Nonlinear Dynamics

, Volume 76, Issue 4, pp 1921–1933 | Cite as

The generalized Hamiltonian model for the shafting transient analysis of the hydro turbine generating sets

  • Yun Zeng
  • Lixiang Zhang
  • Yakun Guo
  • Jing Qian
  • Chenli Zhang
Original Paper


Traditional rotor dynamics mainly focuses on the steady- state behavior of the rotor and shafting. However, for systems such as hydro turbine generating sets (HTGS) where the control and regulation is frequently applied, the shafting safety and stabilization in transient state is then a key factor. The shafting transient state inevitably involves multiparameter domain, multifield coupling, and coupling dynamics. In this paper, the relative value form of the Lagrange function and its equations have been established by defining the base value system of the shafting. Taking the rotation angle and the angular speed of the shafting as a link, the shafting lateral vibration and generator equations are integrated into the framework of generalized Hamiltonian system. The generalized Hamiltonian control model is thus established. To make the model more general, additional forces of the shafting are taken as the input excitation in proposed model. The control system of the HTGS can be easily connected with the shafting model to form the whole simulation system of the HTGS. It is expected that this study will build a foundation for the coupling dynamics theory using the generalized Hamiltonian theory to investigate coupling dynamic mechanism among the shafting vibration, transient of hydro turbine generating sets, and additional forces of the shafting.


Hydro turbine generating sets Shafting Transient The Lagrange relative value system  The generalized Hamiltonian model 

List of symbols


Damping coefficients of the generator rotor


Damping coefficients of the turbine runner


The damping coefficient


Mass eccentricity of the generator rotor


Mass eccentricity of the turbine runner


Output of excitation controller


Internal transient voltage

\(F_{x1}\), \(F_{y1}\)

The \(x\)- and \(y\)-direction additional forces acting on the generator rotor

\(F_{x2}\), \(F_{y2}\)

The \(x\)-and \(y\)-direction additional forces acting on the hydro turbine runner


The Hamiltonian function


Rotary inertia of the HTGS


Rotary inertia of the generator rotor


Rotary inertia of the turbine runner


Stiffness of the upper guide bearing


Stiffness of the lower guide bearing


Stiffness of the hydro turbine bearing


The Lagrange function


Mass of the generator rotor


Mass of the hydro turbine runner


The generator rated torque


The generator magnetic torque


The hydro turbine torque


The generalized momentums

\(Q_{x1}\), \(Q_{y1}\)

The external forces acting on the generator rotor

\(Q_{x2}\), \(Q_{y2}\)

The external forces acting on the hydro turbine runner


Radius of the generator rotor


Radius of the hydro turbine runner


Radial displacement of the generator rotor


Radial displacement of the turbine runner


Radial displacement of the upper guide bearing


Radial displacement of lower guide bearing


Radial displacement of turbine bearing


The generator rated power


Total kinetic energy of the HTGS


Inertia time constant of the generator


Inertia time constant of the generator rotor


Inertia time constant of the turbine runner

\(T_{d0}^{\prime }\)

The time constant


Elastic potential energy of the HTGS


The infinite bus voltage

\(x_{1}\), \(y_{1}\)

Central coordinates of the generator rotor

\(x_{10}\), \(y_{10}\)

Mass coordinates of the generator rotor

\(x_{2}\), \(y_{2}\)

Central coordinates of the turbine runner

\(x_{20}\), \(y_{20}\)

Mass coordinates of the turbine runner


The \(d\)-axis armature reaction reactance


The \(d\)-axis synchronous reactance


The \(d\)-axis transient reactance


The excitation winding reactance


The transmission line reactance


The \(q\)-axis synchronous reactance


Reactance of transformer

\(\delta \)

Rotor angle

\(\varphi \)

Rotation angle of the generator rotor

\(\omega \)

Angular speed of the HTGS

\(\omega _\mathrm{B}\)

Basic value of electrical angular speed

\(\omega _\mathrm{e}\)

Electric angular speed

\(\omega _\mathrm{mB}\)

Basic value of mechanical angular speed



The research reported here is financially supported by the National Natural Science Foundation of China under Grant Nos. 51179079 and 50839003 and part works is financially supported by the Natural Science Foundation of Yunnan Province No. 2013FZ015. The comments made by anonymous reviewers have significantly improved the final version of this paper.


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Yun Zeng
    • 1
  • Lixiang Zhang
    • 1
  • Yakun Guo
    • 2
  • Jing Qian
    • 3
  • Chenli Zhang
    • 4
  1. 1.Department of Engineering MechanicsKunming University of Science and TechnologyKunming China
  2. 2.School of EngineeringUniversity of AberdeenAberdeen UK
  3. 3.College of Electric Power EngineeringKunming University of Science and TechnologyKunming China
  4. 4.Faculty of Metallurgical and EnergyKunming University of Science and TechnologyKunming China

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