Skip to main content

Advertisement

SpringerLink
Log in

Fractional-Order Modeling and Dynamical Analysis of a Francis Hydro-Turbine Governing System with Complex Penstocks

  • Research article
  • Published:
Transactions of Tianjin University Aims and scope Submit manuscript

Abstract

This paper investigates the stability of the Francis hydro-turbine governing system with complex penstocks in the grid-connected mode. Firstly, a novel fractional-order nonlinear mathematical model of a Francis hydro-turbine governing system with complex penstocks is built from an engineering application perspective. This model is described by state-space equations and is composed of the Francis hydro-turbine model, the fractional-order complex penstocks model, the third-order generator model, and the hydraulic speed governing system model. Based on stability theory for a fractional-order nonlinear system, this study discovers a basic law of the bifurcation points of the above system with a change in the fractional-order α. Secondly, the stable region of the governing system is investigated in detail, and nonlinear dynamical behaviors of the system are identified and studied exhaustively via bifurcation diagrams, time waveforms, phase orbits, Poincare maps, power spectrums and spectrograms. Results of these numerical experiments provide a theoretical reference for further studies of the stability of hydropower stations.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Abbreviations

M t :

Mechanical torque of hydro-turbine, N. m

Q :

Discharge of hydro-turbine, m3/s

H :

Head of hydro-turbine, m

n :

Hydro-turbine/rotor speed, rad/s

a :

Guide vane opening, p.u

m t :

Relative deviation of mechanical torque, p.u

q :

Relative deviation of discharge, p.u

h :

Relative deviation of hydro-turbine head, p.u

x :

Relative deviation of rotor speed, p.u

y :

Relative deviation of guide vane opening, p.u

e x , e y , e h :

Partial derivatives of mechanical torque of hydro-turbine with respect to hydro-turbine speed, guide vane opening, and hydro-turbine head, p.u

e qx , e qy , e qh :

Partial derivatives of hydro-turbine discharge with respect to hydro-turbine speed, guide vane opening, and hydro-turbine head, p.u

T wR , T w1 :

Water inertia time constants of main penstock and first bifurcation penstock, s

h w1 :

Characteristic coefficient of first bifurcation penstock, p.u

δ :

Rotor angle, rad

ω :

Relative deviation of rotor speed, p.u

ω 0 :

Base angular speed, rad/s

T ab :

Mechanical starting time, s

m e :

Electromagnetic torque of generator, N. m

P e :

Electromagnetic power of generator, N. m

D :

Damping coefficient, p.u

E ′ q :

Transient electric potential of q-axis, p.u

E ″ q :

Subtransient electric potential of q-axis, p.u

T ′ do :

Transient open circuit time constant of d-axis, s

E f :

Hypothetical no-load electric potential, which depends on field voltage, p.u

x d :

Synchronous reactance of d-axis, p.u

x ′ d :

Transient reactance of d-axis, p.u

i d :

Stator current of d-axis, p.u

V s :

Infinite bus voltage, p.u

x q :

Synchronous reactance of q-axis, p.u

x L :

Reactance of electric transmission line, p.u

x T :

Short-circuit reactance of transformer, p.u

T y :

Engager relay time constant, s

u :

Output signal of the governor, p.u

r :

Reference input of generator speed, p.u

k p :

Adjustment coefficient of proportion, p.u

k i :

Adjustment coefficient of integral, s−1

k d :

Adjustment coefficient of differential, s

References

  1. Ardizzon G, Cavazzini G, Pavesi G (2014) A new generation of small hydro and pumped-hydro power plants: advances and future challenges. Renew Sustain Energy Rev 31:746–761

    Article  Google Scholar 

  2. Westin FF, dos Santos MA, Martins ID (2014) Hydropower expansion and analysis of the use of strategic and integrated environmental assessment tools in Brazil. Renew Sustain Energy Rev 37:750–761

    Article  Google Scholar 

  3. Zhang J, Xu L, Yu B et al (2014) Environmentally feasible potential for hydropower development regarding environmental constraints. Energy Policy 73:552–562

    Article  Google Scholar 

  4. Hoffken JI (2014) A closer look at small hydropower projects in India: social acceptability of two storage-based projects in Karnataka. Renew Sustain Energy Rev 34:155–166

    Article  Google Scholar 

  5. Ren M, Wu D, Zhang J et al (2014) Minimum entropy-based cascade control for governing hydroelectric turbines. Entropy 16(6):3136–3148

    Article  Google Scholar 

  6. Chen D, Ding C, Ma X et al (2013) Nonlinear dynamical analysis of hydro-turbine governing system with a surge tank. Appl Math Model 37(14–15):7611–7623

    Article  MathSciNet  Google Scholar 

  7. Li C, Zhou J (2011) Parameters identification of hydraulic turbine governing system using improved gravitational search algorithm. Energy Convers Manag 52(1):374–381

    Article  MathSciNet  Google Scholar 

  8. Karimi M, Mohamad H, Mokhlis H et al (2012) Under-frequency load shedding scheme for islanded distribution network connected with mini hydro. Int J Electr Power Energy Syst 42(1):127–138

    Article  Google Scholar 

  9. Xu B, Chen D, Zhang H et al (2015) The modeling of the fractional-order shafting system for a water jet mixed-flow pump during the startup process. Commun Nonlinear Sci Numer Simul 29(1–3):12–24

    Article  Google Scholar 

  10. Zeng Y, Zhang L, Guo Y et al (2014) The generalized Hamiltonian model for the shafting transient analysis of the hydro turbine generating sets. Nonlinear Dyn 76(4):1921–1933

    Article  MATH  Google Scholar 

  11. Tian Z, Zhang Y, Ma Z et al (2008) Effect of concrete cracks on dynamic characteristics of powerhouse for giant-scale hydrostation. Trans Tianjin Univ 14(4):307–312

    Article  Google Scholar 

  12. Li H, Chen D, Zhang H et al (2016) Nonlinear modeling and dynamic analysis of a hydro-turbine governing system in the process of sudden load increase transient. Mech Syst Signal Process 80:414–428

    Article  Google Scholar 

  13. Ma Z, Zhang C (2010) Static and dynamic damage analysis of mass concrete in hydropower house of three gorges project. Trans Tianjin Univ 16(6):433–440

    Article  Google Scholar 

  14. Xu B, Wang F, Chen D et al (2016) Hamiltonian modeling of multi-hydro-turbine governing systems with sharing common penstock and dynamic analyses under shock load. Energy Convers Manag 108:478–487

    Article  Google Scholar 

  15. Xiong C, Chen W, Ye Z (2013) Experimental study on calculation of hydro-geological parameters for unsteady flow. Trans Tianjin Univ 19(5):351–355

    Article  Google Scholar 

  16. Pennacchi P, Chatterton S, Vania A (2012) Modeling of the dynamic response of a Francis turbine. Mech Syst Signal Process 29:107–119

    Article  Google Scholar 

  17. Nagode K, Skrjanc I (2014) Modelling and internal fuzzy model power control of a Francis water turbine. Energies 7(2):874–889

    Article  Google Scholar 

  18. Fang H, Chen L, Shen Z (2011) Application of an improved PSO algorithm to optimal tuning of PID gains for water turbine governor. Energy Convers Manag 52(4):1763–1770

    Article  Google Scholar 

  19. Inayat-Hussain JI (2010) Nonlinear dynamics of a statically misaligned flexible rotor in active magnetic bearings. Commun Nonlinear Sci Numer Simul 15(3):764–777

    Article  Google Scholar 

  20. Shen Z (1998) Hydraulic turbine regulation. Hydraulic and Hydroelectricity Press, Beijing (in Chinese)

    Google Scholar 

  21. Zhang H, Chen D, Xu B et al (2015) Nonlinear modeling and dynamic analysis of hydro-turbine governing system in the process of load rejection transient. Energy Convers Manag 90:128–137

    Article  Google Scholar 

  22. Chaoshun Li, Li Chang, Zhengjun Huang et al (2016) Parameter identification of a nonlinear model of hydraulic turbine governing system with an elastic water hammer based on a modified gravitational search algorithm. Eng Appl Artif Intell 50:177–191

    Article  Google Scholar 

  23. Chen D, Ding C, Do Y et al (2014) Nonlinear dynamic analysis for a Francis hydro-turbine governing system and its control. J Franklin Inst 351(9):4596–4618

    Article  Google Scholar 

  24. Li C, Zhou J, Xiao J et al (2013) Hydraulic turbine governing system identification using T-S fuzzy model optimized by chaotic gravitational search algorithm. Eng Appl Artif Intell 26(9):2073–2082

    Article  Google Scholar 

  25. Song F, XU C, Karniadakis GE (2016) A fractional phase-field model for two-phase flows with tunable sharpness: algorithms and simulations. Comput Methods Appl Mech Eng 305:376–404

    Article  MathSciNet  Google Scholar 

  26. Li C, Chen Y, Kurths J (2013) Fractional calculus and its applications. Philos Trans R Soc A Math Phys Eng Sci 371(1990):20130037

    Article  MathSciNet  MATH  Google Scholar 

  27. Sibatov RT, Svetukhin VV (2015) Fractional kinetics of subdiffusion-limited decomposition of a supersaturated solid solution. Chaos Solitons Fractals 81:519–526

    Article  MathSciNet  MATH  Google Scholar 

  28. Atanackovic TM, Janev M, Pilipovic S et al (2014) Convergence analysis of a numerical scheme for two classes of non-linear fractional differential equations. Appl Math Comput 243:611–623

    MathSciNet  MATH  Google Scholar 

  29. Maqbool K, Beg OA, Sohail A et al (2016) Analytical solutions for wall slip effects on magnetohydrodynamic oscillatory rotating plate and channel flows in porous media using a fractional Burgers viscoelastic model. Eur Phys J Plus 131(5):140

    Article  Google Scholar 

  30. Sheng W, Bao Y (2013) Fruit fly optimization algorithm based fractional order fuzzy-PID controller for electronic throttle. Nonlinear Dyn 73(1–2):611–619

    Article  MathSciNet  Google Scholar 

  31. Lu X, Wei C, Liu L et al (2014) Experimental study of the fractional fourier transform for a hollow Gaussian beam. Opt Laser Technol 56:92–98

    Article  Google Scholar 

  32. Jumarie G (2012) Derivation of an amplitude of information in the setting of a new family of fractional entropies. Inf Sci 216:113–137

    Article  MathSciNet  MATH  Google Scholar 

  33. Aghababa MP, Haghighi AR, Roohi M (2015) Stabilisation of unknown fractional-order chaotic systems: an adaptive switching control strategy with application to power systems. IET Gener Transm Distrib 9(14):1883–1893

    Article  Google Scholar 

  34. Deng W, Li C, Lu J (2007) Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn 48(4):409–416

    Article  MathSciNet  MATH  Google Scholar 

  35. Bhalekar S, Daftardar-Gejji V (2010) Fractional ordered Liu system with time-delay. Commun Nonlinear Sci Numer Simul 15(8):2178–2191

    Article  MathSciNet  MATH  Google Scholar 

  36. Guo W, Yang J, Yang W et al (2015) Regulation quality for frequency response of turbine regulating system of isolated hydroelectric power plant with surge tank. Int J Electr Power Energy Syst 73:528–538

    Article  Google Scholar 

  37. Guo W, Yang J, Chen J et al (2015) Time response of the frequency of hydroelectric generator unit with surge tank under isolated operation based on turbine regulating modes. Electr Power Compon Syst 43(20):2341–2355

    Article  Google Scholar 

  38. Xu B, Chen D, Zhang H et al (2015) Dynamic analysis and modeling of a novel fractional-order hydro-turbine-generator unit. Nonlinear Dyn 81(3):1263–1274

    Article  Google Scholar 

  39. Chen Z, Yuan X, Ji B et al (2014) Design of a fractional order PID controller for hydraulic turbine regulating system using chaotic non-dominated sorting genetic algorithm II. Energy Convers Manag 84:390–404

    Article  Google Scholar 

  40. Jin ZO, Xu LA, Xin ZO (2013) Effects of generator electromagnetic process on transient process of hydropower station with isolated load. Eng J Wuhan Univ 46(1):109–112 (in Chinese)

    Google Scholar 

  41. Guo W, Yang J, Wang M et al (2015) Nonlinear modeling and stability analysis of hydro-turbine governing system with sloping ceiling tailrace tunnel under load disturbance. Energy Convers Manag 106:127–138

    Article  Google Scholar 

  42. Guo W, Yang J, Chen J et al (2016) Nonlinear modeling and dynamic control of hydro-turbine governing system with upstream surge tank and sloping ceiling tailrace tunnel. Nonlinear Dyn 84(3):1383–1397

    Article  MathSciNet  Google Scholar 

  43. Diethelm K, Ford NJ, Freed AD (2002) A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 29(1–4):3–22

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This study was supported by the Scientific Research Foundation of the National Natural Science Foundation–Outstanding Youth Foundation (No. 51622906); National Natural Science Foundation of China (No. 51479173); Fundamental Research Funds for the Central Universities (201304030577); Scientific Research Funds of Northwest A&F University (2013BSJJ095); the Scientific Research Foundation for Water Engineering in Shaanxi Province (2013slkj-12); the Science Fund for Excellent Young Scholars from Northwest A&F University (Z109021515); and the Shaanxi Nova Program (2016KJXX-55).

Author information

Authors and Affiliations

  1. Institute of Water Resources and Hydropower Research, Northwest A&F University, Yangling, 712100, China

    Feifei Wang, Diyi Chen, Beibei Xu & Hao Zhang

  2. Department of Mathematics and Statistics, Faculty of Science and Engineering, Curtin University, Perth, WA, 6845, Australia

    Diyi Chen

Authors
  1. Feifei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Diyi Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Beibei Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hao Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Diyi Chen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, F., Chen, D., Xu, B. et al. Fractional-Order Modeling and Dynamical Analysis of a Francis Hydro-Turbine Governing System with Complex Penstocks. Trans. Tianjin Univ. 24, 32–44 (2018). https://doi.org/10.1007/s12209-017-0093-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12209-017-0093-7

Keywords

Search

Navigation