Analysis of multimode oscillations caused by subsynchronous resonance on generator shaft

Over the past century, power systems have grown in a tremendous way which has also looking after about several new problems. One of them is on shaft systems of turbine generator. Capacitors interacts with long transmission lines and causes damage to the shaft which connects turbine with generator. By decreasing the effective reactance, it increases stability of the system and improves the load carrying capability of multiphase power transmission lines. By controlling the line reactance it gives the user a better control over load sharing between parallel transmission lines. Subsynchronous resonance phenomena can occur due to induction generator effect, torsional interaction and transient torque (Padiyar, 2002; Childs and Scharrer, 1988). The first two will occur during the initial stages of the fault and it can be analyzed using Eigen values. Transient torque arises during large short circuit which oscillate at one of the system natural frequencies. Rotor in the turbine generator system is a complex system where many electrical components are involved. The length and area of the rotor will be very large and it weighs over hundred tons (Assenkamp et al., 2017; Wang et al., 2017). Turbine section is a combination of several rotating masses which is mounted on the shaft of turbine generator system. Turbine generator system has six concentrated masses which represent the generator rotor, two numbers of low pressure sections of turbine (LP), an intermediate pressure section of turbine (IP), and a high pressure section of turbine (HP). The sixth mass represents the excitation system mass mounted on the shaft. Torsional oscillations occur when rotors on the shaft starts whirling relative to each other (Mohammadpour et al., 2014). Some of the torsional oscillations can be damped if the magnitude of the torque in the shaft is low. But the main problem of lightly damped oscillation is due to the interaction with the external network. Modal analysis using finite element method is used to determine the vibration characteristics which involves natural frequencies and mode shapes of any structure or a particular rotating machine component. It is the initiating point for a dynamic analysis such as transient dynamic analysis, harmonic or spectrum analysis. Modal analysis can be applied also for a pre-stressed structures like spinning turbine blade. If the structure or a machine component is damped, the analysis is defined as damped modal analysis where the natural frequencies and mode shapes become complex. In a rotating machine component, the gyroscopic effects resulting from rotational velocities are introduced into the modal system which ultimately change the system's damping. The impact of damping changes when bearing is present (Fischer et al., 2010; Mohammadpour & Santi, 2015). The relation between natural frequencies along with rotational velocity can be studied using Campbell diagram. The Campbell diagram gives valuable information about frequency as a function of rotation speed of the shaft.

The main shaft is established by an alloy steel. The main shaft is supported by two main bearings and both bearings are have the capacity to align themselves. The load factor of the shaft line is more and it is greased to provide soft Lubrication system. The bearing seals are maintenance free. The load on each rolling element is calculated by the formulae (1) and (2). Bearings in the actual model is shown in figure 3.

$Q _ { R } = \left[ \frac { 1 } { Z } \sum _ { J = 1 } ^ { z } \left( Q _ { R J} \right) ^ { \mathrm { w } _ { 2 } } \right] ^ { \frac { 1 } { W _ { 2 } } }$ (1)

$Q _ { R } =$ Average load on rotating ring

$Z =$ Number of rolling elements

$W _ { i } =$ Constant

$Q _ { S } = \left[ \frac { 1 } { Z } \sum _ { J = 1 } ^ { z } \left( Q _ { S J } \right) ^ { w _ { c } } \right] ^ { \frac { 1 } { W _ { c } } }$    (2)

$Q _ { S } =$ Average load on stationary rings

$Z =$ Number of rolling elements

$W _ { e } =$ Constant

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Figure 3. Bearings in the experimental setup

There are several kinds of analyses used in engineering and technology. Quasi-static analysis is where the load is applied slowly and make the acceleration zero. Dynamic analysis considers the acceleration as a non-zero element. Compared to both quasi-static and dynamic analysis, modal analysis provides the response of the system. Every rotating structure has an internal frequency or resonant frequency where it starts vibrating. At this frequency, transfer of energy takes place and when this frequency increases, the losses start increasing. The 3D model of the shaft line is modelled in Ansys. Modal analysis is carried out in an efficient manner where point mass, stiffness coefficient is given as per the readings calibrated during compensation. Modal analysis settings are available in table 2.

Table 2. Analysis settings

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Figure 7. Representation of bearing in wireframe form

Bearing on both the sides of the turbine shaft line is modelled in the modal analysis. Torque which is calibrated at 50% of series compensation is related to the stiffness coefficient of the machine. Dampers are not considered in the turbine generator shaft line.

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Figure 8. Meshing of Turbine generator shaft line

Meshing is done in finite element method to discrete the model into different elements. Meshing here is done on the curvature and the relevance centre is kept as fine. High smoothing is required to set the mathematical model in an efficient way. The size of each mesh is made as 0.2m based on the total size of the turbine generator shaft line. Program controlled default settings is followed for transition ratio, maximum layers and growth rate. Inflation algorithm and triangular mesh is defined in the mathematical model. Mesh morphing is disabled as it can save calculation time and increase accuracy. Tetrahedral elements can be a better option for a structure having complex geometry. However, when the shape functions are integrated with the points of mass, it is less accurate than the other shape while meshing. To avoid the complexity and also considering this model, triangular mesh is considered in the meshing. Total number of nodes formed after meshing is around 62722 and elements is around 13055 which is a medium size model and it reduces the processing time and space required for solving the model.

In the proposed work, the generator is loaded with resistive loads. In the meantime series compensators are connected to the system. It is observed that, the shaft vibration increases when the capacitors are connected to the system. This vibration is due to the resonance caused by the capacitors. The shaft line is modelled for its point mass, stiffness coefficient derived from the experimental results. Modal analysis is carried out and multimode oscillations are found out using ANSYS. Finite element method is used to build the mathematical model. Mode 1 becomes unstable which is denoted in the Campbell diagram. From this work, it is clear that finite element can give various modes of oscillation and the deformation of the shaft. It is necessary to use this analysis for any turbine generator shaft line in the design phase. This analysis can increase the stability and reliability of the model.