I-11. Electromagnetic Forces and Mechanical Oscillations 117 the method of images. The consideration of both the original circuits and the additional fictive current distributions as line circuits enables the application of Biot-Savart’s law in the following form: → B = μ 0 I 4π · → dl × → r r 3 (1) In a numerical computation of the flux density B generatedby a three-dimensional assembly of line circuits, these can be modeled by discrete line circuit segments. B is obtained using the principle of superposition after applying a discrete form of (1) to each segment. The three-dimensional distribution of electromagnetic forces can afterward be obtained by computing the vector of force acting on each discrete line circuit segment of the length l and carrying the current I using Lorentz’ law [3]: → F = I · ( → l × → B ) (2) By using line circuits instead of conductors with a finite cross section and a given current density it may be expected that the forces will be computed up to 25% to high [3]. But due to the fact that both the modeling effort and the computing time could be reduced substantially this simplification was found admissible. Thedescribedmethodwasimplemented inanumerical computationprogram andapplied to calculate the three-dimensional distribution of electromagnetic forces acting on the stator end winding of different turbo generators. A typical three-dimensional model of the end region used in the computation is shown in Fig. 2. It consists predominantly of the line circuit segments modeling the bars of the stator end winding. The influence of the rotor end winding is not directly taken into account. Instead of that the magnetic flux generated by the rotor is emulated by line circuits in the area of the retaining ring. The assumption underlying this simplification is, that the stator flux during transients is approximately mirrored by the rotor. Therefore the currents in these additional line circuits are equal to the currents in the stator windings but have opposite direction. Figure 2. End winding model for electromagnetic force computation. 118 Gr¨uning and Kulig The influence of the stator core is considered according to the method of images by mirroring the complete model of line circuits at the end core plane. The impact of other magnetizable parts in the end region, like the rotor shaft, is neglected [3]. Additional investigations performed in [4] and also by the authors showed that both the influence of the rotor shaft and the influence of the end core plane are of minor importance. By repeated computations the applied method is able to calculate discrete time functions of the electromagnetic forces using discrete time functions of currents as input data. Thus it was possible to investigate the transient behavior of the electromagnetic forces, like for example during a three-phase terminal short circuit, which represents the standard check of large electrical machines previous to initial operation. Before accomplishing the force computation the discrete time functions of the currents in the stator windings have to be determined. Due to the fact that measurements with a sufficient time resolution are rare, the currents were computed using the numerical net- work simulation program NETOMAC [5], based on the Park transformation. As a likewise practicable method to obtain discrete time functions of currents, also a two-dimensional finite-difference time-stepping method as described in [6] was applied. Using the finite- difference method more accurate results with differences to measurements of less than 5% can be obtained, but at the price of an extensively longer computation time [1]. In order to be impressed on the nodes of the mechanical finite-element model, the results of the electromagnetic force computation are given by force vectors instead of uniform loads. The magnitude of the force vector acting on a certain node is therefore dependent on the length of the respective line circuit segment, whichhastobeconsideredwhen discussing the force distribution. Depending on the treated generator the number of nodes differs from about 500 to about 2,000, with three time functions of the force components at each node. The electromagnetic forces acting on the end winding of a 90 MVA two-pole turbo generator during a three-phase short circuit are given in Fig. 3, Fig. 4 shows the forces on the end winding of a 1,500 MVA four-pole generator. Figure 3. Forces on the end winding of a two-pole generator. I-11. Electromagnetic Forces and Mechanical Oscillations 119 Figure 4. Forces on the end winding of a four-pole generator. Both figures show the force distribution at a moment 10 ms after short circuit occur- rence, when the currents in the stator windings approximately reach their maximum values. Whereas the maximum forces on the end winding of the two-pole generator occur approxi- mately at the same time, the maximum forces on the four-pole generator end winding occur earlier, approximately 8 ms after short circuit occurrence. The highest magnitudes in the force distribution occur in the area of the involute parts of the windingcoilsnearertothe end core plane.Inbothforcedistributions the number ofpoles of the corresponding generator is visible. Beside radial and tangential force components also strong axial force components occur. The maximumuniformloadsactingon the endwindingofthetwo-polegeneratoramount to approximately 40,000 N/m, in the four-pole generator the uniform loads reach val- ues around 70,000 N/m. This corresponds to the higher power density of the four-pole generator. Figs. 5 and 6 show the force distribution of both the two-pole and the four-pole generator in a plane parallel to the end core plane at different moments around the occurrence of the force maximum. In both figures the force distribution enables the classification of the winding coils corresponding to the respective phase. Indeed the force distribution is non- sinusoidal [1]. As can be seen in Fig. 6, the tangential components in the force distribution of the four- pole generator are much more pronounced than in the force distribution of the two-pole generator. Fig. 7 shows the time functions of the force components acting on a location in the middle of the involute part of the upper layer bar of the first coil of phase a of the two-pole generator, which has to sustain the largest electromagnetic forces during the three-phase terminal short circuit. All components consist of a fraction oscillating with the system frequency and a constant fraction, both decaying exponentially. With the constant fraction decreas- ing the time functions are increasingly dominated by a fraction of twice the system frequency [1]. 120 Gr¨uning and Kulig Figure 5. Plane force distribution in the two-pole generator end region at different moments. Figure 6. Plane force distribution in the four-pole generator end region at different moments. Alternative computation In the case of the two-pole generator the determination of the electromagnetic forces during a three-phase terminal short circuit was also accomplished using an alternative numerical computation method described in [4]. This method is characterized by a modeling of the stator end winding coils with a finite cross section. Furthermore this method considers the magnetizable rotor shaft using the integral equation method based on the separation of the magnetic field H into a zero- divergence fraction and an irrotational fraction [4]. Compared to the computation method introduced above, the rotor end winding is explicitly modeled and the rotor movement is I-11. Electromagnetic Forces and Mechanical Oscillations 121 Figure 7. Time function of force. taken into account. The influence of the stator core is considered similarly by the method of images. The model underlying this computation method is shown in Fig. 8. Duetothefact thatthecomputationtime isextensivelylongerwhenapplyingthismethod, the results of both methods were compared at a single moment 10 ms after short circuit occurrence. In Fig. 9 the force distribution along the upper layer bar of the first coil of phase a calculated by both methods is compared. Regarding the characteristic distribution of electromagnetic forces both methods pro- vide comparable results. The differences correspond widely to the differences between the two end region models like for example the way of modeling the bars of the stator end winding. Figure 8. End winding model for alternative electromagnetic force computation. 122 Gr¨uning and Kulig Figure 9. Comparison of force distribution. Mechanical simulation The simulation method used to compute the vibration behavior of the stator end winding is based on the application of mechanical finite-elements using the results of the electromag- netic simulation as input data [1]. Depending on the manufacturer and the cooling principle the construction of the stator end winding varies. The subsequently introduced mechanical finite-element model of the stator end winding of a 170 MVA air-cooled turbo generator, which represents a character- istic construction, is shown in Fig. 10. The finite-element model comprises the bars of the upper layer and the lower layer which are connected in pairs at the coil nose, the pressure ring which borders the stator core and represents the boundary of the simulation area and the coil support brackets mounted on the pressure ring. The model is completed by the support rings surrounding the end winding and the bandages, which fix the bars to each other, the bars to the support rings and also the Figure 10. Mechanical finite-element model of the stator end winding [2]. I-11. Electromagnetic Forces and Mechanical Oscillations 123 support rings to the support brackets [2]. The components of the model are determined by their masses, stiffness, and damping characteristics. The complete model consists of about 20,000 bricks [1]. Both the bars of the stator end winding and the bandages have a very complex structure, the modeling of which would exceed the calculating capacity. Therefore these components were replacedby substitutesofapproximately the samemechanicalproperties.To determine the parameters of the original bars and bandages, which show a non-linear behavior and also temperature dependence, extensive measurements have been accomplished. The bars of the stator winding consist of multiple twisted conductors, each covered with insulating varnish. Theseareenclosedbyepoxy resinimpregnated glass silktape.To obviate the modeling of such a complex structure the bars of the stator end winding were modeled as massive bodies of comparable mass distribution, stiffness and damping characteristics. To determine the stiffness parameters, bending tests were accomplished. Dynamic param- eters were estimated by additional oscillation tests. Beside the measurements a detailed mechanical finite-element model of a stator bar was implemented and used to compute the parameters of different stator bars numerically. The stiffness parameters of the bandages were determined by conducting static and dynamic tests in a servohydraulic test facility. Tests at different temperatures showed a significant dependency of the stiffness of the bandages on the temperature. The performance of numerical simulations while varying certain model parameters showed that the accuracy of the simulation results is decisively determined by certain model parameters, especially by the stiffness of the bandages [2]. In this connection the precise determination of suchlike parameters still offers a potential for further developments. Results and verification A combined electromagnetic and mechanical simulation of the end winding oscillation behavior under forced vibrations was accomplished for three representative turbo gener- ators differing in cooling principle, power class, and therefore in the stator end winding construction. The deformation due to the electromagnetic forces acting on the stator end winding of a 170 MVA air-cooled turbo generator computed at one moment during a three-phase terminal short circuit is shown in Fig. 11. As can be seen the oscillation behavior owns characteristics of the one of a composite structure. The time functions of the displacement computed at a location in the middle of the involute part of an upper layer bar are shown in Fig. 12, using the Cartesian coordinate system underlying the mechanical simulation method. Reachingvaluesaround1,000 μmintheareaofthecoilnose, thedeformationsarearound 30 times higherthanundersteady-stateconditions.Butdue to the damping characteristics of the end winding structure and according to the time behavior of the electromagnetic forces during the three-phase terminal short circuit the amplitudes of the displacement decrease relatively fast [2]. In the case of a 90 MVA air-cooled turbo generator the computation results of the oscil- lation behavior during a three-phase terminal short circuit were compared to measurements conducted by the manufacturer. At a location on the coil nose of different stator end winding coils acceleration sensors were used to measure the displacement by integrating the signal 124 Gr¨uning and Kulig Figure 11. Deformation of stator end winding during a three-phase terminal short circuit. of the sensors. The comparison of the maximum values of the displacement functions is given in Table 1. As can be seen, except for one location the results of the numerical simulation and the measurements show a good congruence with deviations of mostly noticeable beneath 40% [1]. Beside the combined simulation the mechanical simulation method was applied to de- termine eigenfrequencies and eigenmodes of the treated end winding constructions. There- fore a rotating sinusoidal force distribution with solely a radial force component was impressed on the mechanical finite-element model. By varying the rotational speed of Z-direction Y-direction X-direction 0 –300 –150 0 150 300 –300 –150 0 150 300 –300 –150 0 150 300 0.1 0.2 Time [s] 0.3 0.4 0.5 0.6 Displacement [10 –6 m] Figure 12. Time functions of displacement during a three-phase terminal short circuit [2]. I-11. Electromagnetic Forces and Mechanical Oscillations 125 Table 1. Comparison and displacement [1] Displacement Computed Measured Location of coil nose Tangential ( μ M) Radial ( μ M) Tangential ( μ M) Radial ( μ M) 3 h 934 1,216 — 1,174 6.75 h 1,059 1,055 558 1,215 7.5 h 1,344 1,220 — 877 9 h 931 1,159 879 934 9.75 h 925 1,103 — 1,502 10.5 h 874 1,021 1,068 1,644 11.25 h 1,003 989 — 993 the force distribution different eigenmodes were excited like for example the four-node oscillation of the stator end winding of a 500 MVA hydrogen-cooled generator shown in Fig. 13. The eigenfrequency corresponding to the shown oscillation amounts to 72 Hz. The four-node oscillation represents an eigenform, which can indeed easily be excited in the end winding of turbo generators [1]. A number of eigenmodes with the appendant eigenfrequencies of the 90 MVA air-cooled turbo generator is given in Table 2. Table 2. Eigenmodes and eigenfrequencies Eigenmode Eigenfrequency (Hz) Torsional oscillation 35.2 Two-node oscillation 41.4 Four-node oscillation 73.6 Figure 13. Four-node oscillation excited by rotating sinusoidal force distribution. 126 Gr¨uning and Kulig A resonance measurement accomplished for the 90 MVA generator by using the accel- eration sensors mentioned above showed a clear resonance at 70.5 Hz, which corresponds good to the computed frequency of the four-node oscillation. But repeated measurements at different temperatures showed, that the measured frequency varied slightly with increasing temperature. Conclusions The electromagnetic and mechanical modeling of the turbo generator end region and es- pecially the determination of certain model parameters still contain a great potential of improvement. Anyhow, three-dimensional numerical simulation of the end winding oscil- lation behavior of turbo generators may emerge as a useful tool in design and development of large electrical machines. References [1] O. Drubel, S. Kulig, K. Senske, End winding deformations in different turbo generators during 3-phase short circuit and full load operation, Electr. Eng., Vol. 82, pp. 145–152, 2000. [2] K. Senske, S. Kulig, J. Hauhoff, D. W¨unsch, “Oscillation Behaviour of the End Winding Region of a Turbo Generator During Electrical Failures”, Conference Proceedings CIGRE, Yokohama, October 29, 1997. [3] C G. Richter, Berechnung elektromagnetischer Kr¨afte auf die Spulenseiten im Wickelkopf von Turbogeneratoren, Institute of Electrical Machines and Drives: University of Hannover, 1996. [4] B. Frei-Spreiter,Ein BeitragzurBerechnung derKr¨afteimWickelkopfbereichgroßerSynchron- maschinen, Instituteof Electrical Machines:Swiss Federal Instituteof TechnologyZ¨urich, 1998. [5] B. Kulicke, Digitalprogramm NETOMAC zur Simulation elektromechanischer undmagnetis- cher Ausgleichs-vorg¨ange in Drehstromnetzen, Elektrizit¨atswirtschaft, VDEW, Vol. 78, No. 1, pp. 18–23, 1979. [6] R. Ummelmann, Erweiterung eines Finite-Differenzen-Zeitschritt-Programmsystems auf Syn- chronmaschinen, Institute for Electrical Machines, Drives and Power Electronics: University of Dortmund, 1997. . the middle of the involute part of the upper layer bar of the first coil of phase a of the two-pole generator, which has to sustain the largest electromagnetic forces during the three-phase terminal. Institute of Electrical Machines and Drives: University of Hannover, 1996. [4] B. Frei-Spreiter,Ein BeitragzurBerechnung derKr¨afteimWickelkopfbereichgroßerSynchron- maschinen, Instituteof Electrical. Erweiterung eines Finite-Differenzen-Zeitschritt-Programmsystems auf Syn- chronmaschinen, Institute for Electrical Machines, Drives and Power Electronics: University of Dortmund, 1997.