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

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## Abstract

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.

## Keywords

Hydro turbine generating sets Shafting Transient The Lagrange relative value system The generalized Hamiltonian model## List of symbols

- \(c_{1}\)
Damping coefficients of the generator rotor

- \(c_{2}\)
Damping coefficients of the turbine runner

- \(D\)
The damping coefficient

- \(e_{1}\)
Mass eccentricity of the generator rotor

- \(e_{2}\)
Mass eccentricity of the turbine runner

- \(E_\mathrm{f}\)
Output of excitation controller

- \(E_\mathrm{q}'\)
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

- \(H\)
The Hamiltonian function

- \(J\)
Rotary inertia of the HTGS

- \(J_{1}\)
Rotary inertia of the generator rotor

- \(J_{2}\)
Rotary inertia of the turbine runner

- \(k_{1}\)
Stiffness of the upper guide bearing

- \(k_{2}\)
Stiffness of the lower guide bearing

- \(k_{3}\)
Stiffness of the hydro turbine bearing

- \(L\)
The Lagrange function

- \(m_{1}\)
Mass of the generator rotor

- \(m_{2}\)
Mass of the hydro turbine runner

- \(M_\mathrm{gB}\)
The generator rated torque

- \(M_\mathrm{g}\)
The generator magnetic torque

- \(M_\mathrm{t}\)
The hydro turbine torque

- \(p_{i}\)
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

- \(R_{1}\)
Radius of the generator rotor

- \(R_{2}\)
Radius of the hydro turbine runner

- \(r_{1}\)
Radial displacement of the generator rotor

- \(r_{2}\)
Radial displacement of the turbine runner

- \(r_{3}\)
Radial displacement of the upper guide bearing

- \(r_{4}\)
Radial displacement of lower guide bearing

- \(r_{5}\)
Radial displacement of turbine bearing

- \(S_\mathrm{gB}\)
The generator rated power

- \(T\)
Total kinetic energy of the HTGS

- \(T_{j}\)
Inertia time constant of the generator

- \(T_{j1}\)
Inertia time constant of the generator rotor

- \(T_{j2}\)
Inertia time constant of the turbine runner

- \(T_{d0}^{\prime }\)
The time constant

- \(U\)
Elastic potential energy of the HTGS

- \(U_\mathrm{s}\)
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

- \(X_{\mathrm{a}d}\)
The \(d\)-axis armature reaction reactance

- \(X_{d}\)
The \(d\)-axis synchronous reactance

- \(X_{d}'\)
The \(d\)-axis transient reactance

- \(X_{f}\)
The excitation winding reactance

- \(X_{L}\)
The transmission line reactance

- \(X_{q}\)
The \(q\)-axis synchronous reactance

- \(X_{T}\)
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

## Notes

### Acknowledgments

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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