Energy management strategy for a power-split hydraulic hybrid vehicle based on Lagrange multiplier and its modifications

Zhekang Du, Kai Loon Cheong, Perry Y. Li

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


Lagrange multiplier approach is a computationally efficient method for computing optimal energy management strategy for a hydraulic hybrid vehicle under the assumption that the accumulator dynamics can be ignored and only the net use of storage energy is considered. Although it provides a close estimate to the fuel economy compared to that obtained using dynamic programming, the resulting control strategy does not respect the physical limits of the storage capacity of the hydraulic accumulator. Thus, the synthesized control strategy is not feasible for actual driving. This article investigates the basic Lagrange multiplier approach for real-time control and proposes modifications so that the storage capacity is respected. It is shown that the Lagrange multiplier can be interpreted as an equivalent loss factor which turns out to be the marginal loss associated with the discharge of stored energy. The two proposed modifications are as follows: (1) a moving horizon approach and (2) making the Lagrange multiplier a function of the current state of charge. Both methods are successful in maintaining the accumulator state of charge within limits with modest effect on fuel economy (3%–5% lower).

Original languageEnglish (US)
Pages (from-to)511-523
Number of pages13
JournalProceedings of the Institution of Mechanical Engineers. Part I: Journal of Systems and Control Engineering
Issue number5
StatePublished - May 1 2019

Bibliographical note

Funding Information:
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is performed within the Center for Compact and Efficient Fluid Power, supported by the National Science Foundation of the United States under Grant EEC-0540834.

Publisher Copyright:
© IMechE 2018.


  • Hydraulic hybrid vehicles
  • constrained optimization
  • dynamic programming
  • energy management
  • equivalent consumption minimization strategy
  • power-split vehicles


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