Synchronous Generator Subtransient Reactance
Prediction Using Transient Circuit Coupled
Electromagnetic Analyses & Odd Periodic Symmetry
Joshua Lorenz
Kato Engineering Inc., North Mankato, MN
John T. Fowler
Kato Engineering Inc., North Mankato, MN
Abstract
For synchronous 3 phase electrical generator machine design, the ability to predict the subtransient
reactance of a particular machine design is of prime importance. The subtransient reactance has a
significant impact on the magnitude of the fault currents generated within the machine during an event such
as a 3 phase short-circuit. Power system designers routinely use the generator subtransient reactance as a
key parameter to aid in the design of the complete power generation system. For new generator designs the
subtransient reactance is routinely tested for as part of a thorough evaluation of the generator performance
characteristics. In this paper, a method is presented to calculate the subtransient reactance of a particular
generator design using transient circuit coupled finite element electromagnetic analyses. The finite element
modeling technique presented utilizes odd periodic symmetry along with a ‘moving’ interface at the rotorstator air-gap. The use of odd periodic symmetry allows for the simulations to be accomplished with half
of the model size as compared to even periodic symmetry. The calculated generator subtransient reactance
from the finite element modeling technique presented is compared to generator test data as well as previous
calculation methods. Other key pieces of information which may be generated throughout the process are
also documented, such as calculated voltage harmonics and the open-circuit saturation curve.
Introduction
Kato Engineering primarily manufactures synchronous 3 phase electrical generators. The ability to predict
the subtransient reactance of a particular generator design is of prime importance. Power system designers
routinely use the generator subtransient reactance as a key parameter to aid in the design of the complete
electrical power generation system.
The subtransient reactance, X”d, is the generator internal impedance element that is effective in the first
cycles of a transient load event and determines the magnitude of the instantaneous fault current from the
generator. The transient reactance, X’d, becomes effective after approximately 6 cycles into a transient
load event and determines the amount of voltage change seen at the generator terminals due to the step
change in load. As stated, the reactances of a generator have a direct effect on the transient fault currents
experienced in an electrical power generation system, as well as the motor starting capability of the
generator. The magnitudes of the fault currents need to be calculated so that breakers, etc. can be sized
accordingly. The peak magnitudes of the 3 phase fault currents are inversely proportional to the
subtransient reactance of the generator. For new generator designs, the transient and subtransient
reactances are routinely tested for as part of a thorough evaluation of the generator’s performance
characteristics.
Traditional reactance prediction methods have served engineers well when new designs closely mimic
previous designs which have been tested and their reactance predictions validated. From the perspective of
the machine designer, an uncomfortable level of uncertainty always accompanied a new design that
deviated significantly from standard form. In this paper, a method is presented to calculate the transient
and subtransient reactance of a generator using transient dynamic circuit-coupled finite element
electromagnetic analyses. Finite Element electromagnetic analysis offers a powerful tool to reduce the risk
in new generator design.
A series of Finite Element (FE) electromagnetic simulations were performed in order to generate the
following simulation data:
•
Open-Circuit Saturation Curve
•
Generator Voltage Signal Harmonics
•
Generator Transient and Subtransient Reactance
The open-circuit saturation curve and voltage signal harmonics were obtained from the magnetostatic
model and the reactance calculations were based upon results from a transient circuit-coupled model with a
moving interface at the rotor-stator air gap. The voltage signal harmonics were also calculated with the
transient circuit-coupled model, for comparison. The transient circuit-coupled finite element modeling
technique presented utilizes odd periodic symmetry along with a ‘moving’ interface at the rotor-stator air
gap. The use of odd periodic symmetry allows for the simulations to be accomplished with half of the
model size as compared to even periodic symmetry.
The particular generator design used for these analyses is a Kato Engineering, salient pole, 3 phase, 60 Hz,
13.8 kV synchronous generator. The generator rotor is a 4-pole rotor with a normal operating speed of
1800 RPM. The generator is rated for 4.5 MW of output power at 0.8 Power Factor with a temperature rise
of 95ºC at an ambient temperature of 50ºC. The generator is approximately 7 ft wide by 10 ft tall by 11 ft
long. Figure 1 shows a view of the generator assembly. One of the reasons this design was chosen for
analysis was the fact there is a plethora of good test data in existence for this design.
Figure 1. Generator Assembly View
Model Information
For all of the simulations performed, the portion of the generator assembly modeled is commonly referred
to as the ‘active’ region of the generator rotor and stator. The ‘active’ region is comprised of the generator
rotor and stator ‘stacks’ along with their respective windings. The models are all 2D models with periodic
symmetry. The model covers 1 pole of a 4 pole generator (i.e. 90 degrees). The rotor shaft is included in
the rotor portion of the model, and the stator portion of the model is bounded by the outer diameter of the
laminated stator stack. Figure 2 shows a view of the 2D geometry which was utilized for all of the
simulations. The geometry was created in Pro/Engineer and transferred to ANSYS via an IGES file
transfer. The simulated ‘length’ of the model was 33.5 inches. This is representative of the stack lengths
for the rotor and stator.
Figure 2. Generator Modeled Geometry
The circuit-coupled models utilized the circuit which is shown in Figure 3. The 1-pole model encloses half
of a parallel circuit for the stator windings. The stator phase windings present in the model are A+, B+, Band C-. Therefore the circuit contains four circuit-coupled stator winding elements. For the open-circuit
voltage harmonics simulation, the phase leakage inductors were effectively removed from the coupled
circuit via voltage coupling constraints across each inductor. The rotor field windings are coupled to the
circuit, and so are the rotor damper bars.
Figure 3. Generator Coupled Circuit
The circuit construction and portions of the model building process were handled via an APDL input file
which was read into ANSYS with the use of the /INP command functionality. The element, material
property, and real constant specifications were all defined in the APDL file.
Elements
Three basic element types were used in the models. The table below shows the element types that were
used and a general description of where they were used in the models
Table 1. Element Type Tabular Listing
ANSYS Element Type
Where Used
PLANE53
Rotor shaft, rotor lamination, rotor windings,
rotor damper cage, air regions, stator
lamination, stator windings, stator slot sticks
CIRCU124
All circuit elements
MESH200
For mesh controls and circuit connection
‘dummy’ wires.
The same element mesh was used for the magnetostatic model and the circuit-coupled models. The mesh
contained 9,636 elements and 29,564 nodes (excluding the circuit elements and nodes). Figure 4 shows a
view of the element mesh (excluding the circuit elements). The mesh is not connected at a boundary arc
between the rotor and stator elements. The boundary arc resides at the middle of the ‘air-gap’ between the
rotor and stator.
Figure 4. Generator Element Mesh
Material Properties
For all of the simulations there were 3 unique magnetic materials, each with their own B-H curve. Figure 5
shows the B-H curves for the magnetic materials listed below:
•
Rotor laminations – 16 Gage ASTM A715 Grade 80
•
Stator laminations – 26 Gage M27 C3 (adjusted)
•
Rotor shaft – AISI 1045 steel
Figure 5. Simulation B-H curves
The B-H curve for the stator laminations was adjusted from the vendor material property data in order to
account for the fact the stator core is vented. For this design the stator core has (12) integral vents. The
vents have an axial width of 0.375 inches. Therefore, over the active stack length of the machine (33.5”)
the stator core only has 29” of active lamination material in the axial direction. The fraction of the stack
length which is comprised of stator lamination material is 0.8657 (29/33.5). The actual stator lamination
flux densities in reality are 1.155x those calculated from the 2D model (1/0.8657 = 1.155). The B-H curve
of the stator lamination material is adjusted so that the correct magnetic induction (H) is required in order
to produce the flux densities which would be seen in reality. The mathematical B-H curve adjustment is
based solely upon the fraction of stator stack length active material (i.e. 0.8657). The open-circuit
saturation curve from the magnetostatic model was developed with and without the ‘stator vent correction’
in order to see how the stator vent correction affects the results.
The air regions, rotor and stator winding regions, stator slot stick regions, and damper cage element regions
of the model were all assumed to have a relative magnetic permeability of 1.0.
For the circuit-coupled models, the winding resistances for the circuit coupled winding elements were taken
from one of our in-house generator spreadsheet design programs. The program calculates expected
winding resistances based upon the design input parameters of the generator. The calculated cold winding
resistances were used because the generator reactance test is normally conducted on a ‘cold’ generator.
The ‘fill factors’ of the rotor and stator coils were back-calculated based on the given winding resistance,
winding length, number of turns, geometric area, and cold resistivity in order that the ANSYS calculated
winding resistances match the specified winding resistances from the Kato Engineering in-house design
program. The equation used by ANSYS for winding resistance based upon fill factor, number of turns,
geometric area, length, and resistivity is shown below:
R=
ρ × L × turns × FF
area
The circuit-coupled solid conductor elements, which represent the damper bars of the generator rotor, used
the cold resistivity of copper in the solid conductor element specification.
Boundary Conditions & Model Details
For all models, odd-periodic symmetry constraints were applied to the periodic edges of the model (i.e. the
boundary lines at 45 and 135 degrees). The constraints were applied via the use of the PERBC2D
command. Odd-periodic symmetry was required because the model encompasses only a single polarity for
the rotor poles. A model of twice the angular span (i.e. 180 degrees and 2 rotor poles) would allow for the
use of even-periodic symmetry. However, even-periodic symmetry is not required to obtain all of the
necessary simulation information and the use of odd-periodic symmetry allows for more efficient models.
For all models, An Az = 0 constraint was applied at the OD of the stator laminations. This constraint can
also be described as a ‘flux parallel’ constraint.
For the magnetostatic model a uniform current density is applied to each side of the rotor field winding. A
positive current density was used on the left hand side, and a negative current density was used on the right
hand side (see Figure 2). Fifteen different current densities were used on the magnetostatic model in order
to generate the open-circuit saturation curve. Each unique current density corresponds to a certain amount
of ampere-turns of rotor field excitation.
For the circuit-coupled transient dynamic models, the rotor field excitation was driven by a fixed DC
voltage source connected to the circuit-coupled rotor field windings. The DC source voltage was set such
that the ampere-turns generated in the rotor resulted in the nominal open-circuit voltage at the generator
output terminals. The nominal open-circuit phase voltage is 7,967 Volts line to neutral (i.e. 13.8kV line to
line).
The circuit-coupled transient dynamic models also employ several voltage coupling equations and voltage
constraints. All of the ‘grounded’ nodes in the coupled circuit have a zero voltage DOF constraint. All of
the nodes on the right hand side of the solid conductor elements (i.e. damper cage conductor elements) have
coupled voltage DOF. A coupling equation is also used, for cosmetic purposes, to couple the fixed DC
voltage source to the rotor field winding elements.
The coupled circuit contains a leakage inductor for each phase of the output voltage. The leakage inductor
is used in the coupled-circuit in order to account for the leakage reactance of the generator armature. The
leakage reactance of a generator armature can be categorized by 4 different armature leakage flux
components [1]:
•
Slot leakage flux –The flux in the stator that does not completely encompass the stator slot by
crossing the stator slot. Flux leakage increases with narrower slot and deeper slot.
•
Zig zag leakage flux – The flux that ‘zigzags’ across the air gap from tooth to tooth and never
links the stator winding. Flux leakage increases with smaller air gap and wider teeth with respect
to the slot pitch.
•
Belt leakage flux – Considered negligible in 3 phase synchronous machines.
•
End-connection leakage flux – The stator flux that is generated in air surrounding the stator end
windings. Flux increases with the axial length of the coils with respect to the stack length.
All but the end-connection leakage flux are accounted for in the Finite Element model. For this reason, the
leakage inductors are added to each phase of the output voltage circuit in order to account for the armature
end-connection leakage reactance. The inductance value used for the leakage inductors is based on a
fraction of the total calculated armature leakage reactance. Inductance (L) and reactance (X) are related by
the formula for inductive reactance, as shown below [2]:
L=
X
ω
Kato Engineering’s in-house design programs calculate the total armature leakage reactance based on the
input design parameters of the generator.
For the 3 phase short-circuit simulation the short-circuit fault is simulated by applying a zero voltage DOF
constraint at the output terminals on the coupled circuit. The output terminal nodes correspond to the nodes
on the right hand side of the inductor elements.
For the magnetostatic model, the rotor and stator meshes are connected by the use of the CEINTF
command. The CEINTF command effectively couples the Az DOF between the rotor and stator regions in
the static model.
For the dynamic models, the rotor/stator moving interface is dealt with via the use of a macro titled
‘ODDSLIDE’. The macro was written by the author, and is similar in functionality to the ‘SLIDE’ macro
which is not documented, but supported to an extent by ANSYS. The ‘SLIDE’ macro does not support
odd-periodic symmetry and can only be used for even-periodic symmetry models. The ‘ODDSLIDE’
macro allows for the use of odd-periodic symmetry (i.e. ½ the model size) and is discussed in further detail
in the next section.
For all but one of the transient dynamic models a time step increment of 750 microseconds was used for the
simulations. This equates to an angular step of 8.1 degrees for the 1800 RPM machine, and a ‘sampling
frequency’ of 1.33 kHz. The time stepping used was a compromise between solution fidelity and solution
run time. The period of oscillation for the output voltage and current of the machine is approximately 16.7
milliseconds (i.e. 60 Hz). Therefore, the time stepping used results in roughly 22 solutions per period of
oscillation. Another iteration of the transient dynamic model was performed with a time step increment of
130.2 microseconds (i.e. 7.68 kHz sampling frequency) in order to verify the open-circuit voltage signal
harmonics which were calculated via the magnetostatic model. The generator voltage harmonics test data
is available out to the 39th harmonic which has a frequency of 2.34 kHz.
Dynamic Solution Routine
In order to perform the circuit-coupled transient dynamic simulations using odd-periodic symmetry, a fairly
detailed approach was used to carry out the simulation process. For all of the transient dynamic circuitcoupled simulations the solution process was started with a static solution obtained while the rotor and
stator sections of the model were aligned.
After the static solution, time integration was turned on and the time-stepping solution loop was carried out.
The basic steps of the solution loop employed are listed below:
•
Identify the new rotor position in absolute degrees
•
Enter /prep7
•
Call the ‘oddslide’ macro with the new absolute rotor position
•
Enter /solu
•
Set the analysis type to RESTART
•
Define the time of the new solution
•
Solve
The basic loop shown above was repeated for each time step increment. For the short-circuit simulation the
‘output’ side of the inductor nodes (on the coupled circuit) were grounded at a particular time in the
simulation loop. The grounding was accomplished by constraining the node voltage DOF to zero. The
simulation loop was then continued with the grounded output terminals (i.e. 3 phase short-circuit).
’oddslide’ Macro
The primary function of the ‘oddslide’ macro was to provide connectivity between the rotor and stator
meshes at the air-gap line, given a displaced position of the rotor. The rotor and stator mesh densities need
not be identical at the interface. The dissimilar meshes are handled with the CEINTF command within the
‘oddslide’ macro. Since only a single polarity of the rotor poles is included in the FE mesh, the macro must
consider the absolute location of the displaced rotor in order to determine whether a ‘positive’ or ‘negative’
rotor pole is residing beneath the stationary portion of the FE mesh. In its simplest form, if a ‘positive’
pole is residing beneath the stator mesh the mesh coupling can be described as Az = Az or ‘even’ coupling.
If a negative pole is residing beneath the stator mesh the mesh coupling can be described as Az = -Az or
‘odd’ coupling. This latter type of ‘odd’ coupling essentially reverses the polarity of the rotor pole as seen
by the stator mesh. Figure 6 shows a very simplified flowchart of the process used in the ‘oddslide’ macro.
Figure 6. ‘oddslide’ Macro Flowchart
Analysis Results & Test Data Comparison
All of the analyses were performed with ANSYS 10.0 on an Intel 32-bit system running Windows XP
Professional. The machine was equipped with dual Xeon 3.06 GHz processors and 2.0 GB of RAM.
Open-Circuit Saturation Curve
Figure 7 shows a comparison of the simulation results and test data for the open-circuit saturation curve. A
series of magnetostatic simulations were run with varying field current, to obtain the simulation results
shown below. The generator armature phase voltage is calculated from each of the magnetostatic
simulation results via the following formula [1], where the last two quantities ([Az.2-Az.1] x Ls) are
equivalent to the flux per pole:
VPH = 4.0 × f × f b × k D × k P × N × (A Z.2 − A Z.1 )× L S
•
f – output frequency
•
fb – form factor (1.11 for a perfect sine wave)
•
kD – winding distribution factor
•
kP – pitch (chord) factor
•
N – turns per phase
•
(AZ.2 - AZ.1) – difference in air-gap magnetic vector potential (per unit length) at adjacent
quadrature axes
•
LS – generator stack length
The output frequency, winding distribution factor, pitch factor, turns per phase, and generator stack length
are all parameters which come from the Kato Engineering in-house spreadsheet design program for a given
electrical machine design. The difference in air-gap magnetic vector potential at adjacent quadrature axes
comes from the FE model results.
The predicted field current required to produce rated voltage in the generator was 65.9 amps from the
magnetostatic simulation. The actual field current required to produce rated voltage in the generator during
Factory Acceptance Tests (FAT) was 66.2 amps. The percent error of the simulation prediction was 0.45%. The predicted open-circuit field currents from the Kato Engineering in-house spreadsheet design
programs were 61.4 amps and 78.6 amps. The former prediction comes from our design program which is
used on most new R&D projects and the latter prediction comes from our more standard design program
which employs ‘test data correction’ after a design is built and tested. The R&D spreadsheet program is
generally more accurate for ‘out of the box’ predictions of this type.
Figure 7. Open-Circuit Saturation Curve Comparison
Generator Voltage Signal Harmonics
The simulation no-load voltage signal harmonics can be calculated from the magnetostatic simulation
results via the use of the air-gap radial flux plot. Figure 8 shows the radial flux pattern versus path length
along the air-gap of the machine, from the magnetostatic simulation results. The flux pattern is
transformed from the time domain to the frequency domain via the use of a Fast Fourier Transform (FFT).
The frequency domain magnitudes are normalized by the fundamental frequency (i.e. 1st harmonic). Each
flux wave harmonic fraction is then multiplied by a pitch factor, distribution factor, and skew factor in
order to calculate the individual voltage harmonics, expressed in percent. For this calculation there are
unique pitch, distribution, and skew factors calculated for each individual flux wave harmonic based upon
the number of poles, stator slots, throw, pitch percentage, skew, and phases. The total harmonic distortion
(THD) is calculated by evaluating the square root of the sum of the squares of the individual harmonics
(excluding the fundamental).
Figure 8. Radial Air-Gap Flux versus Path Length
The no-load voltage signal harmonics can also be calculated from the transient open-circuit simulation
results by performing a Fast Fourier Transform directly on the simulation output voltage signal. The
individual harmonic magnitudes from the FFT are once again normalized by the fundamental harmonic
magnitude and the THD is calculated in the same manner as explained above. Figure 9 shows the transient
open-circuit simulation output voltage signals.
Figure 9. Transient Simulation Open-Circuit Voltage
Figure 10 shows a tabular comparison of the no-load line to neutral (L-N) phase voltage signal harmonics
from the simulation predictions and from the actual test data voltage signal harmonics measured in the
generator FAT. The predicted peak individual harmonic was 0.99% and 1.03% for the 5th harmonic from
the magnetostatic and transient simulations, respectively. The peak individual harmonic from the generator
test data was 0.94% for the 5th harmonic. The predicted total harmonic distortion (THD) was 1.336% and
1.490% from the magnetostatic and transient simulations, respectively. The THD from the generator test
data was 1.340%. The R&D spreadsheet program predicted a peak individual harmonic of 1.105% and a
THD of 1.431%.
Figure 10. Generator Voltage Harmonics Comparison
Generator Transient and Subtransient Reactance
Figure 11 shows a tabular comparison of the predicted and tested generator reactance values. The predicted
transient reactance from the simulation was 10.0% and the transient reactance calculated from the actual
test data was 11.2%. The predicted subtransient reactance from the simulation was 8.6% and the
subtransient reactance calculated from the test data was 9.1%. The predicted transient and subtransient
reactances from the R&D spreadsheet program were 15.3% and 12.8%, respectively. It is more convenient
for electrical design engineers to express circuit values in percent or even per unit terms. To convert to
actual ohmic values for the reactances, multiply the percentages (in decimal format) by the ‘base’
impedance (which is derived from rated voltage and current).
Figure 11. Generator Reactance Comparison
For the generator FAT, the transient and subtransient reactances are calculated individually for 3 separate
current traces taken from 3 unique applications of a three phase fault. The results from the 3 calculations
are averaged for the final reported values. The test operators try to achieve a balanced fault current trace as
this makes the calculation process more reliable. For the simulation predictions, the transient and
subtransient reactances are calculated for all 3 phase current traces and the results are averaged to obtain
the final values for comparison.
The process of calculating the reactance values from the short-circuit phase current traces is typically fairly
time consuming and definitely somewhat subjective. The author developed an automated process for
calculating the reactance values in order to reduce the time required to calculate the values based upon the
short-circuit phase current trace data. The automated process was set-up and performed with the software
program Mathcad. Figure 12 shows the phase current trace data from the short-circuit simulation.
Figure 12. Short-Circuit Current Simulation Data
Conclusions
The open-circuit saturation curve predicted from the magnetostatic simulation correlates extremely well
with the open-circuit saturation curve developed from the generator Factory Acceptance Tests. The
predicted field current required to produce rated open-circuit voltage was within 0.5% of the actual tested
field current requirement. The model does an excellent job of predicting the saturation characteristics of
the machine and this capability has proven to be extremely useful. The heating in the field coil winding is
proportional to the square of the current flowing in the winding, and the ability to accurately predict the
current requirements has a big impact on the ability to predict the heat generation in the field coil windings,
among other things. Our in-house spreadsheet design programs generally do a good job of predicting the
linear behavior of the saturation curve (i.e. the air-gap line), but they tend to struggle with the actual
saturation characteristics as well as the effects of the air vents in the stator stack. The excellent correlation
of the FE model saturation curve essentially validates the magnetic material properties used for the
analyses.
The magnetostatic model and transient circuit-coupled model both do a good job of predicting the voltage
signal harmonics for the generator. In this case, the predictions from the magnetostatic model were the
most accurate. The peak individual harmonic prediction from the magnetostatic model was within 5.3% of
the measured peak individual harmonic from the generator Factory Acceptance Tests. The Total Harmonic
Distortion prediction from the magnetostatic model was within 0.3% of the measured Total Harmonic
Distortion. The R&D in-house spreadsheet design program also does a reasonably good job of predicting
the voltage signal harmonics. The spreadsheet predictions were within 17.6% and 6.8% for the peak
individual harmonic and Total Harmonic Distortion, respectively. Note that the simulations show a higher
harmonic distortion content near the 30th harmonic than what is found in the test data. For this generator
design, the slot passing frequency for the rotor is 1.8 kHz. The 30th harmonic has a frequency of 1.8 kHz.
The simulation voltage signal has a 1.8 kHz ripple due to the rotor passing the stator slots. This is typically
reduced in reality by using a slot skew on the stator stack. This particular design has a 1 slot skew and the
skew results in lesser measured distortion near the 30th harmonic.
The transient circuit-coupled simulation provides a powerful tool for predicting the transient and
subtransient reactances of the generator. The subtransient reactance prediction was within 5.5% of the
measured value from the generator Factory Acceptance Tests. The transient reactance prediction was
within 10.7% of the measured value. The reactance calculations themselves are a bit nebulous due to the
nature of how the calculations are carried out. Typically, Kato Engineering guarantees the measured
reactance values within +/-20% of the predicted reactance values. The importance of the ability to predict
the reactance values can be brought to light by considering the history of this particular generator design.
The guaranteed subtransient reactance for this generator design was 9.3% to 12.6% (+/-15% of the standard
spreadsheet design program prediction). This particular customer specification had a slightly narrower
tolerance band for the allowable subtransient reactance (i.e. +/-15%). The measured subtransient reactance
was 9.1% and we had to request a specification deviation from the customer in order to deliver the
generator. Had we been able to use the Finite Element simulation tools presented herein for predicting the
subtransient reactance we would have put forth a guaranteed range of 7.3% to 9.9%, and the measured
subtransient reactance would have been well within the limits of our customer driven specification.
References
1.
John H. Kuhlmann, “Design of Electrical Apparatus”, 3rd ed., John Wiley and Sons, New York,
1950.
2.
James W. Nillson, “Electric Circuits”, 4th ed., Addison-Wesley Publishing Company, 1993.