Proceedings of XLIIIrd International Symposium on Electrical Machines SME 2007, 2 -5 July, Poznań, Poland
ASPECTS OF IDENTIFICATION OF EQUIVALENT
CIRCUIT PARAMETERS OF LARGE SYNCHRONOUS
GENERATORS BY SSFR-TESTS
Michael Freese, Meinolf Klocke
University of Dortmund,
Institute of Electrical Drives and Mechatronics, Prof. Dr.-Ing. Dr.-Ing. Stefan Kulig
Emil-Figge-Str. 70, 44227 Dortmund, Germany, e-mail: michael.freese@uni-dortmund.de
Abstract - A Standstill Frequency Response Test (SSFR) at a
synchronous machine is an alternative method for determining its
dynamic performance. Therefor three measurements at a stopped
machine have to be done. Four voltage, current and frequency
depending values represent the frequency response function of the
machine. This function is reproduced over the equivalent circuits
by Park. A calculation program creates sets of parameters, the
quality of which is tested. The preliminary best sets are determined and new set generations are build up by them over an
evolutionary algorithm. This paper shows an exemplary
application of the SSFR method and its advantages but also some
problematic aspects.
I. INTRODUCTION
A preferably exact prediction of the dynamic performance
of a synchronous machine is the basis of a safe grid operation.
Therefor characteristic machine values have to be determined.
The examination of a three-pole sudden short circuit is a
standard method for determining the equivalent circuit
parameters by Park (see figures 1 and 2). Whereas this
method is preferably applied in the test bay by the manufacturer, such experiments can hardly be conducted with
generators already installed in power plants. In this case the
aforementioned characteristic parameters should also be identifiable by other testing methods like the standstill frequency
response test according to [1] as described below. The
required measurements of this method can be done relatively
easy at a stopped machine and do not produce risk to the
machine because the currents are not as high as those caused
by a sudden short circuit. The results of the measurements
have to be formatted and transformed. The resulting data
build up the frequency response function of the machine and
the target function for a calculation program as well. This calculation program with its evolutionary algorithm for determining the best parameter set will be explained below, too.
Fig.1. Standard equivalent circuit according to IEEE, d-axis
Fig.2. Standard equivalent circuit according to IEEE, q-axis
II. STANDSTILL FREQUENCY RESPONSE TEST
A standstill frequency response test is done at a stopped
machine as the name reveals. This is one of the significant
advantages in comparison to the method of a sudden short
circuit. Just two rotor positions - mostly set by hydraulic help are needed to do three measurements and to get four significant
values. A low-frequency, dual-channel spectrum analyzer is
used to measure the magnitudes and phase-angles of two
signals in the frequency range 0.001 Hz to 1 kHz. Furthermore
an oscillator and a power amplifier are required. The three
measurements to be done are explained in the following.
A. Measurement 1
With the first rotor position the main flux should go at right
angle to the q-axis. That means a minimum value of voltage
ought to be measured at the open rotor winding when injecting
a sinusoidal current into the stator circuit.
Fig.3. Measurement 1
Proceedings of XLIIIrd International Symposium on Electrical Machines SME 2007, 2 -5 July, Poznań, Poland
This measurement provides the first of the four frequency
response significant values, which is the quadrature-axis
operational impedance Zq calculated as:
Zq =
2 u ab
3 ia
Zd =
(1)
T
i f =0
fd
=
1 u ab
2 ia
(3)
u f =0
1 −if 1
⋅
2 ia ü f
(4)
u f =0
B. Measurement 2
The direction of the main flux can be aligned to the d-axis
just by changing the electrical connections as shown in figure
4. The result is a maximum value of voltage induced in the
rotor winding.
III. IDENTIFICATION OF THE EQUIVALENT CIRCUIT
PARAMETERS
A. Considerations, assumptions and reference values
In order to determine a set of equivalent circuit parameters,
that reflect the measured frequency response function and by
that the performance of the machine under investigation, some
considerations and assumptions have to be done as well as
reference values have to be consulted. As one example here,
the stator dc resistance Ra can be appraised: Therefor the curve
of the direct-axis opertional impedance Zd (see figure 6) of
one exclusive machine has to be extrapolated to zero point,
because at a frequency of 0 Hz Zd complies with Ra.
0,006
0,0055
Fig.4. Measurement 2
The quotient of the stator current and the rotor voltage
multiplied by 1 / √ 2 and the transmission ratio üf (between
stator and rotor values) results in the armature-to-field
transfer impedance Zdf:
Z df =
1 uf
⋅üf
2 ia
Zd [mOhm
0,005
0,0045
0,004
0,0035
0,003
(2)
0
0,01
0,02
0,03
0,04
0,05
0,06
0,07
0,08
0,09
0,1
frequency [Hz]
i f =0
Fig.6. Linear diagram of Zd in the range of low frequencies
C. Measurement 3
The third and last measurement of the SSFR test provides
two values. A fixed stator current causes a certain stator voltage
and a short-circuit current in the rotor winding.
Here a stator dc resistance of about 3.6 mΩ can be appraised.
The data sheet of the machine declares a value of 3.6799 mΩ,
so this valuation seems to be respectable. Furthermore
existing parameter lists of several classes and sizes of
machines show similar values at machines of similar power.
This knowledge is an important basis for a reasonable result
of the evolutionary algorithm described below.
B. Evolutionary algorithm
Fig.5. Measurement 3
These values enable the calculation of the direct-axis
operational impedance Zd and the current transfer function Tfd,
where the transmission ratio has to be considered again:
While the frequency response functions can be calculated
over the equivalent circuit parameters, the calculation in the
other direction is not possible. With the aid of a solution
strategy and a calculation program, that realizes this strategy, it
is possible to converge to a set of parameters that reflects the
frequency response function best. The quantity of the potential
parameter sets with the multiplicity of different parameters has
to be limited. Therefore the considerations, suppositions and
consulted reference values mentioned above help to reach a
manageable mass of parameter sets. In the calculation program
sixty starting compositions chosen by random but regarding
user defined boundaries of parameter values are created. These
Proceedings of XLIIIrd International Symposium on Electrical Machines SME 2007, 2 -5 July, Poznań, Poland
compositions are mutated with random numbers. The new
variations are tested by building up the frequency response of
the respective parameters and comparing to the frequency
response that is given by the measurements done at the
machine. The appearing deviations at the performance of the
magnitudes and phases of the four values Zd, Zdf, Tfd and Zq are
weighted by a function that uprates differences at frequencies
near zero and about 50 Hz. The square sum of all deviations is
calculated and the composition of parameters with the lowest
sum is set to the best. Its parameters and its frequency response
are displayed along with that of the measured values for
comparison. As long as a certain accuracy is not achieved the
mutation will be repeated, each time with the 8 best
compositions of parameters determined before that can be seen
as parents producing further children. Only the best will
accomplish and be the basis for a new generation. Worse
compositions are not considered and will be deleted. By this
evolutionary algorithm the compositions of parameters are
getting better with each cycle. In the end of the evolutionary
cycles the putative best equivalent circuit parameters are found.
These parameters are unsaturated values, because the
measurements at the machines are not conducted at rated
values.
IV. RESULTS OF AN EXAMPLARY MACHINE
INVESTIGATION
Determining the equivalent circuit parameters of a
synchronous machine accords with finding one composition of
parameters that builds up a frequency response fitting to that of
the machine best. The identified parameters do not have to
correspond to those determined by other procedures as long as
the electrical behaviour of the machine is reproduced equally.
In the following the parameters that are determined by the
SSFR-procedure for one exclusive machine are listed below in
Table I. Also the frequency responses of the four basic values
Zd, Zdf, Tfd and Zq are diagrammed in Figures 7 and 8.
Fig.7. Absolute values of the frequency response functions
In Figures 7 and 8 the dashed (red) curves show the
frequency response determined on the results of the measurements at the machine while the solid (blue) lines represent the
frequency response of the parameters identified by the SSFR
procedure. The diagrammed curves show the typical frequency
responses of a synchronous machine with a smooth-core rotor
that can be reproduced by the evolutionary algorithm
approximately.
TABLE I
EQUIVALENT CIRCUIT PARAMETERS
DETERMINED BY
THE EVOLUTIONARY ALGORITHM FOLLOWING THE
Parameter
Ra
Xas
Xhd
Xhq
Rf
Xfs
RDq
RDd
XDds
XDqs
Xcd
xd
xq
SSFR TEST
Value [pu, calcul.]
1.65E-03
1.21E-01
1.54
1.46
1E-03
6.48E-02
9E-03
13E-03
2.14E-02
5.58E-02
0
1.66
1.58
Fig.8. Phase angles of the frequency response functions
Proceedings of XLIIIrd International Symposium on Electrical Machines SME 2007, 2 -5 July, Poznań, Poland
V. ASPECTS AND QUALITY OF THE SSFR TEST AND
OF THE IMPLEMENTATION OF THE PARAMETER
IDENTIFICATION
A. Evaluation of the SSFR test
The two significant advantages of the SSFR test in
comparison to the sudden short circuit method are the absence
of risk to the machine because of low measurands and the
resting rotor that makes the measurements easier. However
measurements done at a machine at standstill cannot consider
any rotational effects. Non linear effects are not included in the
investigation, too. The fundamental wave determined by a
fourier analysis is solely considered while all harmonics are
filtered. Furthermore the equivalent circuit by Park is defined
for 50 Hz - the measurements run from 0.01 Hz to 1000 Hz, so
a wide range of frequencies is referred to the model by Park
that will certainly cause failings at the parameter determination.
As prescribed in [1] the frequency range should contain the
small frequencies to 0.001 Hz. Although the measuring setup is
able to implement such low frequencies, the particular
measurements would take more time and cause high
temperatures thus exposing the windings to a considerable risk.
Besides the constant temperature demanded through all
measurements cannot be complied. So a minimum frequency of
0.01 Hz is chosen, lower frequencies are not considered.
B. Evaluation of the parameter identification
The functionality of the calculation program with its
evolutionary algorithm has been confirmed at an exclusive
machine. The frequency response functions calculated from the
equivalent circuit parameters of the data sheet could be rebuild
by the evolutionary algorithm almost congruently. The resulting values for each determined parameter fit to those of the
given parameters.
If the frequency response functions are not determined by
the parameters of the data sheet but by the frequency response
functions of the SSFR test, the deviations in the resulting
parameters are distinctive, so not the calculation program but
the measuring method causes the problems to represent the
performance of the machine by the determined parameters
correctly.
Actual measurements carried out at a machine in service
suffer from usual deviations and in particular from incompleteness concerning the extremely low frequency range.
Thus, from the theoretical point of view the SSFR test as
described in [1] and [2] is an easily understandable and
comprehensive procedure for determining the equivalent circuit
parameters of a synchronous generator.
However, the transfer to practice is quite difficult. Technically uncompliant data are computed from actual measurements unless the search space for the circuit parameters is
restricted in many ways based on the practical experience of a
machine designer. Only such restrictions lead to reasonable
parameter estimations.
VI. CONCLUSION
The equivalent circuit parameters by Park are required for
specifying the characteristics of the machine in order to analyze
the transient behaviour in the system of grid and turbogenerator. Here the measured frequency response functions are
used to determine the d- and q-axis parameters according to the
model by Park. For the optimization of the parameters an
evolutionary algorithm that is currently seen as the most
effective method is used. The research shows that the method
can be applied successfully only if realistic starting values are
chosen and the search space is constrained in a reasonable
manner. For two pole turbo generators these values and wise
restrictions can be set up because of a multiplicity of reference
machine data. For generators with a higher number of pole
pairs not enough reference data can be reverted to, so that the
determination of parameters of those machines implicates more
difficulties. An approach to enhance the results of the SSFR
test method could be the consideration of the non linearities in
the system.
REFERENCES
[1] IEEE Standard IEEE Std 115-1995 (R2002), “IEEE Guide - Test Procedures for Synchronous Machines”.
[2] H. Bissig, “Stillstandsfrequenzgangsmessungen an elektrischen Maschinen”, Zurich, 1991.
[3] H. Hussein, “Bestimmung der Ersatzschaltbildparameter von Synchronmaschinen anhand von Stillstandsfrequenzgangsmessungen”, Dortmund,
2006.