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arxiv: 2307.02004 · v6 · submitted 2023-07-05 · 📡 eess.SY · cs.SY

Wholesale Market Participation via Competitive DER Aggregation

Cong Chen , Ahmed S. Alahmed , Timothy D. Mount , Lang Tong This is my paper

Pith reviewed 2026-05-24 08:09 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords DER aggregationwholesale electricity marketvirtual storagewelfare maximizationdistribution network constraintscompetitive aggregatorcustomer surplus
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0 comments X

The pith

A competitive DER aggregator achieves the same welfare-maximizing wholesale outcome as direct customer participation under identical network access.

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

The paper develops a model in which a profit-seeking aggregator directly controls distributed energy resources to bid into the wholesale electricity market while guaranteeing each customer more surplus than the regulated retail tariff. It derives optimized generation offers, consumption bids, and distribution network access curves for the aggregator operating as virtual storage. The central result is that this aggregated participation produces exactly the same overall welfare maximum as would occur if the same customers submitted bids individually. This equivalence holds when distribution network access constraints remain unchanged. Numerical experiments then compare customer surplus, aggregator profit, and long-run survival of multiple aggregators under varying DER adoption and network limits.

Core claim

The proposed competitive DER aggregator model directly controls local DERs to maximize its profits while ensuring customer surplus exceeds the regulated retail tariff level; when participating in the wholesale market as virtual storage with optimized offers and bids, and under the same distribution network access constraints, it yields the identical welfare-maximizing outcome that would result from direct customer participation in the wholesale market.

What carries the argument

The competitive DER aggregator (DERA) that optimizes its wholesale bids and network access offers while enforcing customer surplus above the retail tariff baseline.

If this is right

  • Customer surplus is strictly higher under aggregation than under the regulated retail tariff.
  • The aggregator can compute explicit bid curves for distribution network access that support its wholesale offers.
  • At long-run equilibrium the number of viable competing aggregators is finite and depends on DER adoption levels.
  • Short-run operations are sensitive to the stringency of distribution network access limits.

Where Pith is reading between the lines

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

  • The equivalence result suggests regulators could treat aggregation as a neutral market entry mechanism rather than a potential source of inefficiency.
  • Extending the model to stochastic DER output or multi-period commitment would test whether the welfare equivalence survives uncertainty.
  • If network access constraints differ in practice between direct and aggregated participation, the claimed equivalence would no longer apply.

Load-bearing premise

The aggregator can directly control every local DER and the distribution network access limits stay identical whether customers bid directly or through the aggregator.

What would settle it

Run a side-by-side market clearing simulation in which the identical set of DERs and network constraints are used once with direct customer bids and once with the aggregator's virtual-storage bids; any difference in cleared quantities, prices, or total welfare would falsify the equivalence claim.

Figures

Figures reproduced from arXiv: 2307.02004 by Ahmed S. Alahmed, Cong Chen, Lang Tong, Timothy D. Mount.

Figure 1
Figure 1. Figure 1: DERA model’s physical and financial interactions. Th [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Expected surplus distribution and market efficiency [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Expected surplus distributions v.s. network access [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Expected surplus distributions v.s. network access [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Long-run competitive equilibrium for multi-interv [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: DERA Benefit function ϕ. (Left: withdrawal access −C; right: injection access C.) E. Long-run competitive equilibrium of DERAs In the long-run competitive equilibrium analysis with multi￾interval aggregation of DERs, we assumed 200 DERAs ini￾tially existed and computed the expected number of surviving DERAs in the long run. For simplicity, we assumed DERAs were homogeneous and had the same setting as Sec. … view at source ↗
Figure 7
Figure 7. Figure 7: Long-run competitive equilibrium. (Left: expected [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Mean and 10,000 scenarios of BTM DG generation from pr [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
read the original abstract

We consider the aggregation of distributed energy resources (DERs), such as solar PV, energy storage, and flexible loads, by a profit-seeking aggregator participating directly in the wholesale market under distribution network access constraints. We propose a competitive DER aggregator (DERA) model that directly controls local DERs to maximize its profits, while ensuring each aggregated customer gains a surplus higher than their surplus under the regulated retail tariff. The DERA participates in the wholesale electricity market as virtual storage with optimized generation offers and consumption bids derived from the propoed competitive aggregation model. Also derived are DERA's bid curves for the distribution network access and DERA's profitability when competing with the regulated retail tariff. We show that, with the same distribution network access, the proposed DERA's wholesale market participation achieves the same welfare-maximizing outcome as when its customers participate directly in the wholesale market. Extensive numerical studies compare the proposed DERA with existing methods in terms of customer surplus and DERA profit. We empirically evaluate how many DERAs can survive in the competition at long-run equilibrium, and assess the impacts of DER adoption levels and distribution network access on short-run operations.

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 competitive DER aggregator (DERA) model in which a profit-seeking entity directly controls local DERs (PV, storage, flexible loads) to maximize its own profit subject to a customer-surplus floor relative to the regulated retail tariff. The DERA is modeled as virtual storage that submits optimized wholesale generation offers and consumption bids; distribution-network access bid curves are also derived. The central result is that, when distribution-network access constraints are held fixed, the DERA's wholesale participation produces the identical welfare-maximizing primal solution as the collection of individual customer problems. Numerical studies compare customer surplus and DERA profit against existing aggregation methods, examine long-run equilibrium survival of multiple DERAs, and assess sensitivity to DER adoption levels and network-access limits.

Significance. If the claimed equivalence is rigorously established, the work supplies a market-design mechanism that allows profit-driven aggregation without welfare loss relative to direct customer bidding, which is relevant for high-DER systems. The derivation of virtual-storage offers, distribution-access curves, and the long-run equilibrium analysis of competing DERAs provide concrete, testable outputs for regulators and market operators.

major comments (2)
  1. [Abstract / §3 (model formulation)] Abstract and main text: the equivalence result (that DERA wholesale participation yields the same welfare-max outcome as direct customer participation under identical network access) is stated as the key contribution, yet the provided manuscript text contains no derivation, explicit statement of the customer-surplus constraint, or proof that the virtual-storage feasible set exactly replicates the aggregate of individual DER feasible sets. This mapping is load-bearing; without it the claim cannot be verified.
  2. [Model formulation and equivalence claim] The model implicitly assumes that aggregator direct control and the single virtual-storage interface impose no additional restrictions beyond those faced by individual customers and that distribution constraints (voltage, line-flow limits) translate identically. No section demonstrates that the DERA profit-max problem plus customer-surplus floor produces bids whose aggregate primal solution coincides with the direct-participation optimum; a counter-example or explicit proof is required.
minor comments (2)
  1. [Abstract] Abstract contains the typo 'propoed' (should be 'proposed').
  2. [Numerical studies section] Numerical studies are described as 'extensive' but the text provides no information on the number of scenarios, robustness checks against forecast error, or sensitivity to the customer-surplus floor parameter.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which identify the need to strengthen the presentation of the central equivalence result. We will revise the manuscript to include the requested derivation, explicit constraint statement, and proof. Our point-by-point responses follow.

read point-by-point responses

  1. Referee: [Abstract / §3 (model formulation)] Abstract and main text: the equivalence result (that DERA wholesale participation yields the same welfare-max outcome as direct customer participation under identical network access) is stated as the key contribution, yet the provided manuscript text contains no derivation, explicit statement of the customer-surplus constraint, or proof that the virtual-storage feasible set exactly replicates the aggregate of individual DER feasible sets. This mapping is load-bearing; without it the claim cannot be verified.

    Authors: We agree the equivalence is the central claim and that the current text does not contain an explicit derivation or proof. In the revision we will add a dedicated subsection (new §3.4) that (i) states the customer-surplus floor constraint explicitly as a set of inequalities ensuring each customer’s net benefit exceeds the regulated-tariff benchmark, (ii) defines the virtual-storage feasible set as the Minkowski sum of the individual DER feasible sets projected onto the wholesale interface, and (iii) proves equivalence by showing that any feasible point of the aggregate direct-participation problem is feasible for the DERA problem and vice versa, with identical objective values under the same network-access limits. The proof proceeds by variable substitution and constraint aggregation. revision: yes

  2. Referee: [Model formulation and equivalence claim] The model implicitly assumes that aggregator direct control and the single virtual-storage interface impose no additional restrictions beyond those faced by individual customers and that distribution constraints (voltage, line-flow limits) translate identically. No section demonstrates that the DERA profit-max problem plus customer-surplus floor produces bids whose aggregate primal solution coincides with the direct-participation optimum; a counter-example or explicit proof is required.

    Authors: The manuscript does assume that direct control by the DERA introduces no extra restrictions beyond the customer-surplus floor; this is intentional and will be stated explicitly. Distribution-network constraints are modeled identically in both settings (same voltage and line-flow limits applied at the same nodes). In the new §3.4 we will prove that the DERA profit-maximization problem subject to the surplus floor yields wholesale bids whose resulting primal solution is identical to the welfare-maximizing solution of the collection of individual customer problems. The proof uses the fact that the surplus-floor constraints bind only at the individual level while the network constraints are shared, so the aggregate feasible set is unchanged. No counter-example exists under the stated assumptions; we will include a short remark confirming this. revision: yes

Circularity Check

0 steps flagged

No circularity: equivalence shown as derived model property under fixed constraints

full rationale

The paper constructs an explicit profit-maximization model for the DERA (with customer-surplus floor), derives its wholesale offers/bids and network-access curves from that model, and then proves equivalence to direct customer participation when network access is held identical. This equivalence is presented as a consequence of the optimization problems having identical feasible sets and objectives under the stated assumptions, not as a definitional identity or a fitted parameter renamed as a prediction. No self-citation load-bearing steps, no ansatz smuggled via prior work, and no renaming of known results appear in the provided text. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no specific free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.0 · 5735 in / 1004 out tokens · 36371 ms · 2026-05-24T08:09:48.951205+00:00 · methodology

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Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    Competit ive DER aggregation for participation in wholesale markets,

    C. Chen, A. S. Alahmed, T. D. Mount, and L. Tong, “Competit ive DER aggregation for participation in wholesale markets,” in Proceedings of the 56th Hawaii International Conference on System Science s. [Online preprint] arXiv:2207.00290, 2023

    work page arXiv 2023

  2. [2]

    (2020) Participation of distributed energy resou rce aggregations in markets operated by regional transmission organization s and inde- pendent system operators, order 2222

    FERC. (2020) Participation of distributed energy resou rce aggregations in markets operated by regional transmission organization s and inde- pendent system operators, order 2222. 2020. Accessed: June 9, 2024. [Online]. https://www.ferc.gov/sites/default/files/2020-09/E-1 0.pdf

    work page 2020

  3. [3]

    TSO/DSO coordination in a context of distributed energy resource pe netration,

    M. Birk, J. P . Chaves- ´Avila, T. G´ omez, and R. Tabors, “TSO/DSO coordination in a context of distributed energy resource pe netration,” Proceedings of the EEIC, MIT Energy Initiative Reports, Cam bridge, MA, USA , pp. 2–3, 2017

    work page 2017

  4. [4]

    Integrating distributed ener gy resources: Optimal prosumer decisions and impacts of net metering tari ffs,

    A. S. Alahmed and L. Tong, “Integrating distributed ener gy resources: Optimal prosumer decisions and impacts of net metering tari ffs,” SIGEN- ERGY Energy Inform. Rev. , vol. 2, no. 2, p. 13–31, Aug. 2022

    work page 2022

  5. [5]

    Order 2222: Observations from a Distributio n Utility,

    J. Nelson, “Order 2222: Observations from a Distributio n Utility,” Southern California Edison (SCE), Tech. Rep., 10 2021

    work page 2021

  6. [6]

    Designing elec tricity rates for an equitable energy transition,

    S. Borenstein, M. Fowlie, and J. Sallee, “Designing elec tricity rates for an equitable energy transition,” Energy Institute at Haas working paper , vol. 314, 2021

    work page 2021

  7. [7]

    Real-time wholesale electricity pricing in Griddy,

    “Real-time wholesale electricity pricing in Griddy,” A ccessed: June 9, 2024. [Online]. https://www.energyogre.com/is-realtime-wholesale- pricing-risky, February 2022

    work page 2024

  8. [8]

    Amber wholesale energy price explained,

    “Amber wholesale energy price explained,” Accessed: Ju ne 9,

    work page

  9. [9]

    https://www.youtube.com/watch?app=desktop& v=DckQbwwWPW A and https://www.amber.com.au/electricity, June 2021

    [Online]. https://www.youtube.com/watch?app=desktop& v=DckQbwwWPW A and https://www.amber.com.au/electricity, June 2021

    work page 2021

  10. [10]

    A hierarchical electr icity market structure for the smart grid paradigm,

    S. D. Manshadi and M. E. Khodayar, “A hierarchical electr icity market structure for the smart grid paradigm,” IEEE Transactions on Smart Grid, vol. 7, no. 4, pp. 1866–1875, 2015

    work page 2015

  11. [11]

    Reinventing the utility for distribut ed energy resources: A proposal for retail electricity markets,

    R. Haider, D. D’Achiardi, V . V enkataramanan, A. Srivas tava, A. Bose, and A. M. Annaswamy, “Reinventing the utility for distribut ed energy resources: A proposal for retail electricity markets,” Advances in Applied Energy, vol. 2, p. 100026, 2021

    work page 2021

  12. [12]

    Multiclass energy mana gement for peer-to-peer energy trading driven by prosumer prefere nces,

    T. Morstyn and M. D. McCulloch, “Multiclass energy mana gement for peer-to-peer energy trading driven by prosumer prefere nces,” IEEE Transactions on Power Systems , vol. 34, no. 5, pp. 4005–4014, 2018

    work page 2018

  13. [13]

    Review of energy sharing: Business models, mechanisms, and prospects,

    Y . Chen and C. Zhao, “Review of energy sharing: Business models, mechanisms, and prospects,” IET Renewable Power Generation , 2022

    work page 2022

  14. [14]

    Analysis of solar energy aggregation under various billin g mecha- nisms,

    P . Chakraborty, E. Baeyens, P . P . Khargonekar, K. Poolla, and P . V araiya, “Analysis of solar energy aggregation under various billin g mecha- nisms,” IEEE Transactions on Smart Grid, vol. 10, no. 4, pp. 4175–4187, 2018

    work page 2018

  15. [15]

    Incentivizing pr osumer coali- tions with energy management using cooperative game theory ,

    L. Han, T. Morstyn, and M. McCulloch, “Incentivizing pr osumer coali- tions with energy management using cooperative game theory ,” IEEE Transactions on Power Systems , vol. 34, no. 1, pp. 303–313, 2018

    work page 2018

  16. [16]

    Energy collectives: A communit y and fairness based approach to future electricity markets,

    F. Moret and P . Pinson, “Energy collectives: A communit y and fairness based approach to future electricity markets,” IEEE Transactions on Power Systems, vol. 34, no. 5, pp. 3994–4004, 2018

    work page 2018

  17. [17]

    Adjustable price-sensitive DER bidding wit hin network envelopes,

    A. Attarha, M. Mahmoodi, S. M. N. R. A., P . Scott, J. Iria, and S. Thi´ ebaux, “Adjustable price-sensitive DER bidding wit hin network envelopes,” IEEE Transactions on Energy Markets, Policy and Regula- tion, vol. 1, no. 4, pp. 248–258, 2023

    work page 2023

  18. [18]

    Aggregating distr ibuted energy resources: efficiency and market power,

    Z. Gao, K. Alshehri, and J. R. Birge, “Aggregating distr ibuted energy resources: efficiency and market power,” Manufacturing & Service Operations Management, 2024

    work page 2024

  19. [19]

    Quantify ing market efficiency impacts of aggregated distributed energy resour ces,

    K. Alshehri, M. Ndrio, S. Bose, and T. Bas ¸ar, “Quantify ing market efficiency impacts of aggregated distributed energy resour ces,” IEEE Transactions on Power Systems , vol. 35, no. 5, pp. 4067–4077, 2020

    work page 2020

  20. [20]

    What is the energy price cap and how does it affect me?

    “What is the energy price cap and how does it affect me?” A c- cessed: June 9, 2024. [Online]. https://www.nerdwallet.com/uk/personal- finance/what-is-the-energy-price-cap/ , April 2023

    work page 2024

  21. [21]

    Turnkey VPPs: Streamlining der management for the dec arbonized grid of the future,

    “Turnkey VPPs: Streamlining der management for the dec arbonized grid of the future,” Accessed: June 9, 2024. [Online]. https://storage. pardot.com/764473/1680019430TuQTtLtN/230327 Turnkey VPP White Paper.pdf, March 2023

    work page 2024

  22. [22]

    On net energy metering X: Opti mal pro- sumer decisions, social welfare, and cross-subsidies,

    A. S. Alahmed and L. Tong, “On net energy metering X: Opti mal pro- sumer decisions, social welfare, and cross-subsidies,” IEEE Transactions on Smart Grid , 2022

    work page 2022

  23. [23]

    (2019) Continuous storage facility participa tion

    ISO-NE. (2019) Continuous storage facility participa tion. Accessed: June 9, 2024. [Online]. https://www.iso-ne.com/static-assets/documents/ 2019/02/20190221-csf.pdf

    work page 2019

  24. [24]

    (2021) Order no

    ——. (2021) Order no. 2222: Participation of distribute d energy resource aggregations in wholesale markets. Accessed: Jun e 9,

    work page 2021

  25. [25]

    https://www.iso-ne.com/static-assets/documents/2021/ 07/a7 order 2222.pdf

    [Online]. https://www.iso-ne.com/static-assets/documents/2021/ 07/a7 order 2222.pdf

    work page 2021

  26. [26]

    Wholesale mar ket partic- ipation of DERAs: DSO-DERA-ISO coordination,

    C. Chen, S. Bose, T. D. Mount, and L. Tong, “Wholesale mar ket partic- ipation of DERAs: DSO-DERA-ISO coordination,” IEEE Transactions on Power Systems , pp. 1–12, 2024

    work page 2024

  27. [27]

    Italy publishes interactive map of substations for en ergy communities,

    “Italy publishes interactive map of substations for en ergy communities,” Accessed: June 9, 2024. [Online]. https://www.pv-magazine.com/2023/ 10/06/italy-publishes-interactive-map-of-substation s-for-energy- communities/, October 2023

    work page 2024

  28. [28]

    Mas-Colell, M

    A. Mas-Colell, M. D. Whinston, J. R. Green et al. , Microeconomic theory. Oxford university press New Y ork, 1995, vol. 1

    work page 1995

  29. [29]

    The nonparametric approach to demand ana lysis,

    H. R. V arian, “The nonparametric approach to demand ana lysis,” Econo- metrica: Journal of the Econometric Society , pp. 945–973, 1982

    work page 1982

  30. [30]

    Elsevier, 1989, vol

    ——, Price discrimination. Elsevier, 1989, vol. 1

    work page 1989

  31. [31]

    2022 distributed energy resources avoided cost calcu lator documentation,

    “2022 distributed energy resources avoided cost calcu lator documentation,” Accessed: June 9, 2024. [Online]. https://www. cpuc.ca.gov/-/media/cpuc-website/divisions/energy-division/documents/ demand-side-management/acc-models-latest-version/20 22-acc- documentation-v1a.pdf, June 2022

    work page 2022

  32. [32]

    Billingsley, Probability and measure

    P . Billingsley, Probability and measure . John Wiley & Sons, 2008

    work page 2008

  33. [33]

    G. A. Jehle and P . J. Reny, Advanced microeconomic theory . Pearson, 2011

    work page 2011

  34. [34]

    Pecan street dataport,

    “Pecan street dataport,” Accessed: June 9, 2024. [Onli ne]. https://www. pecanstreet.org/dataport/

    work page 2024

  35. [35]

    International experiences in retail electrici ty markets,

    A. J. Ros, T. Brown, N. Lessem, S. Hesmondhalgh, J. D. Rei tzes, and H. Fujita, “International experiences in retail electrici ty markets,” The Brattle Group: Sydney, Australia , 2018. 10 VIII. A PPENDIX A. Participation model of prosumers A prosumer in a distribution system can choose to enroll in a NEM X retail program offered by her utility or a DERA p...

    work page 2018

  36. [36]

    For simplicity, we drop the prosumer index n and adopt one representative prosumer

    NEM benchmarks: Considering the benchmark perfor- mance of a regulated utility offering the NEM X tariff, we extend the results in [ 4], [ 21] and present closed-form characterizations of consumer/prosumer surpluses. For simplicity, we drop the prosumer index n and adopt one representative prosumer. The prosumer’s net consumption i s z =d − g, (10) where ...

    work page

  37. [37]

    So, for the above optimization, the domain is D := [max{d,g − C}, min{ ¯d,g +C}]

    at all PoAs. So, for the above optimization, the domain is D := [max{d,g − C}, min{ ¯d,g +C}]. The surplus SNEM-a and the consumption dNEM-a of an active prosumer are given by the following equations. SNEM-a(g,C , C) = U (dNEM-a) − P π(dNEM-a − g) (12) =      U (d−) − π −(d− − g) − π 0, g ≥ d− U (d+) − π +(d+ − g) − π 0, g ≤ d+ U (d0) − π 0, otherwis...

    work page

  38. [38]

    The original pricing scheme keeps the custome r surplus under DERA competitive with that when the customers directly buy energy from the wholesale market

    Two-part pricing in GAB: The optimal DERA two-part pricing scheme is proposed in [ 17] to aggregate BTM DG productions. The original pricing scheme keeps the custome r surplus under DERA competitive with that when the customers directly buy energy from the wholesale market. Here, consid - ering the realistic retail market setting, we revised the DE RA pri...

    work page

  39. [39]

    (c) replies on ζ ≥ 1 and the assumption that S NEM n ≥ 0

    (b) relies on the setting that Kn =ζS NEM n (gn,C n, Cn). (c) replies on ζ ≥ 1 and the assumption that S NEM n ≥ 0. (d) comes from the definition of S NEM n given in ( 17).13(e) comes from Un(d∗ n) − d∗ nπ + ≤ Un(d+ n ) − π +d+ n, (20) which can be derived from the optimality of d+ = arg max d∈D(U (d) − π +d). If gn>d + n , for passive prosumer , we have ω...

    work page

  40. [40]

    (c) replies on ζ ≥ 1 and the assumption that S NEM n ≥ 0

    (b) relies on the setting that Kn =ζS NEM n (gn,C n, Cn). (c) replies on ζ ≥ 1 and the assumption that S NEM n ≥ 0. (d) comes from the definition of S NEM n given in ( 17). (e) replies on gn > d+ n and π − ≥ 0. (f) comes from ( 20). (g) holds because d+ n ≥ 0 and π + ≥ 0. If d+ n <g n ≤ d− n , for active prosumer , we have ω ∗ n (a) ≤ Un(d∗ n) − S NEM n (g...

    work page

  41. [41]

    So we have d∗ i (π LMP,g i) = min {gi +C i, max{ ˆdi,g i − C i}}, (26) ω ∗ i (d∗ i,g i) = Ui(d∗ i ) − K i, (27) where ˆdi := min{ di, max{V −1 i (π LMP),d i}}

    is always binding with χ ∗ i = 1, and the optimal consumption d∗ i equals to V −1 i (π LMP) if it falls into the interval [min{ di,g i +Ci}, max{di,g i − Ci}. So we have d∗ i (π LMP,g i) = min {gi +C i, max{ ˆdi,g i − C i}}, (26) ω ∗ i (d∗ i,g i) = Ui(d∗ i ) − K i, (27) where ˆdi := min{ di, max{V −1 i (π LMP),d i}}. When V −1 i (π LMP) ≥ min{di,g i +Ci} ...

    work page

  42. [42]

    (34) Known that the prosumer utility function is assumed to be concave and continuously differentiable

    becomes π LMP i +ρ⋆ in =Vin(min{ din,g in +C in}). (34) Known that the prosumer utility function is assumed to be concave and continuously differentiable. We have (Vin)−1(π LMP i ) ≥ (Vin)−1(π LMP i + ρ⋆ in) = min { din,g in +Cin}. Similarly, when ρ⋆ in > 0 , we have d⋆ in = max {d in,g in − Cin}, ρ⋆ in = 0 , and (Vin)−1(π LMP i ) ≤ (Vin)−1(π LMP i − ρ ⋆ ...

    work page

  43. [43]

    In the simu- lation, we have N = 50 for each DERA

    Single-interval long-run competitive equilibrium: De- note N as the number of aggregated prosumers. In the simu- lation, we have N = 50 for each DERA. With the quadratic utility of homogeneous prosumer parameterized by α and β in ( 9), the profit of the i-th DERA defined in ( 24) is Πi(C i) = − β (C i +Gi)2 2N +α (Ci +Gi) − π LMPCi − K i, (41) where Gi is t...

    work page

  44. [44]

    For simplicity, we assume homoge- neous DERA with the same expectation of BTM DG genera- tion

    Single-interval long-run competitive equilibrium: We simulated long-run competitive equilibrium for the single interval aggregation by assuming 200 DERAs existed at the beginning and computed the expected number of surviving DERAs in the long run. For simplicity, we assume homoge- neous DERA with the same expectation of BTM DG genera- tion. We sampled 10,...

    work page

  45. [45]

    Note that the number of DERA K is still a scalar applied to all 24 hours

    Multi-interval long-run competitive equilibrium: By adding the 24-hour time dimension to the network access and BTM DG generation, we can extend the derivation of ( 41)(42)(43)(44) from single-interval long-run equilibrium to multi-interval long-run equilibrium. Note that the number of DERA K is still a scalar applied to all 24 hours. We include the simul...

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