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arxiv: 2310.05999 · v3 · pith:3V2AD6Y7new · submitted 2023-10-09 · 📡 eess.SY · cs.SY

Sustainable and Efficient Renewable-Driven Energy Trading via Neural-Enhanced Time-Adaptive Robust Nash Bargaining between Hydrogen-Enriched Gas and Active Distribution Networks

Wenwen Zhang This is my paper

Pith reviewed 2026-05-25 08:24 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords energy tradingNash bargainingADMMrobust optimizationneural network surrogatehydrogen-enriched gasactive distribution networkrenewable integration
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The pith

A neural-enhanced robust Nash bargaining method allows hydrogen-enriched gas and power networks to trade energy with stable social welfare and faster computation despite renewable uncertainties.

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

The paper proposes a strategy for energy trading between hydrogen-enriched compressed natural gas networks and active distribution networks that preserves privacy while handling renewable uncertainty. It uses alternating direction method of multipliers for Nash bargaining in the first stage and robust dispatch in the second to protect against worst-case scenarios. A neural network surrogate then adapts the time resolution based on assessments of social welfare and solving time to speed up the process. If correct, this would enable more efficient and sustainable integration of renewables through cross-sector coordination without sacrificing economic benefits or convergence guarantees.

Core claim

The central claim is that the neural-enhanced time-adaptive robust Nash bargaining strategy, combining ADMM-based privacy-preserved bargaining with robust optimization for worst renewable scenarios and a neural surrogate for time adaptation, achieves stable social welfare of nearly 1.6 percent relative to total cost and reduces runtime by over 69.86 percent to 102.47 seconds, while ensuring benefit stability for both networks and theoretical convergence.

What carries the argument

The neural-enhanced time-adaptive robust Nash bargaining strategy, which uses ADMM for bargaining, robust dispatch for uncertainties, and a neural surrogate to select time scales based on welfare and computation trade-offs.

If this is right

  • The trading scheme converges theoretically.
  • Both networks maintain steady benefits even in worst renewable scenarios.
  • Social welfare stabilizes at nearly 1.6% of total cost.
  • Computation time drops to 102.47 seconds, a 69.86% improvement over fixed time-scale methods.
  • Participation of solid oxide fuel cells and HCNG yields sustainable returns.

Where Pith is reading between the lines

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

  • The method could be applied to other coupled energy systems involving different carriers like heat or water.
  • Neural surrogates trained on resolution effects might improve decision-making in other optimization-heavy energy problems.
  • Real-time implementation in operational networks would test whether the claimed efficiency gains hold beyond numerical cases.

Load-bearing premise

The neural network surrogate, trained on time-resolution effects, accurately predicts and accelerates the process without introducing approximation errors that materially reduce the economic benefits or robustness.

What would settle it

Running the full optimization at the neural-predicted time scales and observing whether the social welfare falls below 1.6% or the runtime savings disappear under the paper's test scenarios.

Figures

Figures reproduced from arXiv: 2310.05999 by Wenwen Zhang.

Figure 1
Figure 1. Figure 1: The proposed two-stage robust benefit sharing framework [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Solving scheme of the proposed two-stage robust benefit shar￾ing model between HCNG-enabled GDN and ADN The MP and SP at iteration r of the C&CG algorithm is formulated as (34)-(35). MP: { 𝐶𝑟 𝐸,𝑆𝑃 = min 𝒚 𝐶 𝐸 (28)-(31) 𝒙 = 𝒙̌𝑟−1 , 𝒖 = 𝒖̌𝑟−1 (34) SP: { 𝐶𝑟 𝐸,𝑆𝑃 = max 𝒖 min 𝒙 𝐶 𝐸 𝒚 = 𝒚̌𝑟 𝑠.𝑡. (13)-(23) (35) where 𝒙̌𝑟−1 , 𝒖̌𝑟−1 are solutions of decision variable in the second stage and the worst case of SP, re… view at source ↗
Figure 3
Figure 3. Figure 3: Configuration diagram of combined optimization model be￾tween electricity and HCNG distribution networks. A. Efficiency analysis of cooperative ADN and GDN To prove the effectiveness of the proposed cooperative op￾erational framework, three competitive models are given: Model 1: The proposed cooperative ADN and GDN. Model 2: No cooperation between ADN and GDN. Follow￾ing [29], ADN configures its own li-ion… view at source ↗
Figure 5
Figure 5. Figure 5: Benefit analysis (Top: Model 1. Bottom: Model 2) [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Marginal profit of the GDN selling HCNG upon the proposed benefit sharing mechanism (Model 1) under the testing robust cases. Tab. V and [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

Integrated hydrogen-enriched compressed natural gas (HCNG) and active distribution network (ADN) is providing efficient and sustainable flexibility for consuming renewable energies. Yet, cross-sector privacy and uncertain high-renewable scenarios block stable coordination. They also worsen decision performance and convergence. To conquer the barrier, a neural enhanced time-adaptive robust Nash bargaining strategy is proposed.In the first stage, to clear energy trading between ADN and gas distribution network (GDN) and promote its sustainability, a privacy preserved Nash Bargaining based on the alternating direction method of multipliers (ADMM) is applied. The next robust dispatch stage explores the worst renewable scenarios and derisks ADNs profit collapse from uncertainties. The convergence of the entire energy trading scheme is theoretically proved. As such, sustainable returns from the participation of solid oxide fuel cell (SOFC) and HCNG are facilitated. Finally, a time complexity and social welfare co-driven neural-enhanced time-adaptive strategy is proposed. The strategy assesses the influence of time resolution on social benefits and solving time in multi-energy trading. Based on the assessment, a neural network surrogate model is trained to accelerate the trading process in a close looped manner. Numerical assessment reveals that, the proposed strategy reaps a stable social welfare of nearly 1.6% to total cost, and benefit-steady situations for both ADN and GDN, even in the worst renewable scenarios. Moreover, it reduces runtime to 102.47s, improving computational efficiency by over 69.86% versus the fixed time-scale baseline, almost without sacrifice in economy.

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 / 1 minor

Summary. The manuscript proposes a neural-enhanced time-adaptive robust Nash bargaining strategy for sustainable energy trading between hydrogen-enriched compressed natural gas (HCNG) networks and active distribution networks (ADN). It combines a privacy-preserving ADMM-based Nash bargaining stage, a robust dispatch stage against worst-case renewables, a theoretical convergence proof for the overall scheme, and a neural network surrogate trained on time-resolution effects to accelerate the closed-loop process, reporting a stable 1.6% social welfare gain relative to total cost and a 69.86% runtime reduction to 102.47 s versus a fixed time-scale baseline, with minimal economic sacrifice.

Significance. If the surrogate predictions prove accurate without materially degrading the robust-stage outcomes, the framework could provide a practical method for improving computational efficiency in cross-sector energy coordination under renewable uncertainty while preserving privacy and sustainability benefits from SOFC and HCNG participation. The presence of a convergence proof for the ADMM bargaining and robust dispatch is a constructive element that, if fully detailed, supports the reliability of the core coordination mechanism.

major comments (2)
  1. [Abstract / neural-enhanced strategy] Abstract and neural-enhanced time-adaptive strategy section: The central numerical claims of 1.6% stable social welfare and 69.86% runtime reduction to 102.47 s are produced by the closed-loop neural surrogate, yet no error metrics, hold-out validation on unseen renewable trajectories, or bounds on how surrogate mispredictions affect the robust ADMM dispatch and welfare outcomes are supplied. This directly undermines the assertion of 'almost without sacrifice in economy.'
  2. [Neural surrogate training] Neural surrogate training description: The surrogate is trained directly on assessments generated by the same optimization model whose performance it accelerates, creating a self-referential dependency. Without independent test cases or sensitivity analysis showing that reported speed-ups and welfare figures hold outside the training distribution, the efficiency and economic claims cannot be confirmed as robust.
minor comments (1)
  1. [Abstract] Abstract: The phrasing 'reaps a stable social welfare of nearly 1.6% to total cost' is imprecise and should be clarified to specify the exact baseline and metric.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment point by point below, indicating where revisions will be made to strengthen the validation of the neural surrogate approach.

read point-by-point responses

  1. Referee: [Abstract / neural-enhanced strategy] Abstract and neural-enhanced time-adaptive strategy section: The central numerical claims of 1.6% stable social welfare and 69.86% runtime reduction to 102.47 s are produced by the closed-loop neural surrogate, yet no error metrics, hold-out validation on unseen renewable trajectories, or bounds on how surrogate mispredictions affect the robust ADMM dispatch and welfare outcomes are supplied. This directly undermines the assertion of 'almost without sacrifice in economy.'

    Authors: We acknowledge that the current version of the manuscript does not report explicit error metrics for the surrogate, hold-out validation on unseen renewable trajectories, or quantitative bounds on the effect of mispredictions on dispatch and welfare. In the revised manuscript we will add a new subsection under the neural-enhanced strategy that presents mean absolute error and maximum deviation of the surrogate predictions, results on a hold-out set of renewable scenarios, and a sensitivity study showing the resulting variation in social welfare and robust dispatch outcomes. These additions will directly support the claim of minimal economic sacrifice. revision: yes

  2. Referee: [Neural surrogate training] Neural surrogate training description: The surrogate is trained directly on assessments generated by the same optimization model whose performance it accelerates, creating a self-referential dependency. Without independent test cases or sensitivity analysis showing that reported speed-ups and welfare figures hold outside the training distribution, the efficiency and economic claims cannot be confirmed as robust.

    Authors: The surrogate is trained on data generated by the optimization model because the objective is to learn the specific mapping between time resolution, social welfare, and computation time for this HCNG-ADN bargaining problem. We agree that robustness outside the training distribution requires further demonstration. In the revision we will include sensitivity analysis that perturbs renewable penetration levels, uncertainty distributions, and network parameters, and we will evaluate the surrogate on scenarios drawn from these perturbed distributions to confirm that the reported runtime reductions and welfare values remain stable. revision: yes

Circularity Check

1 steps flagged

Neural surrogate trained on model evaluations creates self-referential acceleration in reported runtime and welfare

specific steps
  1. fitted input called prediction [Abstract]
    "The strategy assesses the influence of time resolution on social benefits and solving time in multi-energy trading. Based on the assessment, a neural network surrogate model is trained to accelerate the trading process in a close looped manner. Numerical assessment reveals that, the proposed strategy reaps a stable social welfare of nearly 1.6% to total cost, and benefit-steady situations for both ADN and GDN, even in the worst renewable scenarios. Moreover, it reduces runtime to 102.47s, improving computational efficiency by over 69.86% versus the fixed time-scale baseline, almost without牺牲在在"

    The NN surrogate is fitted to performance metrics (social welfare, solving time) produced by running the core optimization model at different resolutions. It is then used in closed-loop to choose those resolutions for the identical model, so the reported speed-up and welfare figures are generated by a surrogate whose training data and predictions are taken from the model's own evaluations.

full rationale

The paper's headline numerical results (1.6% welfare, 69.86% runtime reduction) are generated by the closed-loop neural-enhanced strategy. The strategy first runs the core ADMM bargaining model at varying time resolutions to collect welfare/time data, fits an NN surrogate to those assessments, then deploys the surrogate to select resolutions for the same model. This matches the fitted_input_called_prediction pattern: the acceleration and performance claims are produced by a component whose training data and selection logic are derived directly from the optimization model's own outputs. No other load-bearing steps reduce by construction; the Nash bargaining and robust dispatch stages are presented as independent.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard convergence properties of ADMM for Nash bargaining, the representativeness of chosen worst-case renewable scenarios, and the assumption that the neural surrogate generalizes without retraining; no new physical entities are postulated.

free parameters (1)
  • time-resolution selection parameters
    Chosen via assessment of social welfare versus solving time; used to train the surrogate model.
axioms (2)
  • standard math ADMM converges to the Nash bargaining solution for the privacy-preserved energy trading problem
    Invoked to guarantee stable coordination between ADN and GDN.
  • domain assumption Worst-case renewable scenarios sufficiently capture uncertainty to derisk profit collapse
    Underpins the robust dispatch stage.

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discussion (0)

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Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    Global energy transformation: A roadmap to 2050,

    International Renewable Energy Agency, “Global energy transformation: A roadmap to 2050,” Abu Dhabi, UAE, 2019, Accessed: Jun. 17, 2021. [Online]. Available: https://www.irena.org/publications

    work page 2050

  2. [2]

    A Novel Genera- tion Rescheduling Algorithm to Improve Power System Reliability With High Renewable Energy Penetration,

    M. Fan, K. Sun, D. Lane, W. Gu, Z. Li, and F. Zhang, “A Novel Genera- tion Rescheduling Algorithm to Improve Power System Reliability With High Renewable Energy Penetration,” IEEE Trans. Power Syst ., vol. 33, no. 3, pp. 3349–3357, May 2018

    work page 2018

  3. [3]

    Optimal operation for integrated electricity and natural gas systems considering demand response uncertainties,

    J. Duan, F. Liu, and Y. Yang, “Optimal operation for integrated electricity and natural gas systems considering demand response uncertainties,” Ap- plied Energy, vol. 323, p. 119455, Oct. 2022

    work page 2022

  4. [4]

    Techno-economic analysis of Power -to-Gas plants in a gas and electricity distribution network system with high renewable energy pene- tration,

    G. Fambri, C. Diaz-Londono, A. Mazza, M. Badami, T. Sihvonen, and R. Weiss, “Techno-economic analysis of Power -to-Gas plants in a gas and electricity distribution network system with high renewable energy pene- tration,” Applied Energy, vol. 312, p. 118743, Apr. 2022

    work page 2022

  5. [5]

    Distributed optimization of integrated electricity-natural gas distribution networks considering wind power un- certainties,

    J. Duan, Y. Yang, and F. Liu, “Distributed optimization of integrated electricity-natural gas distribution networks considering wind power un- certainties,” Int. J. Electr. Power Energy Syst. , vol. 135, p. 107460, Feb. 2022

    work page 2022

  6. [6]

    Optimal Operation of an Inte- grated Energy System Incorporated With HCNG Distribution Networks,

    C. Fu, J. Lin, Y. Song, J. Li, and J. Song, “Optimal Operation of an Inte- grated Energy System Incorporated With HCNG Distribution Networks,” IEEE Trans. Sustain. Energy, vol. 11, no. 4, pp. 2141–2151, Oct. 2020

    work page 2020

  7. [7]

    The GRHYD demonstration project ,

    ENGIE, “ The GRHYD demonstration project ,” [Online]. Available: https://www.engie.com/en/businesses/gas/hydrogen/power-to-gas/the- grhyd-demonstration-project

    work page

  8. [8]

    High efficiencies with low fuel utilization and thermally inte- grated fuel reforming in a hybrid solid oxide fuel cell gas turbine system,

    H. Chen, C. Yang, N. Zhou, N. Farida Harun, D. Oryshchyn, and D. Tucker, “High efficiencies with low fuel utilization and thermally inte- grated fuel reforming in a hybrid solid oxide fuel cell gas turbine system,” Applied Energy, vol. 272, p. 115160, Aug. 2020

    work page 2020

  9. [9]

    Effect of blending hydrogen to biogas fuel driven from anaerobic digestion of wastewater on the performance of a solid oxide fuel cell system,

    A. S. Mehr, A. Moharramian, S. Hossainpour, and D. A. Pavlov, “Effect of blending hydrogen to biogas fuel driven from anaerobic digestion of wastewater on the performance of a solid oxide fuel cell system,” Energy, vol. 202, p. 117668, Jul. 2020

    work page 2020

  10. [10]

    Economic performance study of the integrated MR-SOFC-CCHP system,

    Q. Hou, H. Zhao, and X. Yang, “Economic performance study of the integrated MR-SOFC-CCHP system,” Energy, vol. 166, pp. 236–245, Jan. 2019

    work page 2019

  11. [11]

    Effect of a SOFC plant on distri- bution system stability,

    F. Jurado, M. Valverde, and A. Cano, “Effect of a SOFC plant on distri- bution system stability,” J. Power Sources, vol. 129, no. 2, pp. 170 –179, Apr. 2004

    work page 2004

  12. [12]

    Contin- uous-Time Co -Operation of Integrated Electricity and Natural Gas Sys- tems With Responsive Demands Under Wind Power Generation Uncer- tainty,

    A. Nikoobakht, J. Aghaei, M. Shafie -Khah, and J. P. S. Catalao, “Contin- uous-Time Co -Operation of Integrated Electricity and Natural Gas Sys- tems With Responsive Demands Under Wind Power Generation Uncer- tainty,” IEEE Trans. Smart Grid, vol. 11, no. 4, pp. 3156–3170, Jul. 2020

    work page 2020

  13. [13]

    Incentive Mechanism Design for Integrated Microgrids in Peak Ramp Minimization Problem,

    H. K. Nguyen, A. Khodaei, and Z. Han, “Incentive Mechanism Design for Integrated Microgrids in Peak Ramp Minimization Problem,” IEEE Trans. Smart Grid, vol. 9, no. 6, pp. 5774–5785, Nov. 2018

    work page 2018

  14. [14]

    Risk premia in the German day -ahead electricity market revisited: The impact of negative prices,

    N. Valitov, “Risk premia in the German day -ahead electricity market revisited: The impact of negative prices,” Energy Economics, vol. 82, pp. 70–77, Aug. 2019

    work page 2019

  15. [15]

    Energy management for a community - level integrated energy system with photovoltaic prosumers based on bar- gaining theory,

    A. Jiang, H. Yuan, and D. Li, “Energy management for a community - level integrated energy system with photovoltaic prosumers based on bar- gaining theory,” Energy, vol. 225, Jun. 2021

    work page 2021

  16. [16]

    Bargaining -based cooperative energy trading for distribution company and demand response,

    S. Fan, Q. Ai, and L. Piao, “Bargaining -based cooperative energy trading for distribution company and demand response,” Appl. Energy, vol. 226, pp. 469–482, Sep. 2018

    work page 2018

  17. [17]

    Sample Robust Scheduling of Electricity -Gas Systems Under Wind Power Uncertainty,

    R.-P. Liu, Y. Hou, Y. Li, S. Lei, W. Wei, and X. Wang, “Sample Robust Scheduling of Electricity -Gas Systems Under Wind Power Uncertainty,” IEEE Trans. Power Syst., vol. 36, no. 6, pp. 5889–5900, Nov. 2021

    work page 2021

  18. [18]

    Optimal Sizing of Energy Storage Systems Using Frequency -Separation-Based Energy Management for Fuel Cell Hybrid Electric Vehicles,

    J. Snoussi, S. B. Elghali, M. Benbouzid, and M. F. Mimouni, “Optimal Sizing of Energy Storage Systems Using Frequency -Separation-Based Energy Management for Fuel Cell Hybrid Electric Vehicles,” IEEE Trans. Veh. Technol., vol. 67, no. 10, pp. 9337–9346, Oct. 2018

    work page 2018

  19. [19]

    Stability- constrained two-stage robust optimization for integrated hydrogen hybrid energy system,

    Q. Li, Y. Qiu, H. Yang, Y. Xu, W. Chen, and P. Wang, “Stability- constrained two-stage robust optimization for integrated hydrogen hybrid energy system,” CSEE JPES, 2020

    work page 2020

  20. [20]

    Robust optimal dis- patching of integrated electricity and gas system considering refined pow- er-to-gas model under the dual carbon target,

    G. Qin, Q. Yan, D. Kammen, C. Shi, and C. Xu, “Robust optimal dis- patching of integrated electricity and gas system considering refined pow- er-to-gas model under the dual carbon target,” J. Clean. Prod ., vol. 371, Oct. 2022

    work page 2022

  21. [21]

    Robust day -ahead coordinated scheduling of multi -energy systems with integrated heat -electricity de- mand response and high penetration of renewable energy,

    J. Lu, T. Liu, C. He, L. Nan, and X. Hu, “Robust day -ahead coordinated scheduling of multi -energy systems with integrated heat -electricity de- mand response and high penetration of renewable energy,” Renew. Energ., vol. 178, pp. 466–482, Nov. 2021

    work page 2021

  22. [22]

    Adaptive‐robust multi‐resolution generation maintenance scheduling with probabilistic reliability constraint,

    B. Bagheri and N. Amjady, “Adaptive‐robust multi‐resolution generation maintenance scheduling with probabilistic reliability constraint,” IET Gener. Transm. Distrib., vol. 13, no. 15, pp. 3292–3301, Aug. 2019

    work page 2019

  23. [23]

    Steady state analysis of gas networks with distributed injection of alternative gas,

    M. Abeysekera, J. Wu, N. Jenkins, and M. Rees, “Steady state analysis of gas networks with distributed injection of alternative gas,” Appl. Energy, vol. 164, pp. 991–1002, Feb. 2016

    work page 2016

  24. [24]

    Day-Ahead Dispatch of Integrated Electric- ity and Natural Gas System Considering Reserve Scheduling and Renew- able Uncertainties,

    F. Liu, Z. Bie, and X. Wang, “Day-Ahead Dispatch of Integrated Electric- ity and Natural Gas System Considering Reserve Scheduling and Renew- able Uncertainties,” IEEE Trans. Sustain. Energy, vol. 10, no. 2, pp. 646– 658, Apr. 2019

    work page 2019

  25. [25]

    A convex optimization and iterative solu- tion based method for optimal power-gas flow considering power and gas losses,

    L. Yang, X. Zhao, and Y. Xu, “A convex optimization and iterative solu- tion based method for optimal power-gas flow considering power and gas losses,” Int. J. Electr. Power Energy Syst., vol. 121, p. 106023, Oct. 2020

    work page 2020

  26. [26]

    Optimal Scheduling of FTPSS With PV and HESS Considering the Online Degradation of Battery Capacity,

    M. Chen, Z. Liang, Z. Cheng, J. Zhao, and Z. Tian, “Optimal Scheduling of FTPSS With PV and HESS Considering the Online Degradation of Battery Capacity,” IEEE Trans. Transp. Electrific, vol. 8, no. 1, pp. 936 – 947, Mar. 2022

    work page 2022

  27. [27]

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

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

    work page 2022

  28. [28]

    Integrated Electricity and Natural Gas Demand Response for Manufacturers in the Smart Grid,

    F. Dababneh and L. Li, “Integrated Electricity and Natural Gas Demand Response for Manufacturers in the Smart Grid,” IEEE Trans. Smart Grid, vol. 10, no. 4, pp. 4164–4174, Jul. 2019

    work page 2019

  29. [29]

    P -robust energy management of a multi -energy microgrid enabled with energy conversions under various uncertainties,

    V. Khaligh, A. Ghezelbash, M. Mazidi, J. Liu, and J. -H. Ryu, “P -robust energy management of a multi -energy microgrid enabled with energy conversions under various uncertainties,” Energy, vol. 271, p. 127084, May 2023

    work page 2023

  30. [30]

    Flexibility -based improved energy hub model for multi -energy distribution systems,

    N. Tomin, V. Kurbatsky, K. Suslov, D. Gerasimov, E. Serdyukova, and D. Yang, “Flexibility -based improved energy hub model for multi -energy distribution systems,” IET Conference Proceedings, vol. 2022, no. 3. IET, UK, pp. 409–13, 2022

    work page 2022

  31. [31]

    Coordinated Investment Planning of Distributed Multi-Type Stochastic Generation and Battery Storage in Active Distribu- tion Networks,

    A. Ehsan and Q. Yang, “Coordinated Investment Planning of Distributed Multi-Type Stochastic Generation and Battery Storage in Active Distribu- tion Networks,” IEEE Trans. Sustain. Energy , vol. 10, no. 4, pp. 1813 – 1822, Oct. 2019

    work page 2019

  32. [32]

    Distributed Optimization and Statistical Learning via the Alter- nating Direction Method of Multipliers,

    S. Boyd, “Distributed Optimization and Statistical Learning via the Alter- nating Direction Method of Multipliers,” FNT in Machine Learning , vol. 3, no. 1, pp. 1–122, 2010

    work page 2010

  33. [33]

    D. P. Bertsekas, Nonlinear Programming, 2nd ed. Athena-Scientific, 1999

    work page 1999

  34. [34]

    Column Generation Algorithms for Nonlinear Optimization, I: Convergence Analysis,

    R. García, A. Marín, and M. Patriksson, “Column Generation Algorithms for Nonlinear Optimization, I: Convergence Analysis,” Optimization, vol. 52, no. 2, pp. 171–200, Jan. 2003

    work page 2003

  35. [35]

    Solving two -stage robust optimization problems using a column -and-constraint generation method,

    B. Zeng and L. Zhao, “Solving two -stage robust optimization problems using a column -and-constraint generation method,” Oper. Res. Lett. , vol. 41, no. 5, pp. 457–461, Sep. 2013

    work page 2013