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
- Convex Hull Method for Real-time Pricing in Smart Grid with Multiple Generating Units via Smoothing Technique
- MPCE 2026年14卷第5期页码：1909-1920
- 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：
  
  National Natural Science Foundation of China(72071130)
- DOI：10.35833/MPCE.2025.000688
  中图分类号：
- 收稿：2025-07-28，
  
  修回：2025-11-04，
  
  录用：2025-12-21，
  
  网络首发：2026-04-02，
  
  纸质出版：2026-09-15
- 稿件说明：
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Zeyi Zhu, Yan Gao, Xiaodong Ding, 等. Convex Hull Method for Real-time Pricing in Smart Grid with Multiple Generating Units via Smoothing Technique[J]. MPCE, 2026,14(5):1909-1920.

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.
Zeyi Zhu, Yan Gao, Xiaodong Ding, 等. Convex Hull Method for Real-time Pricing in Smart Grid with Multiple Generating Units via Smoothing Technique[J]. MPCE, 2026,14(5):1909-1920. DOI： 10.35833/MPCE.2025.000688.

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

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