Convex Hull Method for Real-time Pricing in Smart Grid with Multiple Generating Units via Smoothing Technique

Zeyi Zhu; Yan Gao; Xiaodong Ding; Youmeng He

doi:10.35833/MPCE.2025.000688

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Convex Hull Method for Real-time Pricing in Smart Grid with Multiple Generating Units via Smoothing Technique

Original Paper | 更新时间：2026-09-28

- Convex Hull Method for Real-time Pricing in Smart Grid with Multiple Generating Units via Smoothing Technique
- Journal of Modern Power Systems and Clean Energy Vol. 14, Issue 5, Pages: 1909-1920(2026)
- 作者机构：
  
  School of Management, University of Shanghai for Science and Technology, Shanghai, China
- 作者简介：
  
  Zeyi Zhu received the M.S. degree from Henan Normal University, Xinxiang, China, in 2020. She is currently working toward the Ph.D. degree with the School of Management, University of Shanghai for Science and Technology, Shanghai, China. Her research interests include demand response, real-time pricing, and distributed generation system. Email:zhuzeyi7969@163.com
  Yan Gao received the Ph.D. degree from the Dalian University of Technology, Dalian, China, in 1996. Since 2001, he has been a Chair Professor with the University of Shanghai for Science and Technology, Shanghai, China. His current research interests include distributed generation system, demand-side management, real-time pricing, and nonsmooth optimization. Email: gaoyan@usst.edu.cn
  Xiaodong Ding received the Ph.D. degree from Donghua University, Shanghai, China, in 2002. He is currently a Professor with the University of Shanghai for Science and Technology, Shanghai, China. His main research interests include modeling and control of complex systems. Email:xdding@ usst.edu.cn
  Youmeng He received the M.S. degree from Qingdao University, Qingdao, China, in 2022. She is working toward the Ph.D. degree with the School of Management, University of Shanghai for Science and Technology, Shanghai, China. Her current research interests include smart grid, real-time pricing, and reinforcement learning. Email:heymyx@126.com
- 基金信息：
- DOI：10.35833/MPCE.2025.000688
  CLC：
- Received：28 July 2025，
  
  Revised：2025-11-04，
  
  Accepted：21 December 2025，
  
  Online First：02 April 2026，
  
  Published：15 September 2026
- 稿件说明：
移动端阅览
Zhu Zeyi,Gao Yan,Ding Xiaodong,et al.Convex Hull Method for Real-time Pricing in Smart Grid with Multiple Generating Units via Smoothing Technique[J].Journal of Modern Power Systems and Clean Energy,2026,14(05):1909-1920.
DOI：

Zhu Zeyi,Gao Yan,Ding Xiaodong,et al.Convex Hull Method for Real-time Pricing in Smart Grid with Multiple Generating Units via Smoothing Technique[J].Journal of Modern Power Systems and Clean Energy,2026,14(05):1909-1920. DOI： 10.35833/MPCE.2025.000688.

摘要

Abstract

The increasing electricity demand imposes higher requirements on the power system to generate electricity and balance the supply and demand. The unit commitment problem is critical in power systems for determining the generation schedules and market prices. However

its inherent nonconvexity and discontinuity pose significant challenges to efficient pricing. The convex hull method addresses this nonconvexity by taking the slope of the convex envelope of cost function for generating units over the convex hull of the feasible set as the price. Nevertheless

due to the nonsmoothness

the convex hull price cannot be computed by using the gradient-based methods

which increases the computational complexity. Smoothing techniques provide an effective way by converting the nonsmooth problem into a smooth one

which enhances the computational efficiency. This paper proposes a real-time pricing scheme for the smart grid that incorporates renewable energy generation and energy storage devices on the demand side and multiple generating units on the supply side. Considering the dynamic demand of users and startup costs of generating units

a social welfare maximization model is formulated. By employing the convex hull method

the convex envelopes of the cost functions for generating units are derived. The smoothing technique is then employed to convert the original model into a smooth one. Furthermore

a distributed iterative algorithm on the basis of gradient projection method is developed by leveraging the separable structure of the variables to solve the model efficiently. Simulation results validate the feasibility and effectiveness of the convex hull method for real-time pricing in smart grid with multiple generating units via smoothing technique.

关键词

Keywords

references

Z. Guo, J. Bai, W. Wei et al. , “Multi-stage robust unit commitment with discrete load shedding based on partially affine policy and two-stage reformulation,” Journal of Modern Power Systems and Clean Energy , vol. 13 , no. 2 , pp. 415 - 425 , Mar. 2025 .

N. Stevens and A. Papavasiliou, “Application of the level method for computing locational convex hull prices,” IEEE Transactions on Power Systems , vol. 37 , no. 5 , pp. 3958 - 3968 , Sept. 2022.

E. S. Ali, S. M. A. Elazim, and A. S. Balobaid, “Implementation of coyote optimization algorithm for solving unit commitment problem in power systems,” Energy , vol. 263 , p. 125697 , Jan. 2023 .

F. Hyder, B. Yan, M. Bragin et al. , “Convex hull pricing for unit commitment: survey, insights, and discussions,” Energies , vol. 17 , no. 19 , p. 4815 , Mar. 2024 .

Y. Liu, D. Liu, and H. Zhang, “Stochastic unit commitment with high-penetration offshore wind power generation in typhoo n scenarios,” Journal of Modern Power Systems and Clean Energy , vol. 12 , no. 2 , pp. 535 - 546 , Mar. 2024 .

J. D. Fuller and M. Pirnia, “Nonconvex multicommodity near equilibrium models: energy markets perspective,” Operations Research Perspectives , vol. 9 , no. 2 , p. 100243 , Jun. 2022 .

S. H. Dolatabadi, M. A. Latify, H. Karshenas et al. , “On pricing issues in electricity markets in the presence of externalities,” Energy , vol. 246 , p. 123273 , May 2022.

L. Yang, Q. Qin, S. Chen et al. , “Fully distributed convex hull pricing based on alternating direction method of multipliers,” Computers and Operations Research , vol. 173 , p. 106823 , Jan. 2025 .

N. Li, Y. Gao, and Y. Yin, “Joint chance-constrained real-time pricing for smart grid with multi-energy generation: a stochastic convex hull-based approach,” Sustainable Energy, Grids and Networks , vol. 43 , p. 101809 , Jul. 2025 .

C. Byers and G. Hug, “Long-run optimal pricing in electricity markets with nonconvex costs,” European Journal of Operational Research , vol. 307 , no. 1 , pp. 351 - 363 , May 2023.

M. S. Ahunbay, M. Bichler, T. Dobos et al. , “Solving large-scale electricity market pricing problems in polynomial time,” European Journal of Operational Research , vol. 318 , no. 2 , pp. 605 - 617 , Oct. 2024 .

C. Gentile, G. Morales-Espana, and A. Ramos, “A tight MIP formulation of the unit commitment problem with start-up and shut-down constraints,” EURO Journal on Computational Optimization , vol. 5 , no. 1-2, pp. 177 - 201 , Mar. 2017 .

H. Xu, B. Feng, G. Huang et al. , “Convex hull based economic operating region for power grids considering uncertainties of renewable energy sources,” Journal of Modern Power Systems and Clean Energy , vol. 12 , no. 5 , pp. 1419 - 1430 , Sept. 2024.

P. R. Gribik, W. W. Hogan, and S. L. Pope. (2024, Nov.). Market-clearing electricity prices and energy uplift. [Online]. Available: https://whogan.scholars.harvard.edu/resource/gribikhoganpopepriceuplift12310 7pdf.

C. Wang, T. Peng, P. B. Luh et al. , “The subgradient simplex cutting plane method for extended locational marginal prices,” IEEE Transactions on Power Systems , vol. 28 , no. 3 , pp. 2758 - 2767 , Aug. 2013 .

G. Wang, U. V. Shanbhag, T. Zheng et al. , “An extreme-point subdifferential method for convex hull pricing in energy and reserve markets – Part II: convergence analysis and numerical performance,” IEEE Transactions on Power Systems , vol. 28 , no. 3 , pp. 2121 - 2127 , Aug. 2013 .

Z. Li and Y. Xu, “Pricing balancing ancillary services for low-inertia power systems under uncertainty and nonconvexity,” Applied Energy , vol. 377 , p. 124591 , Jan. 2025 .

B. Hua and R. Baldick, “A convex primal formulation for convex hull pricing,” IEEE Transactions on Power Systems , vol. 32 , no. 5 , pp. 3814 - 3823 , Sept. 2017.

P. Andrianesis, D. Bertsimas, M. C. Caramanis et al. , “Computation of convex hull prices in electricity markets with nonconvexities using Dantzig-Wolfe decomposition,” IEEE Transactions on Power Systems , vol. 37 , no. 4 , pp. 2578 - 2589 , Jul. 2022 .

Y. Wang, Z. Yang, J. Yu et al. , “Pricing in non-convex electricity markets with flexible trade-off of pricing properties,” Energy , vol. 274 , no. 1 , p. 127382 , Jul. 2023 .

Z. Lu, J. Wang, M. Shahidehpour et al. , “Convex-hull pricing of ancillary services for power system frequency regulation with renewables and carbon-capture-utilization-and-storage systems,” IEEE Transactions on Power Systems , vol. 39 , no. 5 , pp. 6615 - 6635 , Sept. 2024.

N. Sugishita, A. Grothey, and K. McKinnon, “Use of machine learning models to warm-start column generation for unit commitment,” INFORMS Journal on Computing , vol. 36 , no. 4 , pp. 1129 - 1146 , Jul. 2024 .

W. Lin and Z. Yang, “Nonconvexity pricing and allocating costs in stochastic electricity markets,” CSEE Journal of Power and Energy Systems , vol. 10 , no. 4 , pp. 1466 - 1477 , Jul. 2024 .

N. Ito, A. Takeda, and T. Namerikawa, “Convex hull pricing for demand response in electricity markets,” in Proceedings of 2013 IEEE International Conference on Smart Grid Communications , Vancouver, Canada, Oct. 2013 , pp. 151 - 156 .

Y. Wang, J. Li, Y. Gao et al. , “Monitoring power consumption with automated process control strategy and adjustment considering price sensitivity and multi-source collaboration,” Computers and Industrial Engineering , vol. 196 , p. 110480 , Oct. 2024 .

Y. Yang, S. Du, and Y. Chen, “Real-time pricing method for smart grid based on social welfare maximization model,” Journal of Industrial and Management Optimization , vol. 19 , no. 3 , pp. 2206 - 2225 , Mar. 2023 .

S. Barua and N. Mohammad, “Energy storage scheduling considering day-ahead time of use pricing to minimize electricity consumption cost of prosumer,” Journal of Energy Storage , vol. 113 , p. 115613 , Mar. 2025 .

D. Qu, J. Li, and M. Yong, “Real-time pricing for smart grid considering energy complementarity of a microgrid interacting with the large grid,” International Journal of Electrical Power and Energy Systems , vol. 141 , p. 108217 , Oct. 2022 .

Y. Wang, J. Li, D. Qu et al. , “Low-carbon economic operation strategy for a multi -microgrid system considering internal carbon pricing and emission monitoring,” Journal of Process Control , vol. 143 , p. 103313 , Sept. 2024.

O. E. Williamson, “Peak-load pricing and optimal capacity under indivisibility constraints,” The American Economic Review , vol. 56 , no. 4 , pp. 810 - 827 , Jan. 1966 .

B. Liu, F. Xie, and L. Chai, “A scalable energy management mechanism for peer-to-peer electricity market,” IEEE Transactions on Industrial Informatics , vol. 20 , no. 7 , pp. 9255 - 9265 , Jul. 2024 .

K. Parandeh, A. Bagheri, and S. Jadid, “Optimal day-ahead dynamic pricing of grid-connected residential renewable energy resources under different metering mechanisms,” Journal of Modern Power Systems and Clean Energy , vol. 11 , no. 1 , pp. 168 - 178 , Jan. 2023 .

D. Qu, J. Li, and X. Ma, “Distributed real-time pricing of smart grid considering individual differences,” Omega , vol. 127 , p. 103109 , Sept. 2024.

L. Song and G. Sheng, “A smoothing Newton method for real-time pricing in smart grids based on user risk classification,” Mathematics , vol. 13 , no. 5 , p. 822 , Feb. 2025 .

H. Zhu, H. Yin, X. Feng et al. , “Demand response management via real-time pricing for microgrid with electric vehicles under cyber-attack,” Electronics , vol. 12 , no. 6 , p. 1321 , Mar. 2023 .

Z. Xu, L. Guo, Y. Gao et al. , “Real-time pricing of smart grid based on piece-wise linear functions,” Journal of Systems Science and Information , vol. 7 , no. 4 , pp. 295 - 316 , Aug. 2019 .

N. Li and Y. Gao, “Real-time pricing based on convex hull method for smart grid with multiple generating units,” Energy , vol. 285 , p. 129543 , Dec. 2023 .

S. Lyaqini, A. Hadri, and L. Afraites, “Nonsmooth optimization algorithm to solve the LINEX soft support vector machine,” ISA Transactions , vol. 153 , pp. 322 - 333 , Oct. 2024 .

M. A. Karapetyants, “A fast continuous time approach for nonsmooth convex optimization using Tikhonov regularization technique,” Computational Optimization and Applications , vol. 87 , pp. 531 - 569 , Oct. 2024 .

R. T. Rockafellar and R. J.-BWets, Variational Analysis . New York: Springer-Verlag, 1998.

F. Wu and W. Bian, “Smoothing accelerated proximal gradient method with fast convergence rate for nonsmooth convex optimization beyond differentiability,” Journal of Optimization Theory and Applications , vol. 197 , pp. 539 - 572 , Mar. 2023 .

A. Bohm and S. J. Wright, “Variable smoothing for weakly convex composite functions,” Journal of Optimization Theory and Applications , vol. 188 , pp. 628 - 649 , Feb. 2021 .

Y. Li, J. Li, Z. Yu et al. , “A cosh-based smoothing Newton algorithm for the real-time pricing problem in smart grid,” International Journal of Electrical Power and Energy Systems , vol. 135 , p. 107296 , Oct. 2022 .

J. J. Moreau, “Proximité et dualité dans un espace Hilbertien,” Bulletin de la Société Mathématique de France , vol. 93 , pp. 273 - 299 , Mar. 1965 .

G. Yuan, Y. Gao, and B. Ye, “Optimal dispatching strategy and real-time pricing for multi-regional integrated energy systems based on demand response,” Renewable Energy , vol. 179 , pp. 1424 - 1446 , Dec. 2021 .

L. Zhang, Y. Gao, H. Zhu et al. , “Bi-level stochastic real-time pricing model in multi-energy generation system: a reinforcement learning approach,” Energy , vol. 269 , p. 121926 , Oct. 2022 .

R. Wang, C. Zhang, L. Wang et al. , “A stochastic Nesterov’s smoothing accelerated method for general nonsmooth constrained stochastic composite convex optimization,” Journal of Scientific Computing , vol. 93 , p. 52 , Oct. 2022 .

D. P. Bertsekas, Nonlinear Programming , 3rd ed. Belmont, USA: Athena Scientific, 2016.

P. M. Teferra, L. M. Ngoo, and G. N. Nyakoe, “Fuzzy-based prediction of solar PV and wind power generation for microgrid modeling using particle swarm optimization,” Heliyon , vol. 9 , p. e12802, Jan. 2023 .

D. Samadi, A.-H. Mohsenian-Rad, R. Schober et al. , “Optima l real-time pricing algorithm based on utility maximization for smart grid,” in Proceedings of 2020 1st IEEE International Conference on Smart Grid Communications , Gaithersburg, USA, Oct. 2010 , pp. 415 - 420 .

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