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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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References

Davis, S. C. , Diegel, S. W. , and Boundy, R. G. , 2013, Transportation Energy Data Book, 32 ed., U.S. Department of Energy, Washington, DC.
U.S. Department of Transportation, 2014, “ 2012 Commodity Flow Survey United States,” Technical Report No. EC12TCF-US.
Saltsman, B. , 2014, “ Impacts of Connectivity and Automation on Vehicle Operations,” Global Symposium on Connected Vehicle and Infrastructure, University of Michigan Transportation Research Institute.
Caveney, D. , 2010, “ Cooperative Vehicular Safety Applications,” IEEE Control Syst. Mag., 30(4), pp. 38–53. [CrossRef]
Hooker, J. N. , 1988, “ Optimal Driving for Single-Vehicle Fuel Economy,” Transp. Res. Part A, 22(3), pp. 183–201. [CrossRef]
Monastyrsky, V. V. , and Golownykh, I. M. , 1993, “ Rapid Computation of Optimal Control for Vehicles,” Transp. Res. Part B, 27(3), pp. 219–227. [CrossRef]
Lattemann, F. , Neiss, K. , Terwen, S. , and Connolly, T. , 2004, “ The Predictive Cruise Control: A System to Reduce Fuel Consumption of Heavy Duty Trucks,” SAE Paper No. 2004-01-2616.
Hellström, E. , 2010, “ Look-Ahead Control of Heavy Vehicles,” Ph.D. thesis, Linköping University, Linköping, Sweden.
Mensing, F. , Bideaux, E. , Trigui, R. , and Tattegrain, H. , 2013, “ Trajectory Optimization for Eco-Driving Taking Into Account Traffic Constraints,” Transp. Res. Part D, 18, pp. 55–61. [CrossRef]
Jiménez, F. , López-Covarrubias, J. L. , Cabrera, W. , and Aparicio, F. , 2013, “ Real-Time Speed Profile Calculation for Fuel Saving Considering Unforeseen Situations and Travel Time,” IET Intell. Transp. Syst., 7(1), pp. 10–19. [CrossRef]
Jiménez, F. , and Cabrera-Montiel, W. , 2014, “ System for Road Vehicle Energy Optimization Using Real Time Road and Traffic Information,” Energies, 7(6), pp. 3576–3598. [CrossRef]
Kohut, N. , Hedrick, K. , and Borrelli, F. , 2009, “ Integrating Traffic Data and Model Predictive Control to Improve Fuel Economy,” 12th IFAC Symposium on Control in Transportation Systems, pp. 155–160.
Xu, S. , Li, S. E. , Deng, K. , Li, S. , and Cheng, B. , 2014, “ A Unified Pseudospectral Computational Framework for Optimal Control of Road Vehicles,” IEEE/ASME Trans. Mechatronics, 20(4), pp. 1–12.
Schwarzkopf, A. B. , and Leipnik, R. B. , 1977, “ Control of Highway Vehicles for Minimum Fuel Consumption Over Varying Terrain,” Transp. Res., 11(4), pp. 279–286. [CrossRef]
Chang, D. J. , and Morlok, E. K. , 2005, “ Vehicle Speed Profiles to Minimize Work and Fuel Consumption,” J. Transp. Eng., 131(3), pp. 173–182. [CrossRef]
Fröberg, A. , Hellström, E. , and Nielsen, L. , 2006, “ Explicit Fuel Optimal Speed Profiles for Heavy Trucks on a Set of Topographic Road Profiles,” SAE Technical Paper No. 2006-01-1071.
Saerens, B. , Rakha, H. , Diehl, M. , and Van den Bulck, E. , 2013, “ A Methodology for Assessing Eco-Cruise Control for Passenger Vehicles,” Transp. Res. Part D, 19, pp. 20–27. [CrossRef]
Kamal, M. A. S. , Mukai, M. , Murata, J. , and Kawabe, T. , 2011, “ Ecological Vehicle Control on Roads With Up-Down Slopes,” IEEE Trans. Intell. Transp. Syst., 12(3), pp. 783–794. [CrossRef]
Kamal, M. S. , Mukai, M. , Murata, J. , and Kawabe, T. , 2013, “ Model Predictive Control of Vehicles on Urban Roads for Improved Fuel Economy,” IEEE Trans. Control Syst. Technol., 21(3), pp. 831–841. [CrossRef]
Wang, M. , Daamen, W. , Hoogendoorn, S. P. , and van Arem, B. , 2014, “ Rolling Horizon Control Framework for Driver Assistance Systems—Part I: Mathematical Formulation and Non-Cooperative Systems,” Transp. Res. Part C, 40, pp. 271–289. [CrossRef]
Wang, M. , Daamen, W. , Hoogendoorn, S. P. , and van Arem, B. , 2014, “ Rolling Horizon Control Framework for Driver Assistance Systems—Part II: Cooperative Sensing and Cooperative Control,” Transp. Res. Part C, 40, pp. 290–311. [CrossRef]
Hartl, R. F. , Sethi, S. P. , and Vickson, R. G. , 1995, “ A Survey of the Maximum Principles for Optimal Control Problems With State Constraints,” SIAM Rev., 37(2), pp. 181–218. [CrossRef]
Vinter, R. , 2000, Optimal Control (Systems & Control: Foundations & Applications), Springer Science & Business Media, Heidelberg, Germany.
Clarke, F. , and De Pinho, M. , 2010, “ Optimal Control Problems With Mixed Constraints,” SIAM J. Control Optim., 48(7), pp. 4500–4524. [CrossRef]
Osmolovskii, N. , and Maurer, H. , 2012, Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control, Vol. DC 24, SIAM Publications, Philadelphia, PA.
Maurer, H. , Büskens, C. , Kim, J.-H. , and Kaya, Y. , 2005, “ Optimization Methods for the Verification of Second Order Sufficient Conditions for Bang-Bang Controls,” Optim. Control Appl. Methods, 26(3), pp. 129–156. [CrossRef]
Maurer, H. , and Pickenhain, S. , 1995, “ Second-Order Sufficient Conditions for Control Problems With Mixed Control-State Constraints,” J. Optim. Theory Appl., 86(3), pp. 649–667. [CrossRef]
Graichen, K. , and Petit, N. , 2008, “ Constructive Methods for Initialization and Handling Mixed State-Input Constraints in Optimal Control,” J. Guid. Control Dyn. 31(5), pp. 1334–1343. [CrossRef]
Wyczalkowski, M. , and Szeri, A. J. , 2003, “ Optimization of Acoustic Scattering From Dual-Frequency Driven Microbubbles at the Difference Frequency,” J. Acoust. Soc. Am., 113(6), pp. 3073–3079. [CrossRef] [PubMed]
Alam, A. , 2014, “ Fuel-Efficient Heavy Duty Vehicle Platooning,” Ph.D. thesis, Kungliga Tekniska Högskolan, Stockholm, Sweden.
Orosz, G. , and Shah, S. P. , 2012, “ A Nonlinear Modeling Framework for Autonomous Cruise Control,” ASME Paper No. DSCC2012-MOVIC2012-8871.
Navistar, 2011, “ Maxxforce 11 and 13 Liter Engines,” Navistar, Lisle, IL.
Heywood, J. B. , 2002, Internal Combustion Engine Fundamentals, McGraw-Hill, New York.
Ozatay, E. , Özgüner, Ü. , Onori, S. , and Rizzoni, G. , 2012, “ Analytical Solution to the Minimum Fuel Consumption Optimization Problem With the Existence of a Traffic Light,” 5th Annual Dynamic Systems and Control Conference and 11th Motion and Vibration Conference, pp. 837–846.
Guzzella, L. , and Onder, C. H. , 2004, Introduction to Modelling and Control of Internal Combustion Engine Systems, Springer, Heidelberg, Germany.
Orosz, G. , Wilson, R. E. , and Stépán, G. , 2010, “ Traffic Jams: Dynamics and Control,” Philos. Trans. R. Soc., A, 368(1928), pp. 4455–4479. [CrossRef]
Hestenes, M. R. , 1966, Calculus of Variations and Optimal Control Theory, Wiley, Hoboken, NJ.
Krener, A. J. , 1977, “ The High Order Maximal Principle and Its Application to Singular Extremals,” SIAM J. Control Optim., 15(2), pp. 256–293. [CrossRef]
Stoer, J. , Bulirsch, R. , Bartels, R. , Gautschi, W. , and Witzgall, C. , 1993, Introduction to Numerical Analysis, Vol. 2, Springer, Heidelberg, Germany.
Wächter, A. , and Biegler, L. T. , 2006, “ On the Implementation of an Interior-Point Filter Line-Search Algorithm for Large-Scale Nonlinear Programming,” Math. Program., 106(1), pp. 25–57. [CrossRef]
Gay, D. M. , and Kernighan, B. , 2002, AMPL: A Modeling Language for Mathematical Programming, second edition, Duxbury Press/Brooks/Cole, Boston, MA.
Allgower, E. E. L. , and Georg, K. , 2003, Introduction to Numerical Continuation Methods (Classics in Applied Mathematics), Vol. 45, Society for Industrial and Applied Mathematics, Philadelphia, PA.
Rao, A. V. , 2009, “ Survey of Numerical Methods for Optimal Control,” Adv. Astronaut. Sci., 135(1), pp. 497–528.
He, C. R. , and Orosz, G. , 2014, “ Fuel Consumption Optimization of Heavy-Duty Vehicles: An Analytical Approach,” ASME Paper No. DSCC2014-6362.
Roose, D. , and Szalai, R. , 2007, “ Continuation and Bifurcation Analysis of Delay Differential Equations,” Numerical Continuation Methods for Dynamical Systems, B. Krauskopf , H. Osinga , and J. Galán-Vioque , eds., Springer, Heidelberg, Germany, pp. 359–399.
Mittelmann, H. D. , 1986, “ A Pseudo-Arclength Continuation Method for Nonlinear Eigenvalue Problems,” SIAM J. Numer. Anal., 23(5), pp. 1007–1016. [CrossRef]
Jorge, N. , and Wright, S. J. , 1999, Numerical Optimization (Operations Research and Financial Engineering), Vol. 2, Springer, Heidelberg, Germany.

Figures

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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