Abstract
Shaft voltage is present in all electrical machines and can be useful as a diagnosis variable once the effect of some defects on this quantity is well quantified. 2D numerical models have already been used to tackle this point but 3D effects due to end windings can also have a significant influence on this voltage. This paper deals with the study of a turbo-generator using a 3D finite element model in order to investigate this task. In this aim, two finite element models in 2D and 3D of a high power turbo-generator are performed. The effect of three common defects on the shaft voltage is investigated while taking into account the nonlinear behaviour of ferromagnetic materials and the armature winding of the stator. Results with both models are analyzed and compared.
Introduction
Fault detection in turbo-alternators of high power plants is necessary to ensure the availability of the electrical energy production. In this aim, diagnostic tools are required for effective predictive maintenance in order to detect the common faults. Some techniques are already used to detect defects such as the use of the leakage field analysis for the detection of inter turn short-circuit [1–3] the measurement of the vibration to detect eccentricities [4,5], the analysis of the air gap magnetic flux density [6]. However, whatever the adopted method, it generally needs specific measurements which, sometimes, can be intrusive.
On the other hand, in electrical machines, slight inherent construction imperfections, air gaps introduced by the implementation of the sectors of sheet steels of the magnetic circuit or mechanical frictions can generate an electromotive force between both ends of the shaft, called shaft voltage. The latter contributes to a large current in bearings [7] that makes it obviously an undesirable quantity. Therefore, the design of electrical machines takes these factors into consideration in order to limit, as much as possible, the occurrence and/or magnitude of shaft voltage and the latter is always measured in high-power machines to protect bearings from potential large currents.
It turns out that shaft voltage can also be due to numerous other causes that are internal to the machine or from external sources [8]. Among these causes, we can quote rotor short circuit or eccentricities that are quite common defects. Therefore, the analysis of shaft voltage could be interesting to detect those [9]. However, in order to use this variable for reliable detection, it is first necessary to know precisely the contribution on this voltage of each defect aimed to be detected. Based on numerical approach with a 2D model, studies have already tackled the analysis of the shaft voltage of synchronous generators as a potential diagnosis tool [10,12,13]. Thus, the shaft voltage is calculated as the potential difference at the terminals of a resistance that represents the oil film [11] or as a resultant voltage of an equivalent magnetic torus [15]. However, to our best knowledge, no analysis was conducted to determine the impact of 3D modelling on the shaft voltage in the case of some common defects.
The present paper aims to get deeper in the analysis of the shaft voltage of high power synchronous generators by considering the effect of end windings when classical defects occur while taking into account the armature winding of the stator. Static and dynamic eccentricities will be investigated as well as rotor short-circuits on a 4-pole turbo-generator that is one of the wide used machines in high power plants.
The paper is organized in three parts. First, a short introduction of the principal causes that generate a shaft voltage and the principal common defects in turbo-generators, i.e. static and dynamic eccentricities and inter turn short circuit of the field winding, is given.
The second part is dedicated to the presentation of the approach chosen for the study of the shaft voltage along with the numerical approach based on FE model of the turbo-generator. Then, 2D and 3D models and the armature winding of the stator of the studied machine are presented.
Finally, simulations are carried out with both models in the case of the three common defects. The time curve profile of the shaft voltage for each defect as well as currents induced in the parallel windings are analyzed and the obtained results are summarized.
Some conclusions and ideas of future works and investigation close the paper.
Circumferential magnetic flux and shaft voltage
Shaft voltage is the electrical potential between both ends of the rotor shaft of electrical machines [17]. The causes that generate such quantity can be of several kinds of either capacitive or inductive origins and may be internal to the machine or from external sources [8]. In this paper, as the diagnosis is realized from magnetic defects, only magnetic asymmetries are studied. In this case, the alternation of the magnetic flux distribution in the machine generates a resultant circumferential magnetic flux Φ d and then induces a shaft voltage as shown in Fig. 1.
Circumferential magnetic flux.
In healthy conditions, this voltage can be due to material imperfection or to the manufacturing process. However some defects can also induce this potential such as eccentricities or rotor short-circuits which constitute the most common defects that can occur during the operation of large turbo-generators. Then, the analysis of the shaft voltage can be useful in fault detection [9].
Generally, two types of eccentricities can be distinguished. Static eccentricity is defined when the geometrical axes of the rotor and the stator do not coincide while the rotor rotates around its own axis. Thus, for a given rotor point, the air gap width is no longer uniform, but varies with respect to position. In this case, for a given stator point, the air gap is constant but it does not present the same width all around the machine. Dynamic eccentricity occurs when the rotor rotation axis is different from its geometrical one which is the same as the one of the stator. In this case, for a given rotor point, the air gap is constant.
Field winding inter turn short circuit is mainly due to a failure of the insulation between some rotor conductors.
At low amplitude, these defects do not present a real risk for the machine and many generators operate then without a significant effect on their performance. However, these defects can evolve and generate additional constraints that may eventually provoke damages.
Previous studies have tackled the interest of the shaft voltage as a diagnosis tool. Using FEA, shaft voltage is determined by considering an external circuit coupling between the shaft and the casing. Then, the shaft voltage is determined as the voltage at the ends of the resistance that models the oil film of the bearing [11]. All these studies were conducted with a 2D finite element model.
Main geometrical characteristics of the studied structure
Main geometrical characteristics of the studied structure
To go deeper, simulations have been carried out, in 2D and 3D, using a homemade FEM code (code_Carmel) [19] to study a 4-pole 1300 MW turbogenerator (Table 1). Hence, a 2D model is built in a careful manner, i.e. in the form of a regular mapped star mesh, in order to avoid any possible numerical error that can be introduced by the quality of the mesh. Furthermore, a large air area around the machine is modelled in order to allow the possibility to a leakage flux to be established. Figure 2 gives a view of the 2D mesh which holds 80 000 prismatic elements.
View of the 2D model.
A 3D model is also built up. With the aim to reduce the calculation time with the real shape of end windings, the latter is obtained by extruding the previous mesh with different thicknesses along the rotation axis (z-axis) while taking into account the symmetry along this same axis, leading to 800 000 prismatic elements regularly distributed. Then, end windings are modelled as straight conductors as presented in Fig. 3. Finally, a large volume of air all around the machine is also taken into account.
View of the 3D model with end windings.
As the movement is modelled through the locked step procedure in code_Carmel, a static or a dynamic eccentricity can be taken into account just by moving stator or rotor elements respectively [18]. This allows the use of the same mesh to study the machine either in healthy conditions or with eccentricities. In the case of a field winding short circuit, the defect is simply modelled by decreasing the number of the ampere turns in a slot of the rotor excitation circuit.
The shaft voltage is determined by considering the shaft and the casing as two different inductors and the induced emf represents the shaft voltage.
Finally, the coupling of the stator windings, constituted of four coils connected in parallel is taken into account through an external electrical circuit (Fig. 4). Each phase is constituted of four windings connected in parallel and two neutral points. R1 and L1 represent the resistance and leakage flux inductance respectively, R2 is the resistance between two windings, R L is the phase load resistance and R n is the resistance that connects the neutral points.
Armature winding of the stator.
Using the numerical models, different calculations are carried out with vector magnetic potential formulation at no load (R L = 106 Ω), rated field current (1800 A) and speed 1500 rpm in healthy conditions and in the case of 3 different defects:
20% static eccentricity 20% dynamic eccentricity 12% rotor inter-turn short circuit
These defects are obviously unrealistic but they are choses to allow understanding the physical phenomena. Calculations are achieved under magnetostatic hypothesis (no currents induced in the damper bars) while taking into account the nonlinear behavior of the magnetic material. Whatever the case studied, since R L is of very high value, there is almost no current flowing through thus simulating no load operating.
In order to highlight the impact of the modelling of end windings on the shaft voltage, results obtained by both models are compared and discussed.
Healthy conditions
In healthy conditions, using the 2D numerical model, the distribution of the magnetic flux density in each pole is symmetric. Therefore, there is no circumferential magnetic flux and the shaft voltage is then practically zero as shown in Fig. 5. The small variation of the air gap permeance seen by the windings of each phase induces circulating currents of very low magnitude (Fig. 6) with no effect on the shaft voltage. In the case of 3D simulations, the effect of the magnetic field density in the end winding part of the excitation circuit induces a non-negligible shaft voltage whose waveform takes a sinusoidal shape of 8 V and 50 Hz (Fig. 5) while the circulating currents remain of similar low magnitude as in the 2D case calculations (Fig. 6).
Shaft voltage in healthy conditions.
Circulating currents in the armature winding of the stator in healthy conditions.
To highlight these 3D effects, the magnetic flux density is shown on a cross section in the middle of the machine (Fig. 7a) and in the end winding part (Fig. 7b). When compared, we observe the effect of the current in the end windings which leads to an asymmetry of the magnetic flux density (Fig. 7b) with regard to its distribution in the middle of the machine (Fig. 7a). The difference between both cases can be seen on the areas of the machine pointed by arrows.
Magnetic flux density in the middle of the machine (a) and in the end winding side (b).
In the case of a 20% static eccentricity, the distribution of the magnetic flux density is no more symmetrical in the cross section of the machine. The air gap permeance variation generates emfs in each coil connected in parallel which leads to circulating currents at the electrical frequency (50 Hz) in the armature windings (Fig. 8). In the case of 2D simulations, the combined effect of the air gap permeance variation and of the circulating currents leads to a shaft voltage with a main component at 250 Hz and in a second one in a lesser extent at 50 Hz. When using 3D model, the component at 50 Hz shown in healthy conditions still be present, with a slightly lower magnitude, but also the occurrence of a component at 250 Hz (Fig. 9). The magnitude of the latter is slightly lower in 3D case due to the lower magnitude of the circulating current when end windings are taking into account.
Circulating currents in the armature winding of the stator with a static eccentricity.
Shaft voltage in the case of a static eccentricity.
In summary, as in healthy conditions, end windings have a non-negligible effect and induce a shaft voltage with a main rated frequency component (50 Hz). This component is not present when only 2D calculations are performed. On the other hand, there is no modification on the signature of this defect whatever the model used. The variation of the air gap permeance introduce by a such defect generate circulating current at 50 Hz which induce a shaft voltage with a component at 250 Hz.
Previous study has shown that dynamic eccentricity does not have any inherent impact on the shaft voltage [16]. However, when the stator winding have coils connected in parallel, the variation of the air gap permeance induces circulating currents at (p −1)f∕p and (p +1)f∕p. Where p is the pole pair number and f is the electrical frequency. In this case, circulating currents induced at 25 Hz and 75 Hz and lead to the generation of a 300 Hz component on the shaft voltage (see Fig. 11).
Simulations with the same conditions are performed using the 3D model. As previously, the component at 50 Hz due to the end winding modelling is still present along with the signature of the defect at 300 Hz whose magnitude is smaller than the case of 2D calculations. This is due to the same causes as in the case of static eccentricity Fig. 10.
Circulating currents in the armature winding of the stator with a dynamic eccentricity.
Shaft voltage in the case of a dynamic eccentricity.
In 2D as in 3D, dynamic eccentricity leads to a shaft voltage component at 300 Hz, of almost constant magnitude, due to circulating currents in the armature windings while the end winding effect adds a component at 50 Hz of much higher amplitude.
The last studied defect is relative to a 12% inter-turn short circuit in one pole of the field winding. The time waveform of the shaft voltage obtained by both models is given in Fig. 12a along with its harmonic content in Fig. 12b and the waveforms of the circulating current in the windings are presented in Fig. 13. In a global manner this defect leads to effects similar to those due to dynamic eccentricity, i.e. circulating currents in the armature windings at 25 and 75 Hz and a shaft voltage with a specific component at 300 Hz.
Shaft voltage in the case of a rotor short circuit.
Circulating currents in the armature winding of the stator in the case of a rotor short circuit.
As for previous cases, end winding effect leads to an additive component at 50 Hz with higher magnitude.
In summary, end winding effect induces a shaft voltage at the electrical frequency of 50 Hz whatever the operating conditions of a 4-pole turbo-generator. In case of faulty operating, this component is added to the signature of the defect that is a 250 Hz component in case of static eccentricity and a 300 Hz component in case of dynamic eccentricity or rotor inter-turn short circuit. The latter can be obtained by simple 2D modelling as the 3D aspect does not induce a significant effect on their magnitude.
In this paper, a finite element analysis is carried out in order to investigate the impact of end windings on the shaft voltage in the presence of some defects at no load. To conduct such analysis, two finite element models, in 2D and 3D, of a 4-pole high power turbo-generator were built up. In 3D case, end windings have been modelled as straight conductors in order to reduce the calculation time.
From the obtained results, it can be concluded that the end winding effect generates an inherent shaft voltage component at 50 Hz frequency whose magnitude is quite independent from the operating conditions of the machine. This component in mainly due to the asymmetry of the magnetic flux density in the end winding side of the machine compared with the flux density distribution in a cross section in the middle of the machine. It is afterwards added to the component of the shaft voltage relative to a considered defect, i.e. a 250 Hz component for a static eccentricity and a 300 Hz component for the dynamic eccentricity and the rotor inter turn short circuit. Moreover, the end winding effect seems to have no influence on the magnitude of these components that means they can be obtained from 2D modelling.
Future works will focus on complementary points such as the effects of the currents induced in the damping bars and also the one of the load on the shaft voltage under healthy conditions and in the case of the three common defects.
Footnotes
Acknowledgements
The authors would like to thank EDF R&D for their financial support and research participation.