Research Papers

Fuel Consumption Optimization of Heavy-Duty Vehicles With Grade, Wind, and Traffic Information

[+] Author and Article Information
Chaozhe R. He

Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: hchaozhe@umich.edu

Helmut Maurer

Institute of Computational and
Applied Mathematics,
University of Münster,
Münster D-48149, Germany
e-mail: maurer@math.uni-muenster.de

Gábor Orosz

Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: orosz@umich.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 23, 2015; final manuscript received May 17, 2016; published online July 27, 2016. Assoc. Editor: Paramsothy Jayakumar.

J. Comput. Nonlinear Dynam 11(6), 061011 (Jul 27, 2016) (12 pages) Paper No: CND-15-1343; doi: 10.1115/1.4033895 History: Received October 23, 2015; Revised May 17, 2016

In this paper, we establish a mathematical framework that allows us to optimize the speed profile and select the optimal gears for heavy-duty vehicles (HDVs) traveling on highways while varying parameters. The key idea is to solve the analogous boundary value problem (BVP) analytically for a simple scenario (linear damped system with quadratic elevation profile) and use this result to initialize a numerical continuation algorithm. Then, the numerical algorithm is used to investigate how the optimal solution changes with parameters. In particular, we gradually introduce nonlinearities (air resistance and engine saturation), implement different elevation profiles, and incorporate external perturbations (headwind and traffic). This approach enables real-time optimization in dynamic traffic conditions, therefore may be implemented on-board.

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Grahic Jump Location
Fig. 1

Elevation h as a function of distance d and arc-lengths s

Grahic Jump Location
Fig. 2

Contours in the plane of speed v and control input ad. (a) Optimal gear ratios; (b) experimental fuel consumption contours, with unit (g/s); (c) fitted fuel consumption contours, with unit (g/s); and (d) BSFC with optimal gear ratios applied, with unit (g/(kW hr)). The black crosses represent points with minimal BSFC value. In all the four panels, black-dashed curves indicate the boundaries of the domain accessible by the engine for all the gears.

Grahic Jump Location
Fig. 3

Traffic penalty given by Eqs. (14)(16)

Grahic Jump Location
Fig. 4

Six possible scenarios involving one switch. For trajectories 1 and 2, ad switches from maximum to minimum (i.e., ζ=pv+λv switches from negative to positive). For trajectories 4 and 5, ad switches from minimum to maximum (i.e., ζ=pv+λv switches from positive to negative). Trajectories 1 and 5 represent transverse scenarios, while trajectories 2 and 4 show tangential scenarios. Trajectories 3 and 6 are for the grazing scenarios, where the minimum and the maximum control input is maintained, respectively. Bold segments and crosses indicate ζ = 0.

Grahic Jump Location
Fig. 5

Time evolution of the speed v (left column), the corresponding control input ad (middle column), and the switching variable ζ (right column) for different values of the parameters σ as indicated. The rows correspond to the points A–F marked in Fig. 6. Red-dashed curves represent analytical solutions, while solid curves represent numerical solutions and they match very well.

Grahic Jump Location
Fig. 6

The upper panel gives the terminal time tf as a function of σ, while the lower panel shows the fuel consumption Q(44) as a function of σ, for v(tf)=v(0)=25 m/s, amax=0.6 m/s2 for the systems (27) and (40). The dashed vertical lines separate six regions of qualitatively different solutions, with the numbers corresponding to those in Fig. 4. Points A–F correspond to the cases in Fig. 5.

Grahic Jump Location
Fig. 7

Visualization of the nonsmooth functions (12) and (27) (black dashed) and the corresponding smooth functions (45) and (46) (solid)

Grahic Jump Location
Fig. 8

Time evolution of the speed v (left column), the corresponding control input ad (center column), and the switching variable ζ for different values of the parameter σ (right column) as indicated. The rows correspond to the points A, G, H, I, and J marked in Fig. 9. The solid curves are associated with the trajectories acquired by our BVP solver based on pseudo-arclength collocation method using the smoothed controllers (45) and (46), and the dashed curves are those generated by direct method.

Grahic Jump Location
Fig. 9

The terminal time tf (upper panel) and the fuel consumption Q(44) (lower panel) as a function of σ for v(tf)=v(0)=25 m/s. The corresponding trajectories are shown in Fig. 8 for the points marked A, G, H, I, and J.

Grahic Jump Location
Fig. 10

Effect of the speed limit penalty with different values of vlim as indicated. The same notation is used as that in Fig. 8.

Grahic Jump Location
Fig. 11

Traffic flow penalty with different values of vtraf as indicated. The same notation is used as that in Fig. 8.

Grahic Jump Location
Fig. 12

The ratio between the traffic cost and the total cost (1) as a function of the weight ρ2

Grahic Jump Location
Fig. 13

Concept of the pseudo-arclength method



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