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Nonlinear modeling and dynamic control of hydro-turbine governing system with upstream surge tank and sloping ceiling tailrace tunnel

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Abstract

Using the Hopf bifurcation theory, the nonlinear dynamic characteristics of hydro-turbine governing system of hydropower station with upstream surge tank and sloping ceiling tailrace tunnel are studied. Firstly, a novel and rational nonlinear mathematical model of the hydro-turbine governing system is proposed. This model contains the nonlinear dynamic equation of pipeline system, which can accurately describe the motion characteristics of the interface of free surface-pressurized flow in sloping ceiling tailrace tunnel. According to the nonlinear mathematical model, the existence and direction of Hopf bifurcation of the nonlinear dynamic system are analyzed. Furthermore, the algebraic criterion of the occurrence of Hopf bifurcation is derived. Then the stability domain and bifurcation diagram of hydro-turbine governing system are drawn by the algebraic criterion. Finally, the dynamic characteristics under different state parameters are investigated and the dynamic control method is proposed. The results indicate that: For the example in this paper, the Hopf bifurcation of hydro-turbine governing system is supercritical. The phase space trajectories of characteristic variables stabilize at the equilibrium points and stable limit cycles when the system state parameter point locates in the stable domain and unstable domain, respectively. The dynamic response processes of the characteristic variables of the hydro-turbine governing system under load disturbance show an obvious feature of wave superposition. To make the dynamic response process of the hydro-turbine governing system with upstream surge tank and sloping ceiling tailrace tunnel more stable and attenuate faster, the governor parameters \((K_{p}, K_{i})\) should locate in the stable domain and keep away from the bifurcation point as much as possible.

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References

  1. Chaudry, M.H.: Applied Hydraulic Transients. Van Nostrand, New York (2014)

    Book  Google Scholar 

  2. Liu, Q.Z., Peng, S.Z.: Surge Tank of Hydropower Station. China Waterpower Press, Beijing (1995)

    Google Scholar 

  3. Wu, R.Q., Chen, J.Z.: Hydraulic Transients of Hydropower Station. China Water Power Press, Beijing (1997)

    Google Scholar 

  4. Guo, W.C., Yang, J.D., Chen, J.P., Teng, Y.: Study on the stability of waterpower-speed control system for hydropower station with air cushion surge chamber. In: 27th IAHR Symposium on Hydraulic Machinery and Systems, Montreal (2014)

  5. Yang, J.D., Chen, J.Z., Lai, X.: Study on the stability of water level fluctuation for upstream and downstream surge chambers system. J. Hydraul. Eng. 7, 50–56 (1993)

    Google Scholar 

  6. Chen, J.P., Yang, J.D., Guo, W.C., Teng, Y.: Study on the stability of waterpower-speed control system for hydropower station with upstream and downstream surge chambers based on regulation modes. In: 27th IAHR Symposium on Hydraulic Machinery and Systems, Montreal (2014)

  7. Krivehenko, G.I., Kvyatkovskaya, E.V., Vasilev, A.B., Vladimirov, V.B.: New design of tailrace conduit of hydropower plant. Hydrotech. Construct. 19, 352–357 (1985)

    Article  Google Scholar 

  8. ASCE: Guide Book for Civil Engineering in the Planning and Design of Hydropower Engineering, Vol. II (1990)

  9. Yang, J.D., Chen, J.Z., Chen, W.B., Li, S.X.: Study on the configuration of hydropower station tailrace tunnel with sloping ceiling. J. Hydraul. Eng. 3, 9–12 (1998)

    Google Scholar 

  10. Lei, Y., Yang, J.D., Lai, X.: Experimental study on the hydraulic operating characteristic of hydropower plant tailrace tunnels with sloped top. J. Wuhan Univ. Hydraul. Electric Eng. 32, 23–27 (1999)

    Google Scholar 

  11. Cheng, Y.G., Yang, J.D., Zhang, S.H., Chen, J.Z.: Hydraulic transients in hydropower plant tailrace tunnels with sloped top. J. Hydrodyn. 3, 1–6 (1998)

    Google Scholar 

  12. Li, J.P., Yang, J.D.: Flow regime analysis of free surface-pressurized mixed flow in tailrace tunnel with sloping ceiling. Water Resourc. Power 28, 86–87 (2010)

    Google Scholar 

  13. Miu, M.F., Zhang, Y.L.: Approximate formula for vacuum degree at the inlet of a draft tube in a tailrace with inclined ceiling. J. Hydroelectr. Eng. 30, 130–135 (2011)

    Google Scholar 

  14. Lai, X., Chen, J.Z., Yang, J.D.: Stability analysis of hydropower station with inclined ceiling tailrace. J. Hydroelectr. Eng. 4, 102–107 (2001)

    Google Scholar 

  15. Zhou, J.X., Zhang, J., Liu, D.Y.: Study on small fluctuation in system of two units and common tail tunnel with sloping ceiling. Water Resourc. Hydropower Eng. 35, 64–67 (2004)

    Google Scholar 

  16. Li, J.P., Yang, J.D.: Effect of tail water level fluctuation on stable operation of units in hydropower station. Eng. J. Wuhan Univ. 37, 28–32 (2004)

    Google Scholar 

  17. Wang, M.J., Yang, J.D., Wang, H.: Analysis of stability of tailrace open channel in small fluctuation governing system. J. Hydroelectr. Eng. 34, 161–168 (2015)

    Google Scholar 

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

    Article  Google Scholar 

  19. Agüero, J.L., Barbieri, M.B., Beroqui, M.C., Lastra, R.E., Mastronardi, J., Molina, R.: Hydraulic transients in hydropower plant impact on power system dynamic stability. In: Power and Energy Society General Meeting - Conversion and Delivery of Electrical Energy in the 21st Century, 2008 IEEE, Pittsburgh (2008)

  20. Ling, D.J., Shen, Z.Y.: The nonlinear model of hydraulic turbine governing systems and its PID control and Hopf bifurcation. Proc. CSEE 25, 97–102 (2005)

    Google Scholar 

  21. Ling, D.J., Tao, Y.: An analysis of the Hopf bifurcation in hydro-turbine governing system with saturation. IEEE Trans. Energy Convers. 21, 512–515 (2006)

    Article  Google Scholar 

  22. Fang, H.Q., Chen, L., Dlakavu, N., Shen, Z.Y.: Basic modeling and simulation tool for analysis of hydraulic transients in hydroelectric power plants. IEEE Trans. Energy Convers. 23, 834–841 (2008)

    Article  Google Scholar 

  23. Xu, B.B., Chen, D.Y., Zhang, H., Zhou, R.: Dynamic analysis and modeling of a novel fractional-order hydro-turbine-generator unit. Nonlinear Dyn. 79, 2066–2070 (2015)

    Google Scholar 

  24. Zeng, Y., Guo, Y.K., Zhang, L.X., Xu, T.M., Dong, H.K.: Nonlinear hydro turbine model having a surge tank. Math. Comput. Model. Dyn. 19, 12–28 (2013)

    Article  MATH  Google Scholar 

  25. Chen, D.Y., Ding, C., Do, Y.H., Ma, X.Y., Zhao, H., Wang, Y.C.: Nonlinear dynamic analysis for a Francis hydro-turbine governing system and its control. J. Franklin Inst. 351, 4596–4618 (2014)

    Article  Google Scholar 

  26. Chen, D.Y., Ding, C., Ma, X.Y., Yuan, P., Ba, D.D.: Nonlinear dynamical analysis of hydro-turbine governing system with a surge tank. Appl. Math. Model. 37, 7611–7623 (2013)

    Article  MathSciNet  Google Scholar 

  27. Huang, C.D., Cao, J.D.: Hopf bifurcation in an n-dimensional Goodwin model via multiple delays feedback. Nonlinear Dyn. 79, 2541–2552 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Cheng, Z., Cao, J.: Bifurcation and stability analysis of neural network model with distributed delays. Nonlinear Dyn. 46, 363–373 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, R., Xiao, D.: Bifurcations and chaotic dynamics in a 4-dimensional competitive Lotka–Volterra system. Nonlinear Dyn. 59, 411–422 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Li, J.Y., Chen, Q.J.: Nonlinear dynamical analysis of hydraulic turbine governing systems with nonelastic water hammer effect. J. Appl. Math. 2014, 1–11 (2014)

    Google Scholar 

  31. Li, F., Jin, Y.: Hopf bifurcation analysis and numerical simulation in a 4D-hyperchaotic system. Nonlinear Dyn. 67, 2857–2864 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Moradi, H., Alasty, A., Vossoughi, G.: Nonlinear dynamics and control of bifurcation to regulate the performance of a boiler-turbine unit. Energy Convers. Manag. 68, 105–113 (2013)

    Article  Google Scholar 

  33. Wei, Z.C., Yu, P., Zhang, W., Yao, M.H.: Study of hidden attractors, multiple limit cycles from Hopf bifurcation and boundedness of motion in the generalized hyperchaotic Rabinovich system. Nonlinear Dyn. (2015). doi:10.1007/s11071-015-2144-8

  34. Zheng, Y., Wang, Z.: Stability and Hopf bifurcations of an optoelectronic time-delay feedback system. Nonlinear Dyn. 57, 125–134 (2009)

    Article  MATH  Google Scholar 

  35. Xiao, M., Daniel, W., Cao, J.: Time-delayed feedback control of dynamical small-world networks at Hopf bifurcation. Nonlinear Dyn. 58, 319–344 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Luongo, A., Paolone, A., Di Egidio, A.: Multiple timescales analysis for 1:2 and 1:3 resonant Hopf bifurcations. Nonlinear Dyn. 34, 269–291 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhang, L., Zhang, Z., Huang, L.: Double Hopf bifurcation of time-delayed feedback control for maglev system. Nonlinear Dyn. 69, 961–967 (2012)

    Article  MathSciNet  Google Scholar 

  38. Liao, X.F., Xie, T.T.: Local stability and Hopf bifurcation of two-dimensional nonlinear descriptor system. Nonlinear Dyn. (2015). doi:10.1007/s11071-015-2164-4

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

    Article  Google Scholar 

  40. Jiang, C.W., Ma, Y.C., Wang, C.M.: PID controller parameters optimization of hydro-turbine governing systems using deterministic-chaotic-mutation evolutionary programming. Energy Convers. Manag. 47, 1222–1230 (2006)

    Article  Google Scholar 

  41. Chen, Z.H., Yuan, X.H., Ji, B., Wang, P.T., Tian, H.: 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 (2014)

    Article  Google Scholar 

  42. Kishor, N., Singh, S.P., Raghuvanshi, A.S.: Dynamic simulations of hydro turbine and its state estimation based LQ control. Energy Convers. Manag. 47, 3119–3137 (2006)

    Article  Google Scholar 

  43. Wang, W., Zeng, D.L., Liu, J.Z., Niu, Y.G., Cui, C.: Feasibility analysis of changing turbine load in power plants using continuous condenser pressure adjustment. Energy 64, 533–540 (2014)

    Article  Google Scholar 

  44. Chen, Z.H., Yuan, X.H., Tian, H., Ji, B.: Improved gravitational search algorithm for parameter identification of water turbine regulation system. Energy Convers. Manag. 78, 306–315 (2014)

    Article  Google Scholar 

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

    Article  Google Scholar 

  46. Kishor, N.: Nonlinear predictive control to track deviated power of an identified NNARX model of a hydro plant. Expert Syst. Appl. 35, 1741–1751 (2008)

    Article  Google Scholar 

  47. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42. Springer Science & Business Media, New York (1983)

    Book  MATH  Google Scholar 

  48. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, Hoboken (2008)

    MATH  Google Scholar 

  49. Luongo, A., Paolone, A.: Perturbation methods for bifurcation analysis from multiple nonresonant complex eigenvalues. Nonlinear Dyn. 14(3), 193–210 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  50. Parker, T.S., Chua, L.O.: Practical Numerical Algorithms for Chaotic Systems. Springer-Verlag World Publishing Corp, New York (1989)

  51. Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, London (1981)

    MATH  Google Scholar 

  52. Li, J.B., Feng, B.Y.: Stability, Bifurcations and Chaos. Yunnan Science and Technology Press, Kunming (1995)

    Google Scholar 

  53. Nguyen, L.H., Hong, K.S.: Hopf bifurcation control via a dynamic state-feedback control. Phys. Lett. A 376, 442–446 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  54. Zhang, R., Wang, Y., Zhang, Z.D., Bi, Q.S.: Nonlinear behaviors as well as the bifurcation mechanism in switched dynamical systems. Nonlinear Dyn. 79, 465–471 (2015)

    Article  Google Scholar 

  55. Peng, Z.W., Hu, G.G., Han, Z.X.: Power System Voltage Stability Analysis Based on Bifurcation Theory. China Electric Power Press, Beijing (2005)

    Google Scholar 

  56. Srivastava, K.N., Srivastava, S.C.: Application of Hopf bifurcation theory for determining critical value of a generator control or load parameter. Electr. Power Energy Syst. 17, 347–354 (1995)

    Article  Google Scholar 

  57. Wang, H.O., Abed, E.H., Hamdan, A.M.A.: Bifurcations, chaos, and crises in voltage collapse of a model power system. IEEE Trans. Circuits I(41), 294–302 (1994)

  58. Alvarez, J., Curiel, L.E.: Bifurcations and chaos in a linear control system with saturated input. Int. J. Bifurc. Chaos 7, 1811–1822 (1997)

    Article  MATH  Google Scholar 

  59. Cui, F.S., Chew, C.H.: Bifurcation and chaos in the duffing oscillator with a PID controller. Nonlinear Dyn. 12, 251–262 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  60. Guo, W.C., Yang, J.D., Chen, J.P., Teng, Y.: Effect mechanism of penstock on stability and regulation quality of turbine regulating system. Math. Probl. Eng. 2014, 1–13 (2014)

    Google Scholar 

  61. Guo, W.C., Yang, J.D., Yang, W.J., Chen, J.P., Teng, Y.: Regulation quality for frequency response of turbine regulating system of isolated hydroelectric power plant with surge tank. Int. J. Electr. Power 73, 528–538 (2015)

    Article  Google Scholar 

  62. IEEE Working Group: Hydraulic turbine and turbine control model for system dynamic studies. IEEE Trans. Power Syst. 7, 167–179 (1992)

    Google Scholar 

  63. Li, J.P.: Study of model test and numerical simulation on the tailrace tunnel with sloping ceiling in the underground hydropower station. Doctor Thesis in Engineering. Wuhan University, Wuhan (2005)

  64. Wei, S.P.: Hydraulic Turbine Regulation. Huazhong University of Science and Technology Press, Wuhan (2009)

    Google Scholar 

  65. Guo, W.C., Yang, J.D., Yang, W.J.: Comparative study on stability of three hydraulic turbine regulation modes. J. Hydroelectr. Eng. 33, 255–262 (2014)

    Google Scholar 

  66. Guo, W.C., Yang, J.D., Chen, J.P., Yang, W.J., Teng, Y., Zeng, W.: 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), 2340–2354 (2015)

    Article  Google Scholar 

  67. Guo, W.C., Yang, J.D., Chen, J.P.: Research on critical stable sectional area of surge chamber considering the fluid inertia in the penstock and characteristics of governor. In: Proceedings of the ASME 26th symposium on fluid machinery, Chicago (2014)

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (Project no. 51379158) and the China Scholarship Council (CSC).

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Authors and Affiliations

  1. State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, 430072, China

    Wencheng Guo, Jiandong Yang, Jieping Chen & Mingjiang Wang

  2. HYDROCHINA Northwest Engineering Corporation, Xi’an, 710065, China

    Mingjiang Wang

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  1. Wencheng Guo
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  2. Jiandong Yang
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Correspondence to Wencheng Guo.

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Guo, W., Yang, J., Chen, J. et al. Nonlinear modeling and dynamic control of hydro-turbine governing system with upstream surge tank and sloping ceiling tailrace tunnel. Nonlinear Dyn 84, 1383–1397 (2016). https://doi.org/10.1007/s11071-015-2577-0

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