Projektbericht ZID 04-372-1System of Hydro Generators
Erich Schmidt Institute of Electrical Drives and Machines, Vienna University of Technology
A–1040 Vienna, Austria, Gusshausstrasse 25–29 Phone: +43-1-58801-37221, Email: erich.schmidt@tuwien.ac.at
Introduction
The stator core laminations of large hydro generators are compressed with stator clamping plates, clamp- ing bolts and clamping fingers. The typical industrial design as shown in Fig.1 uses clamping bolts which go through the stator yoke as well as the upper and lower clamping plates. Thus, the tension of the clamping bolts established for fixing the stator core laminations will be absorbed with the clamping plates. Due to this fact, laminations are impossible and solid materials have to be utilized with clamping plates and clamping fingers.
Consequently, the magnetic field in the end region on both sides of the machine causes eddy currents in these conductive parts of the stator clamping system [1, 2]. With large hydro generators, the specific losses in these regions can reach high values and can cause local thermal problems. For this reason, an accurate calculation of these eddy current losses is a matter of interest with all design phases of large hydro generators.
A fully 3D approach will be presented for the computation of eddy current losses in the stator clamping parts of a 450 MVA hydro generator. Both time-harmonic and nonlinear transient finite element analyses are carried out. With the indent of an inclusion with design review and design optimization, the results obtained from both analysis methods are compared regarding local and total eddy current losses. Table1 lists the main data of the investigated three-phase synchronous generator.
Fig. 1: Stator core with clamping system using bolts through core
Table 1: Main Data of the Synchronous Generator
Rated power 450 MVA
Rated voltage 21000 V
Rated current 12370 A
Rated power factor 0.9
Rated frequency 50 Hz
Rated speed 428 rpm
Runaway speed 800 rpm
Number of poles 14
Finite Element Solvers
The eddy current analyses utilize a 3D vector potential formulation with an incorporated Coulomb gauge
curl ( ν ∼ · curl ~A
with appropriate Neumann and Dirichlet boundary conditions
( ν ∼ · curl ~A
~A × ~n = ~0 on ΓB , (2b)
where ν ∼
denotes an anisotropic reluctivity tensor, σ ∼
an anisotropic conductivity tensor, ~J0 an applied source current density, ΓH the boundary where ~n× ~H is specified and ΓB the boundary where ~n · ~B is specified [3, 4, 5, 6, 7].
Thus, the unknown degrees of freedom U of the magnetic dominant eddy current problem are calculated from the nonlinear system of ordinary differential equations
( C d
dt + K
U = P , (3)
where P, C, K are the generalized loads, the damping matrix and the nonlinear stiffness matrix [5, 6]. This nonlinear system is solved with a fixed time step according to 20 time steps within one power frequency cycle. The desired convergence is achieved by modified Newton-Raphson iterations with an updated nonlinear stiffness matrix for each time step [6, 8]. In case of the time-harmonic analyses, the corresponding linear complex system is solved with an ICCG algorithm [7, 9].
Finite Element Modeling
The 3D finite element model includes one pole pitch of the generator. The required periodicity of the solution is obtained using anti-periodic boundary conditions along the appropriate boundary surfaces in moving direction. Second order pentahedral and hexahedral elements are used throughout all conducting regions because they give better results than tetrahedral elements [10].
Slitting the stator teeth is a well-known design feature and has been shown to reduce the stator core losses because of limiting the width of eddy currents flow areas in the stator laminations [11]. The nonlinear anisotropic material behaviour of the laminated stator core is included with an anisotropic reluctivity tensor and appropriate constraints for the magnetic vector potential [12]. Fig. 2 depicts the stator tooth discretization in detail.
To encounter for the high saturation of the nonlinear clamping plate in the surface regions with the linear time-harmonic analyses, different permeabilities are used in the center regions and the regions nearby the surfaces. Thereby, reduced values compared to those listed in Table 2 are used in r and z planes with a thickness of 10% regarding thickness and radial height.
Table 2: Linear Material Data
σ [106S/m] µrrr [1] µr [1] µrzz [1]
Stator core 3.50 300 300 15
Clamping finger 1.33 1 1 1
Clamping plate 5.00 250 250 250
Fig. 2: Finite element discretization of the stator tooth region
The excitating field is represented with its fundamental harmonic wave by using 1D line current ele- ments. There are 64 current paths in radial direction with sinusoidal currents of power frequency and an appropriate phase shift according to the pole pitch.
In addition to the binary boundary constraints, Dirichlet boundary conditions are modelled on both surfaces in axial direction and at the outer stator boundary. In the airgap, a Neumann boundary condition takes into account for the rotor surface. The data of the finite element model are listed in Table 3.
Table 3: Characteristics of the Finite Element Model
Number of Elements 92620
Number of Nodes 79980
Number of Equations 441540
Analysis Results
With regard to commissioning tests, the most significant case of the eddy current losses in the clamping system is the no-load condition.
Fig. 3 and Fig. 4 show the time-dependent power losses in the clamping fingers and clamping plate. It can be seen that the losses are nearly constant in the steady state. Thus, higher harmonics due to slots and saturation will effect only the local power loss density but are negligible regarding the total power losses in the clamping system.
Table4 and Table5 list the total eddy current power losses at the time values of t = 0.230 s and t = 0.235 s. This time difference represents an angular shift of = −π/2 with regard to the propagation of the excitating fundamental wave. According to the number of nine stator slots and clamping finger pairs along one pole pitch, this is equivalent to the propagation of 4.5 slot pitches in circumferential direction.
This fact is clearly represented with the listed data for all regions of clamping fingers and clamping plate with both the nonlinear transient and linear time-harmonic analyses. In comparison of the analyses, the latter results in larger differences between minimum and maximum values of the power losses. This is
0.00 0.04 0.08 0.12 0.16 0.20 0.24
Time (s)
Fig. 3: Power losses in the clamping fingers versus time
0.00 0.04 0.08 0.12 0.16 0.20 0.24
Time (s)
Fig. 4: Power losses in the clamping plate versus time
due to the fact, that higher local eddy currents simultaneously cause a higher local saturation in the nonlinear clamping plate. Consequently, the local skin depth increases which yields reduced local eddy current losses. Nevertheless, the results obtained from both analysis methods show a good agreement regarding the total power losses.
Table 4: Eddy Current Losses [W] at Different Time Steps in Steady State, Nonlinear Transient Analysis
Clamping Fingers Clamping Plate
t = 0.230 s t = 0.235 s t = 0.230 s t = 0.235 s
Region 1 34 146 150 1005
Region 2 35 145 182 990
Region 3 62 119 441 820
Region 4 102 80 742 580
Region 5 136 44 950 258
Region 6 149 31 1019 126
Region 7 135 46 919 295
Region 8 100 82 711 598
Region 9 60 121 420 864
Summary 813 814 5534 5536
Table 5: Eddy Current Losses [W] at Different Time Steps in Steady State, Linear Time-Harmonic Analysis
Clamping Fingers Clamping Plate
t = 0.230 s t = 0.235 s t = 0.230 s t = 0.235 s
Region 1 27 156 130 1056
Region 2 31 151 163 1023
Region 3 64 119 397 789
Region 4 110 73 723 463
Region 5 146 36 988 198
Region 6 157 25 1068 118
Region 7 138 45 926 260
Region 8 96 86 628 558
Region 9 52 130 314 872
Summary 821 821 5337 5337
Fig. 5 and Fig. 6 depict the power loss density obtained from the nonlinear transient and the linear time-harmonic analysis at the time values of t = 0.230 s and t = 0.235 s. This time difference represents an angular shift of = −π/2 with regard to the propagation of the excitating fundamental wave. According to the number of nine stator slots and clamping finger pairs along one pole pitch, this is equivalent to the propagation of 4.5 slot pitches in circumferential direction.
It can be seen, that the power loss distributions from both analysis methods are very similar with both time values. But with the clamping plate, the regions of lower and higher local power losses are more different against both analyses.
TransientResponse Analysis PowerLossDensity Frequency 50Hz TimeStep 0.230s
FrequencyResponse Analysis PowerLossDensity Frequency 50Hz TimeStep 0.230s
500.000 - 1581.1
1581.1 - 5000.0
5000.0 - 15811.4
15811.4 - 50000.0
1.6e+05 - 5.0e+05
5.0e+05 - 1.6e+06
1.6e+06 - 5.0e+06
5.0e+06 - 1.6e+07
1.6e+07 - 5.0e+07
Fig. 5: Power loss density in the clamping system, time value of t = 0.230 s, nonlinear transient analysis (upper part) and linear time-harmonic analysis (lower part)
TransientResponse Analysis PowerLossDensity Frequency 50Hz TimeStep 0.235s
FrequencyResponse Analysis PowerLossDensity Frequency 50Hz TimeStep 0.235s
500.000 - 1581.1
1581.1 - 5000.0
5000.0 - 15811.4
15811.4 - 50000.0
1.6e+05 - 5.0e+05
5.0e+05 - 1.6e+06
1.6e+06 - 5.0e+06
5.0e+06 - 1.6e+07
1.6e+07 - 5.0e+07
Fig. 6: Power loss density in the clamping system, time value of t = 0.235 s, nonlinear transient analysis (upper part) and linear time-harmonic analysis (lower part)
Conclusion
The paper presents 3D finite element analyses for the computation of eddy current losses in the stator clamping system of large hydro generators. Both time-harmonic and nonlinear transient calculations are carried out. With the indent of including the numerical analyses with design review and design optimization of the generators, the results obtained from both analysis methods are compared regarding the total eddy current losses as well as their local distributions.
Due to the nonlinear behaviour of the clamping plate, the eddy currents nearby the surfaces cause high saturated surface regions. To achieve comparable results with the time-harmonic analyses, the clamping plate has to be modeled with different permeabilities in the center regions and the regions nearby the surfaces. With this modeling strategy, the total eddy current losses evaluated from both analyses show a good agreement. Nevertheless, the time-harmonic solution in comparison with the nonlinear transient solution yields different local eddy current distributions in particular with the clamping plate. On the other hand, the time-harmonic analyses are much faster and can therefore be introduced into the design process of large hydro generators.
References
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