Transactions of Tianjin University

, Volume 24, Issue 1, pp 32–44 | Cite as

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

Research article
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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.

Keywords

Hydro-turbine governing system Complex penstocks Fractional-order Nonlinearity Bifurcation 

Abbreviations

Mt

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

mt

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

ex, ey, eh

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

eqx, eqy, eqh

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

TwR, Tw1

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

hw1

Characteristic coefficient of first bifurcation penstock, p.u

δ

Rotor angle, rad

ω

Relative deviation of rotor speed, p.u

ω0

Base angular speed, rad/s

Tab

Mechanical starting time, s

me

Electromagnetic torque of generator, N. m

Pe

Electromagnetic power of generator, N. m

D

Damping coefficient, p.u

Eq′

Transient electric potential of q-axis, p.u

Eq″

Subtransient electric potential of q-axis, p.u

Tdo′

Transient open circuit time constant of d-axis, s

Ef

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

xd

Synchronous reactance of d-axis, p.u

xd′

Transient reactance of d-axis, p.u

id

Stator current of d-axis, p.u

Vs

Infinite bus voltage, p.u

xq

Synchronous reactance of q-axis, p.u

xL

Reactance of electric transmission line, p.u

xT

Short-circuit reactance of transformer, p.u

Ty

Engager relay time constant, s

u

Output signal of the governor, p.u

r

Reference input of generator speed, p.u

kp

Adjustment coefficient of proportion, p.u

ki

Adjustment coefficient of integral, s−1

kd

Adjustment coefficient of differential, s

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Notes

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).

References

  1. 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 31:746–761CrossRefGoogle Scholar
  2. 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 37:750–761CrossRefGoogle Scholar
  3. 3.
    Zhang J, Xu L, Yu B et al (2014) Environmentally feasible potential for hydropower development regarding environmental constraints. Energy Policy 73:552–562CrossRefGoogle Scholar
  4. 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 34:155–166CrossRefGoogle Scholar
  5. 5.
    Ren M, Wu D, Zhang J et al (2014) Minimum entropy-based cascade control for governing hydroelectric turbines. Entropy 16(6):3136–3148CrossRefGoogle Scholar
  6. 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):7611–7623MathSciNetCrossRefGoogle Scholar
  7. 7.
    Li C, Zhou J (2011) Parameters identification of hydraulic turbine governing system using improved gravitational search algorithm. Energy Convers Manag 52(1):374–381MathSciNetCrossRefGoogle Scholar
  8. 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):127–138CrossRefGoogle Scholar
  9. 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):12–24CrossRefGoogle Scholar
  10. 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):1921–1933CrossRefMATHGoogle Scholar
  11. 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):307–312CrossRefGoogle Scholar
  12. 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 80:414–428CrossRefGoogle Scholar
  13. 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):433–440CrossRefGoogle Scholar
  14. 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 108:478–487CrossRefGoogle Scholar
  15. 15.
    Xiong C, Chen W, Ye Z (2013) Experimental study on calculation of hydro-geological parameters for unsteady flow. Trans Tianjin Univ 19(5):351–355CrossRefGoogle Scholar
  16. 16.
    Pennacchi P, Chatterton S, Vania A (2012) Modeling of the dynamic response of a Francis turbine. Mech Syst Signal Process 29:107–119CrossRefGoogle Scholar
  17. 17.
    Nagode K, Skrjanc I (2014) Modelling and internal fuzzy model power control of a Francis water turbine. Energies 7(2):874–889CrossRefGoogle Scholar
  18. 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):1763–1770CrossRefGoogle Scholar
  19. 19.
    Inayat-Hussain JI (2010) Nonlinear dynamics of a statically misaligned flexible rotor in active magnetic bearings. Commun Nonlinear Sci Numer Simul 15(3):764–777CrossRefGoogle Scholar
  20. 20.
    Shen Z (1998) Hydraulic turbine regulation. Hydraulic and Hydroelectricity Press, Beijing (in Chinese) Google Scholar
  21. 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 90:128–137CrossRefGoogle Scholar
  22. 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 50:177–191CrossRefGoogle Scholar
  23. 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):4596–4618CrossRefGoogle Scholar
  24. 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):2073–2082CrossRefGoogle Scholar
  25. 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 305:376–404MathSciNetCrossRefGoogle Scholar
  26. 26.
    Li C, Chen Y, Kurths J (2013) Fractional calculus and its applications. Philos Trans R Soc A Math Phys Eng Sci 371(1990):20130037MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Sibatov RT, Svetukhin VV (2015) Fractional kinetics of subdiffusion-limited decomposition of a supersaturated solid solution. Chaos Solitons Fractals 81:519–526MathSciNetCrossRefMATHGoogle Scholar
  28. 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 243:611–623MathSciNetMATHGoogle Scholar
  29. 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):140CrossRefGoogle Scholar
  30. 30.
    Sheng W, Bao Y (2013) Fruit fly optimization algorithm based fractional order fuzzy-PID controller for electronic throttle. Nonlinear Dyn 73(1–2):611–619MathSciNetCrossRefGoogle Scholar
  31. 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 56:92–98CrossRefGoogle Scholar
  32. 32.
    Jumarie G (2012) Derivation of an amplitude of information in the setting of a new family of fractional entropies. Inf Sci 216:113–137MathSciNetCrossRefMATHGoogle Scholar
  33. 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):1883–1893CrossRefGoogle Scholar
  34. 34.
    Deng W, Li C, Lu J (2007) Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn 48(4):409–416MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Bhalekar S, Daftardar-Gejji V (2010) Fractional ordered Liu system with time-delay. Commun Nonlinear Sci Numer Simul 15(8):2178–2191MathSciNetCrossRefMATHGoogle Scholar
  36. 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 73:528–538CrossRefGoogle Scholar
  37. 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):2341–2355CrossRefGoogle Scholar
  38. 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):1263–1274CrossRefGoogle Scholar
  39. 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 84:390–404CrossRefGoogle Scholar
  40. 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. 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 106:127–138CrossRefGoogle Scholar
  42. 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):1383–1397MathSciNetCrossRefGoogle Scholar
  43. 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):3–22MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Tianjin University and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Institute of Water Resources and Hydropower ResearchNorthwest A&F UniversityYanglingChina
  2. 2.Department of Mathematics and Statistics, Faculty of Science and EngineeringCurtin UniversityPerthAustralia

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