vol. 28, no. 3, pp. 3114–3125, 2013

    work page 2013

  29. [29]

    Guaranteeing a physically realizable battery dispatch without charge-discharge complementarity constraints,

    N. Nazir and M. Almassalkhi, “Guaranteeing a physically realizable battery dispatch without charge-discharge complementarity constraints,” IEEE Trans. on Smart Grid , vol. 14, no. 3, pp. 2473–2476, 2021

    work page 2021

  30. [30]

    F. M. Dekking, A Modern Introduction to Probability and Statistics: Un- derstanding why and how . Springer Science & Business Media, 2005

    work page 2005

  31. [31]

    An 8-zone test system based on iso new england data: Development and application,

    D. Krishnamurthy, W. Li, and L. Tesfatsion, “An 8-zone test system based on iso new england data: Development and application,” IEEE Trans. on Power Systems , vol. 31, no. 1, pp. 234–246, 2015

    work page 2015

  32. [32]

    Forecast error data from elia,

    Elia, “Forecast error data from elia,” 2024. [Online]. Available: https://www.elia.be/en/grid-data APPENDIX A. Lagrange Function and KKT Conditions The Lagrange function and KKT conditions of problem (1) are formulated in (7). L= E X t G(gt+φtdt)+ M(pt+ψtdt) −αtbt −βtpt (7a) +αt(bt −ψtbdt −P )+ βt(pt+ψtedt −P )−πt(φt+ψt −1) −λt(gt+pt −bt −Dt)+ θt(et+1 −et...

    work page 2024