From Offline to Adaptive Online Energy Management Strategy of Hybrid Vehicle Using Pontryagin’s Minimum Principle

  • Nadir OuddahEmail author
  • Lounis Adouane
  • Rustem Abdrakhmanov


This paper details the development of an energy management strategy (EMS) for real-time control of a multi hybrid plug-in electric bus. The energy management problem has been formulated as an optimal control problem in order to minimize the fuel consumption of the bus drivetrain for a typical day of operation. Considering the physical characteristics of the studied hybrid electric bus and its well-known daily tour, the Pontryagin’s minimum principle (PMP) is firstly used as the mean to obtain offline optimal EMS. Afterward, in order to adapt the proposed strategy for real-time implementation, the proposed control parameters are adapted online using feedback from the battery state of energy (SOE) which allows us to accurately control the battery SOE in the presence of wide range of uncertainties. The work proposed in this paper is conducted on a dedicated high-fidelity dynamical model of the hybrid bus, that was developed on MATLAB/TruckMaker software. The performance evaluation of the proposed strategy is carried out using a normalized driving cycles to represent different driving scenarios. Obtained results show that among the investigated methods, it is reasonable to conclude that the proposed adaptive online strategy based on PMP is the most suitable to design the targeted EMS.

Key words

Optimal control Heavy hybrid vehicle Energy management Pontryagin’s minimum principle 



bus frontal area


drag coefficient


displacement of the hydraulic motor


displacement of the hydraulic pump


electric motor


maximum energy stored in the battery


aerodynamic force


gravity force


rolling resistance


tractive force


gravity acceleration


Hamiltonian function


hydraulic motor


hydraulic pump


internal combustion engine


mass of the bus


fuel flow rate


power delivered by the battery


power consumed by the electric motor


instantaneous power of the fuel


lower heating value of the fuel


static loaded radius of the wheel


battery state of charge


battery state of energy


torque of the hydraulic motor


torque of the engine


torque of the wheel


admissible control set


velocity of the bus


acceleration of the bus

γ, σ

lagrange multipliers used to introduce constraints


mechanical efficiency of the hydraulic motor


mechanical efficiency of the hydraulic pump


volumetric efficiency of the hydraulic motor


volumetric efficiency of the hydraulic pump


efficiency of the battery


slope of the road


slope of the road


initial values of the costate


maximum values of the costate


minimum values of the costate


rolling resistance coefficient


maximum hydraulic torque variation rate


density of the air

ρ1, ρ2

gearbox’ reduction ratios


rotational speed of the hydraulic motor


rotational speed of the engine


rotational speed of the wheel


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Copyright information

© The Korean Society of Automotive Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Nadir Ouddah
    • 1
    Email author
  • Lounis Adouane
    • 1
  • Rustem Abdrakhmanov
    • 1
  1. 1.Institut Pascal, IMobS3, UCA/SIGMA UMR CNRS 6602Clermont-FerrandFrance

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