Energy management of hybrid vehicles with state constraints: a penalty and implicit Hamiltonian minimization approach

Marcelino Sánchez (Corresponding author), Sébastien Delprat, Theo Hofman

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19 Citations (Scopus)
31 Downloads (Pure)


When designing hybrid vehicles, the energy management is formulated as an optimal control problem. The Pontryagin's minimum principle represents a powerful methodology capable of solving the energy management offline. Moreover, the Pontryagin's minimum principle has been proved useful in the derivation of online energy management algorithms, such as the equivalent consumption minimization strategy. Nevertheless, difficulties on the application of the Pontryagin's minimum principle arise when state constraints are included in the definition of the problem. A possible solution is to combine the Pontryagin's minimum principle with a penalty function approach. This is done by adding functions to the Hamiltonian, which increase the value of the Hamiltonian whenever the optimal trajectory violates its constraints. However, the addition of penalty functions to the Hamiltonian makes it harder to compute its minimum. This work proposes an effective penalty approach through an implicit Hamiltonian minimization. The proposed method is applied to solve the energy management for a hybrid electric vehicle modeled as a mixed input-state constrained optimal control problem with two states: the battery temperature and state-of-energy. It is demonstrated to be up to 46 times faster than the dynamic programming method while taking benefits of state-of-the-art boundary value problem solvers and avoiding any issue related to state quantization.

Original languageEnglish
Article number114149
Number of pages12
JournalApplied Energy
Publication statusPublished - 15 Feb 2020


  • Energy management
  • Hybrid electric vehicles
  • Mixed input-state constraints
  • Penalty function approach
  • Pontryagin's minimum principle


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