Optimal congestion pricing with day-to-day evolutionary flow dynamics: A mean–variance optimization approach

Qixiu Cheng, Jun Chen, Honggang Zhang, Zhiyuan Liu

Research output: Journal article publicationJournal articleAcademic researchpeer-review

3 Citations (Scopus)


This paper investigates the optimal congestion pricing problem that considers day-to-day evolutionary flow dynamics. Under the circumstance that traffic flows evolve from day to day and the system might be in a non-equilibrium state during a certain period of days after implementing (or adjusting) a congestion toll scheme, it is questionable to use an equilibrium-based index under steady state as the objective to measure the performance of a congestion toll scheme. To this end, this paper proposes a mean–variance-based congestion pricing scheme, which is a robust optimization model, to consider the evolution process of traffic flow dynamics in the optimal toll design problem. More specifically, in the mean–variance-based toll scheme, travelers aim to minimize the variance of expected total travel costs (ETTCs) on different days to reduce risk in daily travels, while the average ETTC over the whole planning period is restricted to being no larger than a predetermined target value set by the authorities. A metaheuristic approach based on the whale optimization algorithm is designed to solve the proposed mean–variance-based day-to-day dynamic congestion pricing problem. Finally, a numerical experiment is conducted to validate the effectiveness of the proposed model and solution algorithm. Results show that the used 9-node network can reach a steady state within 18 days after implementing the mean–variance-based congestion pricing, and the optimal toll scheme can be also obtained with this toll strategy.

Original languageEnglish
Article number4931
JournalSustainability (Switzerland)
Issue number9
Publication statusPublished - 1 May 2021
Externally publishedYes


  • Day-to-day dynamics
  • Dynamic congestion pricing
  • Mean–variance optimization
  • Robust optimization
  • Whale optimization algorithm

ASJC Scopus subject areas

  • Geography, Planning and Development
  • Renewable Energy, Sustainability and the Environment
  • Environmental Science (miscellaneous)
  • Energy Engineering and Power Technology
  • Management, Monitoring, Policy and Law


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