Eddy Current Heating in Large Salient Pole Generators

Transcription

1 Eddy Current Heating in Large Salient Pole Generators C.P.Riley and A.M. Michaelides Vector Fields Ltd., 24 Bankside, Kidlington, Oxford OX5 1JE, UK phone: (+44) , fax: (+44) Abstract Pole face heating in large salient pole generators has been observed for many years, leading to high trailing edge temperatures and degredation of insulation. Electromagnetic 3D finite element based software including rotation coupled to thermal analysis has been recently developed, enabling the mechanisms causing the heating to be examined. Accurate estimates of the ventilation and convective cooling are shown to be particularly important. I. INTRODUCTION Rotors of large salient pole generators in the MW range are usually constructed from solid steel forgings for mechanical strength. Load imbalance or fault conditions will induce significant negative sequence eddy currents in the pole faces. However, even under synchronous operation, significant eddy currents can be induced in the pole face resulting in unacceptable temperature rise. The eddy currents result from both Changes in distribution of flux linking the pole due to stator slotting and Harmonics in the armature reaction MMF waveform due to the discrete turns in the stator winding. Analytical expressions for the losses resulting from these eddy currents have been known for many years, but are only approximate and assume an infinitely long machine. Numerical solutions for electromagnetic fields have now advanced sufficiently to allow a more realistic distribution of the losses to be computed. In particular, three dimensional dynamic simulation of the machine including eddy currents and rotation is now available. Multi-physics capabilities allow the computed losses to be exported to numerical thermal analyses to compute corresponding temperature rise. This paper describes finite element based simulations of the dynamic electromagnetic problem in an idealized 60 Hz, 6-pole, 3-phase generator (shown in Fig. 1), and includes coupling to three dimensional finite element thermal analysis. The importance of accurate knowledge of heat transfer coefficients on the temperature prediction is particularly demonstrated. II. DYNAMIC ELECTROMAGNETIC SIMULATION Numerical solutions of electromagnetic field problems play an important role in the design of electrical machines. Developments in finite element methods and the improved performance of computers now allow non-linear transient analysis with eddy currents and rotation to be routinely undertaken [1], [2]. Figure 1. Finite element model of 6-pole generator To obtain accurate evaluation of the eddy currents and resulting losses, three dimensional analysis is needed, as there are substantial end effects including transfer of currents between poles. However, it is more efficient to establish the steady state operating conditions using a two dimensional analysis. The OPERA-2d RM software [1] solves the 2D transient electromagnetic equation for the normal component of the magnetic vector potential, A z 1 A µ z A σ t z = J The windings of the machine are connected to external circuits, rather than prescribing the value of source current, J s, and the rotation is accommodated by re-meshing a band of elements in the air gap at each time step. This introduces no additional equations at each time step and only requires modification of the matrix sparsity pattern and equation terms from the newly created elements. The rotor winding is supplied with DC voltage and each phase of the stator windings is connected to the appropriate load for the required terminal conditions, 13.2 MW at 9 kv and 0.85 p.f. lag, at 1200 rpm. The transient equation is then solved until steady state operation is achieved. Fig. 2 shows the solution of the field and the current density values in the windings after 2 seconds when steady state has been achieved. The armature reaction can be seen, causing the flux density to rise towards the trailing edge of the pole. It is necessary to compute 3 poles of the machine as the generator has a nonintegral number of slots / pole / phase. s (1)

2 In Fig. 4, the modified shape in the final 60 degree cycle results from evaluating losses with a finer angular discretisation (2.5 ). Earlier in the simulation (before steady state was achieved) they were only evaluated every 20 degrees. However, internally the analysis used the same time step throughout. Figure 2. Steady state field and current distribution The higher flux density at the trailing edge and the effects of slotting can be more easily observed from the radial component of the air-gap flux density, shown in Fig. 3. Figure 4. Power dissipation in rotor The variation in loss through each 60 degree rotation period mainly occurs because of considerable differences in the harmonic content of the armature reaction MMF waveform. In this machine, this is particularly strong due to the non-integral number of slots/pole/phase (2.67). Fig. 5 shows the distribution of eddy currents at the maximum loss during each 60 degree rotation. As can be seen, the current density is highest (about 2.8 MA/m²) on the trailing edge of the pole, which corresponds to practical experience of the area where overheating occurs. It can also be seen that there is a small transfer of eddy current between poles, which can only be quantified accurately by the three dimensional calculation. Figure 3. Radial air-gap flux density at steady state The values of field current and armature phase current computed by the two dimensional analysis are exported to the three dimensional model. The three dimensional dynamic analysis including rotation of the rotor and eddy currents, CARMEN, was originally developed to model superconducting synchronous generators [2]. The large air-gap in these machines allowed a lock step method of rotation to be implemented for the motion [3], and this is also suitable for the 6-pole machine examined here, without requiring the step size to be extremely small. CARMEN is also a transient analysis, so although the steady state operation currents were already known, there are switch-on transients that must decay before the steady-state eddy current pattern can be determined. However, as shown in Fig. 4, the variation of power dissipation during each 60 degree period quickly becomes cyclical, as the stator and rotor fields are synchronous. Obtaining the steady state conditions from the two dimensional simulation is then seen to be very advantageous. The two dimensional analysis required 2 seconds of simulation to achieve steady state while the three dimensional analysis is operating in steady state after only 0.05 seconds. Figure 5. Eddy current density in pole face (A/m²) at maximum loss Figure 6 shows the eddy current distribution at a time when the minimum loss during the 60 degree cycle occurs. The maximum current density is now reduced to about 1.8 MA/m 2 and no longer occurs on the trailing edge of the pole.

3 Figure 6. Eddy current density in pole face (A/m 2 ) at minimum loss III. THERMAL CALCULATIONS The steady-state temperature rise in the rotor has been computed using the TEMPO-ST software [4], which solves the final temperature equation k T = Q (2) The distribution of losses in the rotor are computed by averaging the values resulting from the eddy currents computed at each time step over one 60 degree period. I²R losses in the rotor winding are also included. The finite element mesh in the thermal problem is confined to only one pole of the rotor, as all poles experience the same heat production during one 60 degree period, with the other poles implied by periodic boundary conditions. Even within this pole, the finite element mesh for the thermal analysis is not the same as that used for the electromagnetic. In the electromagnetic modeling, skin effect confines the eddy currents to the surface of the rotor, requiring a fine discretization at the surface but allowing larger elements in the body. For the thermal analysis, although the heat generation is confined to the skin (and the rotor winding), heat flow will be experienced throughout the pole and the discretization must support this. The field winding insulation, which can be ignored during the electromagnetic analysis, must also be included in the thermal simulation. The discretization for the thermal analysis along with the averaged heat intensity distribution computed from the electromagnetic analysis is shown in Fig. 7. The higher loss intensity from the pole face eddy currents at the trailing edge is clearly visible. The peak value is 380 kw / m 3. Heat transfer is assumed to occur due to rotation of the rotor in the surrounding air. Forced ventilation from both ends of the machine creates an axial temperature gradient in the air from 25 C at the ends of the machine to 50ºC in the centre. Fig. 8 shows the resulting temperature at the surface of the pole. The trailing edge is warmer than the leading edge with a peak temperature of about 187 C. This temperature will be sufficient to cause long term damage to the insulation on the rotor winding. Figure 7. Average heat intensity distribution in pole face Figure 8. Temperature on pole face An accurate knowledge of the heat transfer coefficients is vital. It has been assumed in this model that the leading edge half of the pole face and the leading face of the winding have a 50% better heat transfer coefficient than the other surfaces where heat is lost to the cooling gas. Reducing the disparity to 25% and assuming that only a negligible amount of heat is transferred through the trailing edge winding face significantly modifies the distribution, as shown in figure 9. The temperature on the trailing edge of the pole now rises to about 215 C. IV. Three dimensional effects DISCUSSION The idealized generator modelled in this paper has been examined principally to determine the significant factors affecting pole face heating calculation and to make recommendations to obtain solutions efficiently. It is very clear from the eddy current distributions shown in Figs. 5 and 6 that only a three dimensional analysis is adequate. Although the currents are induced principally in an axial direction, they turn at the end to form closed loops. Even more importantly there is transfer of current from pole to pole through the rotor yoke. This can be observed clearly

4 when the windings are removed from the picture, as shown in Fig. 10. Figure 11. Field current following load change Figure 9. Temperature on pole face with modified heat transfer coefficients Developments planned in the CARMEN software should enable the analysis speed to be improved. Periodic boundary conditions would allow only 3 (half length axially) poles to be modelled instead of 6 probably giving a factor of between 2.5 and 3.5 improvement with the same level of discretization. Changing from a lockstep method to re-meshing the air-gap, similar to OPERA- 2d RM, should enable a reduced number of elements without loss of accuracy. Even so, it would still seem probable that obtaining steady state operating conditions in the three dimensional analysis would require several weeks of CPU time. Loss calculation Figure 10. Eddy current transfer between poles Steady state operation However, obtaining the steady state operating conditions using the three dimensional analysis is probably not viable. As described above, the two dimensional analysis only reached stability after 2 seconds of simulation time. This is primarily due to the long field time constant. Figure 11 illustrates the transient variation of the rotor current during the time to achieve steady state, following a load change. The two dimensional analysis requires about 16 hours CPU time on a 1.4 GHz PC to achieve the steady state solution after 2 seconds of simulation. The three dimensional analysis took more than 4 days CPU time on a 2.65 GHz PC to complete a single revolution of the rotor (0.05 seconds of simulation time). Obtaining the steady state operating conditions in three dimensions would take more than 5 months! This is a reflection on the number equations solved in the 2D model (4429) and the 3D model (188902). Both models were using a similar time step (0.43 milliseconds in 2D, 0.31 milliseconds in 3D). The non-sinusoidal 60 degree periodic variation of the loss intensity shows that a time-stepping analysis of the machine is necessary to obtain the correct average loss. Analytic formulae for computing pole face losses assume that they can be obtained from a steady state solution based on the stator slot pitch and the harmonic content of the armature reaction MMF. Thermal analysis The absolute accuracy of the thermal calculations is very dependent on the heat transfer that is assumed from different surfaces of the rotor. Fairly small changes in value can change the pole face temperature by 25 ºC or more. Accurate heat transfer values could only be obtained using a very sophisticated model in a computational fluid dynamics analysis software package. The complexity of the stator end winding alone makes this a daunting challenge, without the inclusion of the rotation effects and stator cooling ducts. At this stage, it is probably more useful to use the thermal analysis to observe trends. When making design modifications to overcome excessive trailing edge heating, heat transfer coefficients could be obtained empirically from an existing machine by adjusting values until a reasonable agreement with observed temperature rise is obtained.

5 V. CONCLUSIONS Dynamic electromagnetic finite element analysis allows the calculation of pole face loss in large salient pole synchronous generators during steady state operation. Three dimensional effects are significant and require modelling. However, obtaining steady state behaviour using a 3D analysis is probably not viable yet. The synchronous operating conditions can be obtained from a two dimensional analysis. Loss production during rotation varies due to changing harmonic content of the armature reaction MMF and alignment of the rotor pole with the stator teeth. However, the loss production is cyclical and an average value can be determined. Exporting these losses to a steady-state thermal analysis leads to the typical temperature profile observed in large synchronous generators. Absolute values of temperature are difficult to predict due to the sensitivity of the results to the accuracy of heat transfer coefficient values. However, the coupling of the electromagnetic and thermal analyses allows trends associated with design changes to be evaluated. VI. REFERENCES [1] Biddlecombe C., Simkin J., Jay A., Sykulski J., Lepaul S., Transient electromagnetic analysis coupled to electric circuits and motion, IEEE Trans. Mag., vol. 34, no. 5, September 1998 [2] Emson C., Riley C., Walsh D., Ueda K., Kumano T., Modelling eddy currents induced by rotating systems, IEEE Trans. Mag., vol. 34, no. 5, September 1998 [3] Preston, T., Reece, A., Sanga, P., Induction motor analysis by time stepping techniques, IEEE Trans. Mag., vol. 24, no. 1, January 1988 [4]