Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Conservation law-based air mass flow calculation in engine intake systems



In current engine controls, a number of control methods are based on the air charge estimation in engine intake systems. Since the derivative of the air mass flow through the throttle valve goes to infinity when the intake pressure is close to the upper stream pressure, the relatively large numerical error or oscillation occurs near the singularity point when using common algorithms. This paper develops an effective algorithm for calculating the air mass flow in engine intake systems. Utilizing the high-level model description (HLMD), the system is described by mass and energy conservation laws and therefor the singularity issue at the zero pressuredifference point is transformed into a singularity issue at the corresponding energy point. Then, the implicit midpoint rule, a special symplectic discrete method, is selected to integrate the energy and mass conservation system. The simulation results show that the numerical behaviour of the air mass flow is significantly improved at the singularity point by using the proposed algorithm. The experimental results also verify that the qualitative behaviour of the air mass flow calculated by the proposed algorithm is consistent with the actual physical system.


本文提出了一种有效的发动机进气系统进气流量计算方法。基于HLM方法, 应用能量和质量守恒定律建立进气系统模型, 从而将零压差点的奇异问题转换成相应能量点的奇异问题。提出了一种隐式的离散求解方法, 消除了数字仿真技术带来的奇异点震荡问题。仿真结果表明, 应用此算法进气流量的数值行为在奇异点得到有效改善, 实验结果也验证了由此算法计算的进气流量的定性行为与实际的物理系统相一致。

This is a preview of subscription content, log in to check access.


  1. 1

    Saraswati S, Agarwal P K, Chand S. Neural networks and fuzzy logic-based spark advance control of SI engines. Expert Syst Appl, 2011, 38: 6916–6925

  2. 2

    Yang J, Shen T, Jiao X. Model-based stochastic optimal air-fuel ratio control with residual gas fraction of spark ignition engines. IEEE Trans Contr Syst Technol, 2014, 22: 896–910

  3. 3

    Xie H, Song K, He Y. A hybrid disturbance rejection control solution for variable valve timing system of gasoline engines. ISA Trans, 2014, 53: 889–898

  4. 4

    Wu Y, Kumar M, Shen T. A stochastic logical system approach to model and optimal control of cyclic variation of residual gas fraction in combustion engines. Appl Therm Eng, 2016, 93: 251–259

  5. 5

    Guzzella L, Onder C. Introduction to Modeling and Control of Internal Combustion Engine Systems. Berlin: Springer Science & Business Media, 2009. 30–39

  6. 6

    Bowns D E, Tomlinson S P, Dorey R E. Computer simulation techniques for the dynamic performance assessment of fuild power system. In: Proceedings of the 7th International Fluid Power Symposium, Bath, 1986. 81–88

  7. 7

    Krus P. The simulation of fluid power system with complex load dynamics. Int J Model Simul, 1986, 6: 52–57

  8. 8

    Ellman A, Vilenius M J. Methods for Simulating Steady-State and Dynamic Behavior of Two-Way Cartridge Valve Circuits. SAE Technical Paper 901584. 1990

  9. 9

    Ellman A, Piché R. A two regime orifice flow formula for numerical simulation. J Dyn Sys Meas Control, 1999, 121: 721–724

  10. 10

    Borutzky W, Barnard B, Thoma J. An orifice flow model for laminar and turbulent conditions. Simul Model Pract Theory, 2002, 10: 141–152

  11. 11

    Hairer E, Lubich C, Wanner G. Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations. Berlin: Springer Science & Business Media, 2006. 204–211

  12. 12

    Ito H, Yohata H, Kako J, et al. Development of High Level Modeling Method for Rapid Modeling Process. SAE Technical Paper 2013-01-0244. 2013

  13. 13

    Kako J, Sata K, Ohata A, et al. Effect of transient residual gas fraction for gasoline engines. In: Proceedings of the 32nd Chinese Control Conference (CCC), Xi’an, 2013. 7762–7767

  14. 14

    Edelberg K, Hedrick J K. A high level approach to mean value modeling of an automotive engine during cold-start. In: Proceedings of 2014 American Control Conference (ACC), Portland, 2014. 3165–3170

  15. 15

    Ohata A. Physical turbocharger model on high level model descrition (HLMD). In: Proceedings of the 7th IFAC Symposium on Advances in Automotive Control, Tokyo, 2013. 289–294

  16. 16

    Eriksson L. Modeling and control of turbocharged SI and DI engines. Oil Gas Sci Technol, 2007, 62: 523–538

  17. 17

    Marsden J E, West M. Discrete mechanics and variational integrators. Acta Numer, 2001, 10: 357–514

  18. 18

    Chyba M, Hairer E, Vilmart G. The role of symplectic integrators in optimal control. Optim Contr Appl Methods, 2009, 30: 367–382

Download references

Author information

Correspondence to Tielong Shen.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fan, J., Wu, Y., Ohata, A. et al. Conservation law-based air mass flow calculation in engine intake systems. Sci. China Inf. Sci. 59, 112210 (2016).

Download citation


  • air mass flow
  • engine intake system
  • conservation law
  • singularity
  • numerical method


  • 进气流量
  • 发动机进气系统
  • 守恒律
  • 奇异
  • 数值方法