Standstill Frequency Response
Test Analyzer by Using Excel
Macros for Educational
Purposes
B. VAHIDI, M. R. BANK TAVAKOLI
Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran
Received 17 June 2009; accepted 2 January 2010
ABSTRACT: Accurate knowledge characteristics of a synchronous generator are very essential to have an idea
of its operation conditions. In the presented paper generator standstill frequency response tests is used as an
approach to derive dynamic models and equivalent circuits descriptive of machine behavior for dynamic studies
by using Excel Macros. The method is described and applied to a case study. The extraction process for obtaining
the parameters is demonstrated. Evaluation of the method with 20 senior under graduate students and 20 junior
power system engineers is very positive in terms of their developing confidence in and understanding of this test
and deriving the dynamic model. ß 2010 Wiley Periodicals, Inc. Comput Appl Eng Educ; Published online in Wiley
InterScience (www.interscience.wiley.com); DOI 10.1002/cae.20415
Keywords: synchronous generator; dynamic model; equivalent circuit; standstill frequency response
INTRODUCTION
Computer simulation plays an important role in engineering
course teaching. Nowadays, a variety of software tools are
available to simulate electrical circuits. Many simulations of
different aspects of a power system and engineering application
using software have been presented by different researchers
[17]. Some researchers used other methods for teaching
electrical machine [810].
A synchronous generator, being a vital piece of equipment
in an electric system, requires critical attention from the
standpoint of its performance under dynamic conditions; therefore, it is highly desirable to derive the dynamic model of a
synchronous generator [1116].
This subject is the duty of power system engineers;
therefore, the students of power system engineering should be
familiar with it. Our course on electric machine III is taken by
students in 15 weeks (3 h per week). In this course tests on
synchronous generator and finding the parameters of equivalent
circuit are taught. Before taking electric machine III they
should pass two other courses called electric machine I and
Correspondence to B. Vahidi (vahidi@aut.ac.ir).
ß 2010 Wiley Periodicals Inc.
electric machine II. These two courses are about DC machines,
transformer, and induction machines. In our university, the power
system students, after these three courses, which are mandatory,
can take another course which is called special machine.
The standstill frequency response test (SSFR) is one of the
most descriptive tests which is performed by power system
engineers to derive the dynamic model of generator. It would be
quiet helpful for students of power system engineering to become
familiar with this test and also procedure of handling the data and
extracting the parameter out of raw test results. Moreover,
analyzing the test results and procedure of parameter derivation
should create better sight trough the theory of synchronous
generators.
STANDSTILL FREQUENCY RESPONSE TEST
SSFR testing has received much industry attention as a mean of
identifying the dynamic characteristics of machine rotor flux
behavior. Methods have been developed to establish equivalent
circuits for the d- and q-axis of synchronous machines which
closely match the measured frequency response characteristics in
standstill tests [14].
Frequency response methods using the protocols of IEEE
standard 115A [17] appear to have more adherents than those
supporting the time method of IEEE standard 115 [18], if
1
2
VAHIDI AND TAVAKOLI
set of parameters in such equations to comply well with test
results.
one judges popularity by the number of papers employing the
frequency response approach [11]. In IEEE 115 [18] methods are
described for obtaining synchronous machine parameters in the
form of reactances and time constants. These are the familiar
synchronous, transient and subtransient reactances as well as
open circuit and short circuit, transient and subtransient
time constants [17]. Difficulties with IEEE standard 115 for
obtaining synchronous machine parameters in order to do
dynamic modeling of machine are well discussed in introduction
of this standard.
The results of SSFR tests are now accepted as an alternative
means to time—honored sudden short circuit test for the
determination of parameters of synchronous machines. Whilst
the sudden short circuit test can only provide information on
the parameters of second order models in direct axis
whereas SSFR tests provide information on both the direct and
quadrature axis parameters, there are also limitations in the
application of SSFR tests. The excitation is limited to very low
levels and because the tests are carried out at standstill, the
damper winding connections do not have any centrifugal force
on them making the contact resistance different to that with
the machine running [16]. This must affect the damper
winding time constants and therefore the measured frequency
response. Accepting these limitations, SSFR tests still have an
important role to play in the determination of the machine
parameters [16].
ð2Þ
sGðsÞ ¼ Gð0Þ
ð1 þ Tg1 sÞ
ð1 þ Tg2 sÞð1 þ Tg3 sÞ
ð3Þ
ra
ð4Þ
pffiffiffi
3 Ifd
2 Id
ð5Þ
Vq
Iq gq
ð7Þ
For configuration of Figure 2, gq ¼ 1.5.
It is possible to calculate Rd and Rq out of measured Zd(s)and
Zq(s) which is different from armature resistance because of
jX fd
jX c 2
rfd
I fd
jX d 2
E fd
rd 1
rd 2
jX q1
jX q 2
jX q 3
rq1
rq 2
rq 3
jX l
jX aq
Figure 1
Vd
Id gd
Zq ðsÞ ¼
jX ad
Vd
Vq
ð1 þ T1q sÞð1 þ T2q sÞ
ð1 þ T3q sÞð1 þ T4q sÞ
The value of gd depends on the configuration of armature
windings during the test. For configuration of Figure 2, gd ¼ 2.
For d- and q-axis tests assume the field winding is in open
circuit condition.
Therefore the armature to field transfer impedance (Zaf0(s))
and the q-axis operational impedance (Zq(s)) are
pffiffiffi
3 Efd
Zaf0 ðsÞ ¼
ð6Þ
2 Id
jX d 1
Iq
Lq ðsÞ ¼ Lq ð0Þ
sGðsÞ ¼
jX c1
jX l
ð1Þ
Zd ðsÞ ¼
Structures of synchronous machines can be represented by the
standard, SSFR2 (second order), and SSFR3 (third order) models.
The SSFR3 model is shown in Figure 1. SSFR2 model structure
can be obtained from SSFR3 model by reducing the number of
damper windings and neglecting JXc2. The models of Figure 1 are
based on the reciprocal per unit system in which all parameters
are referred to the stator [13].
Following formulas for characteristics transfer functions are
relevant for SSFR2 equivalent circuit. Our aim is to find the best
ra
ð1 þ T1d sÞð1 þ T2d sÞ
ð1 þ T3d sÞð1 þ T4d sÞ
where, s ¼ jo ¼ j2pf
Windings connections for SSFR tests are shown in Figure 2.
Assume that for d-axis test (field winding is in short circuit
condition), voltage and currents are measured:where Vd is the
armature voltage; Id is the armature current; Ifd is the exciting
winding current.
then,
EQUIVALENT CIRCUITS AND
PARAMETERS DERIVATION
Id
Ld ðsÞ ¼ Ld ð0Þ
Third order (SSFR3) models for d- and q-axis.
STANDSTILL FREQUENCY RESPONSE TEST
a
Nfd
1
Zaf0
¼
lim
Lad
Na
s
Nfda ¼
Iq
A
3
ð14Þ
s!0
I fd
Rfd ¼
lim
sGðsÞ
s
ð15Þ
s!0
E fd
Vq
Lad
2
3 Nfda
Initialization
bI
To start curve fitting procedure, the initial values should be set in
advance. Based on the fact that there are some values provided by
manufacturer (or typical values) for standard machine parameters
(transient and subtransient reactances and time constants), by
using (16) to (26) good initial values for parameters of equivalent
circuits in Figures 1 and 2 would be calculated.
In order to obtain feasible initial values for the parameters,
the values of Canay reactances (Fig. 1) are assumed to be zero.
Following equations are utilized to set the initial values for
equivalent circuit parameters by use of available machine
transient and subtransient parameters.
B
C
d
A
I fd
Xfd ¼
ðXd Xl ÞðXd0 Xl Þ
Xd Xd0
ð16Þ
Xd1 ¼
ðXd0 Xl ÞðXd00 Xl Þ
Xd0 Xd00
ð17Þ
Xd2 ¼
ðXd00 Xl ÞðXd000 Xl Þ
Xd00 Xd000
ð18Þ
rd1 ¼
1 Xd00 ðXd0 Xl Þ2
2pfTd00 Xd0 ðXd0 Xd00 Þ
ð19Þ
rd2 ¼
1 Xd000 ðXd00 Xl Þ2
2pfTd000 Xd00 ðXd00 X 000 Þ
ð20Þ
Xq1 ¼
ðXq Xl ÞðXq0 Xl Þ
Xq Xq0
ð21Þ
ð9Þ
Xq2 ¼
ðXq0 Xl ÞðXq00 Xl Þ
Xq0 Xq00
ð22Þ
In next step, the values of Ld(0) and Lq(0) are extracted and
from these values the Lad and Laq are computed
Xq3 ¼
ðXq00 Xl ÞðXq000 Xl Þ
Xq00 Xq000
ð23Þ
rq1 ¼
1 Xq0 ðXq Xl Þ2
2pfTq0 Xq ðXq Xq0 Þ
ð24Þ
rq2 ¼
2
1 Xq00 ðXq0 Xl Þ
2pfTq00 Xq0 ðXq0 Xd00 Þ
ð25Þ
2
1 Xq000 ðXq00 Xl Þ
2pfTq000 Xq00 ðXq00 Xq000 Þ
ð26Þ
E fd
Vd
C
B
Figure 2 Windings connections for SSFR test: a) q-axis test and
b) d-axis test.
temperature in test conditions and test wiring resistances
Rd ¼ lim Zd ðsÞ; s ! 0
Rq ¼ lim Zq ðsÞ; s ! 0
ð8Þ
The d-axis operational inductance (Ld(s)) and q-axis
operational inductance (Lq(s)) are then computed using following
equations
d
Ld ðsÞ ¼ Zd ðsÞR
s
Zq ðsÞRq
Lq ðsÞ ¼
s
Ld ð0Þ ¼ lim Ld ðsÞ
ð10Þ
s!0
Lq ð0Þ ¼ lim Lq ðsÞ
ð11Þ
s!0
Lad ¼ Ld ð0Þ Ll
ð12Þ
Laq ¼ Lq ð0Þ Ll
ð13Þ
where, Ll is the leakage inductance of armature which is available
from manufacturer’s data.
The armature to field transfer ratio and field circuit
resistance at test conditions (Rfd) are as follows. These values
will be kept constant during curve fitting procedure.
rd3 ¼
In order to start the curve fitting process all resistances and
inductances are converted to actual values according to following
bases.
4
VAHIDI AND TAVAKOLI
Vn2
Sn
ð27Þ
Vn2
on Sn
ð28Þ
Rbase ¼
Lbase ¼
where Sn is the rated power of generator; Vn is the rated voltage of
generator; on ¼ j2pfn.
The overall procedure of parameter derivation from test
results is shown in Figure 3. The sinusoidal voltage and current
curves needs to be pre-processed to obtain the main component
phase and amplitude. This can be done by using Fourier
Transformation. Characteristic transfer functions, that is, Zd(s),
sG(s), Zaf0(s), Zq(s), Ld(s), and Lq(s) are then calculated for all
tested frequency range. Before starting curve fitting procedure,
some independent parameters should be derived based on abovementioned characteristic transfer functions. These parameters
are Rd, Rq, Ld(0), Lq(0), Lad, Laq, Rfd, and Nfda. Using these fixed
values and initial values for other parameters of equivalent circuit
of Figure 1, the curve fitting procedure started. Finally, after the
procedure reached a reasonable mean square error in all points of
tested frequency range between predicted values of transfer
functions in model and test results, final values are obtained and
corrected and machine transient and subtransient parameters are
calculated.
Curve Fitting Procedure
The curve fitting procedure is required here to tune the machine
parameters in such a way to represent a suitable frequency
response of the machine in tested frequency range. Because the
direct and quadrature axis are independent, two goal functions are
defined. For direct axis, the sum of all squared errors between the
model predictions (which depends on equivalent circuit parameters) and test results of Zd(s) and sG(s) for all tested points is
defined as the goal function. Similarly, the square error between
model and test results of Zq(s) for all tested frequency range is
defined as the goal function for quadrature axis curve fitting
procedure.
A direct searching method is utilized to find the parameter
set which results in minimum error based on the above-mentioned
goal functions [19]. In this method, all the parameters are
changed one by one in small steps around its starting values while
other parameters are kept constant. The goal functions are
evaluated until reaching minimum (local optimum) on this
parameter. Nevertheless, the initial values in this method are
very important because bad initialization may lead to very high
iterations [19]. Therefore, it is important to use proper initial
values for parameters, that is, the procedure which is described in
the Initialization Section. Figure 4, shows the curve fitting
algorithm.
Calculating the Transient and Subtransient Parameters
The outcome of the curve fitting procedure would be the values of
equivalent circuit parameters of synchronous generator. From
these values, transient and subtransient reactances and time
constants of the machine should be calculated because they are
most common representing parameters of the machine. To do so,
following formulas are used to calculate the reactances and time
constants.
Xd ¼ Xl þ Xad
ð29Þ
Xd0 ¼ Xl þ Xad kXfd
ð30Þ
Xd00 ¼ Xl þ Xad kfXc1 þ Xd1 kXfd g
ð31Þ
Xd000 ¼ Xl þ Xad kfXc1 þ Xd1 k½Xc2 þ Xd2 kXfd g
ð32Þ
Xq ¼ Xl þ Xaq
ð33Þ
Xq0 ¼ Xl þ Xaq Xq1
ð34Þ
Xq00 ¼ Xl þ Xaq Xq1 Xq2
ð35Þ
Xq000 ¼ Xl þ Xaq Xq1 Xq2 Xq3
ð36Þ
Xfd þ Xl kXad
2pfrfd
ð37Þ
Xfd kðXc1 þ Xl kÞðXad Þ þ Xd1
2pfrd1
ð38Þ
Td0 ¼
Figure 3
The flowchart of overall parameter derivation from test results.
Td00 ¼
STANDSTILL FREQUENCY RESPONSE TEST
Figure 4
Td000 ¼
Curve fitting procedure.
Xfd kbXc2 þ Xd1 kðXc1 þ Xl kXad Þc þ Xd2
2pfrd2
ð39Þ
Xl kXaq þ Xq1
2pfrq1
ð40Þ
Tq0 ¼
Xl kXaq Xq1 þ Xq2
00
Tq ¼
2pfrq2
Tq000 ¼
Xl kXaq Xq1 Xq2 þ Xq3
2pfrq3
5
ð41Þ
on air gap line of no-load curve. The modified rfd (rfd mod) is also
computed from (44).
Tcoef þ Top 3 Na 2
rfd mod ¼
rfd meas
ð44Þ
Tcoef þ Ttest 2 Nfd
where Top is the field winding temperature in normal operation;
Ttest is the field winding temperature during SSFR test; rfd meas is
the measured value of field winding resistance; Tcoef is the
correction factor, that is, 234.5 for copper winding and 225 for
aluminum winding.
ð42Þ
After curve fitting Xad and rfd should be corrected due to the
low excitation level of the test and also the effect of deviation of
test temperature from normal operational temperature. Xadu is the
corrected value for Xad (which is computed from the test results in
low excitation level) and is computed from (43).
rffiffiffi
3 Na Vn
Xadu ¼
ð43Þ
2 Nfd Ifdo on
where Ifdo is the field current which can produce the rated voltage
METHOD VALIDATION BASED THE TEST RESULTS
OF AN ACTUAL CASE
In order to further confirm the accuracy of the proposed method
the SSFR test is conducted on a power plant generator and the test
results are used to extract the machine parameters. The Microsoft
Excel Macro utility is used to create a stand-alone module for
SSFR test result analyzing. This Utility is then used by students
and junior engineers to evaluate the test results. Nominal
parameters of tested generator are shown in Table 1.
6
VAHIDI AND TAVAKOLI
Table 1
Nominal Parameters of Tested Generator
Rated power
Rated voltage
Rated frequency
Field current for producing rated voltage on
air gap line
Leakage inductance
147.775
MVA
13.8
50
kV
Hz
491
0.095
A
pu
A test circuit same as Figure 2a is prepared for the generator.
The rotor position is changed until the induced voltage in open
circuit exciting winding be <30 mV due to the armature current
of 10 A (in this condition the rotor is in q-axis position). Then,
armature current kept constant (10 A) and test is repeated at
different frequencies. After q-axis test, without changing the rotor
position, d-axis test is conducted according to Figure 2b for two
cases:
Table 3
cd
cq
Correction Factors of Impedances
2
1.5
Table 4 Generator Measured Resistances and Temperature During
Test
Armature resistance per phase
0.00141 X
Direct measuring of field resistance
0.1015 X
Environment temperature during test
278C
Generator windings temperature during normal operation
1008C
Winding conductor material
cu
(1) Field winding was in open circuit condition.
(2) Field winding was in short circuit condition.
Recorded voltages and currents are imported to MATLAB
for pre-processing and de-noising. Afterwards, the amplitudes,
phases of voltages and currents in all tested frequencies are
imported to the program which is prepared in Microsoft Excel.
Base values for converting the currents, Impedances and
inductances are shown in Table 2. With regard to Figure 2, the
correction factors for characteristic impedances of d- and q-axis
are as Table 3, see Equations (4) and (7) for more details.
The generator measured resistances and also site temperature during test are shown in Table 4 which are used for final
correction of results.
After evaluating the Zd, Zq, sG(s) and Zaf0, Nafd, and Rfd can
be computed by using Figures 5 and 6 and Equations (14) and
(15) (slopes obtained out of these figures should be divided by 2p
to calculate the values in Eqs. 14 and 15). Focusing on low
frequency values of Zd and Zq and extrapolation (Fig. 7), Rd and
Rq will be extracted.
After computing Rd and Rq the inductances of d- and q-axis
(Ld(s) and Lq(s)) can be computed from Equation (9). Ld(0) and
Lq(0), Lad, Nafd, and Rfd are then extracted in a similar manner
from Equations (12), (14), and (15). These parameters are
tabulated in Table 5. Only in this step, curve fitting procedure
begins. The procedure is continued until the goal function
changes in consequent iteration become small enough (1e-3 in
our case).
Typical transient and subtransient parameters which are
used for initialization and corresponding computed equivalent
circuit parameters (see Eqs. 1626) are shown in Table 6.
Table 2
Figure 5
Zaf0 amplitude versus frequency.
Figure 6
sG(s) amplitude versus frequency.
Base Values
Parameters
Base value
Armature current
Armature impedance
Armature inductance
Field current (nominal turn ratio)
Field impedance (nominal turn ratio)
Field inductance (nominal turn ratio)
6182.46 A
1.289 X
0.0041 H
1044.186 A
135.53 X
0.431 H
STANDSTILL FREQUENCY RESPONSE TEST
Figure 7
Calculating the resistances from Zd and Zq in low frequencies.
7
Figure 8
model).
The actual and fitted curves for Ld(s) magnitude (second order
Figure 9
model).
The actual and fitted curves for Ld(s) angle (second order
The outcome of the above procedure which is repeated for
SSFR2 and SSFR3 equivalent circuits reveals this fact that test
results of the generator are better achieved by second order model
for d-axis and third order model for q-axis. Figures 813 show
the results. The actual and per-unit values for the parameters of
equivalent circuits are shown in Table 7. The reactances and time
constants of machine are shown in Table 8.
STUDENTS FEEDBACK
The methodology described in the present paper has explained for
20 senior undergraduate students in power system group and 20
Table 5 Values of Parameters That Are Computed From Transfer
Functions of Machine
Rd
0.0014
Ohm
Rq
0.0017
Ohm
Ld(0)
0.0077
Henry
Lq(0)
0.0073
Henry
Lad
0.00728
Henry
lim[sG(s)/s]
0.3407
1/Henry
lim[Zaf0/s]
0.0915
sec
Nafd
12.56
Rfd
0.00255
Ohm
Table 6
Initial Values for Computation
Parameter
Value
Unit
Parameter
Value
Unit
Xd
Xd0
Xd00
Xd000
Xq
Xq0
Xq00
Xq000
Td0
Td00
Td000
Tq0
Tq00
Tq000
2.29
0.28
0.19
0.15
2.12
0.39
0.19
0.12
0.88
0.021
0.02
0.15
0.021
0.01
pu
pu
pu
pu
pu
pu
pu
pu
s
s
s
s
s
s
Lfd
Rd1
Ld1
Rd2
Ld2
Lc1
Lc2
Rq1
Lq1
Rq2
Lq2
Rq3
Lq3
0.00083
0.0504
0.00080
0.0365
0.00053
0
0
0.0119
0.00142
0.0414
0.00057
0.0334
0.00014
Henry
Ohm
Henry
Ohm
Henry
Henry
Henry
Ohm
Henry
Ohm
Henry
Ohm
Henry
Figure 10 The actual and fitted curves for sG(s) amplitude (second
order model).
8
VAHIDI AND TAVAKOLI
Figure 11
model).
Figure 12
model).
The actual and fitted curves for sG(s) angle (second order
The actual and fitted curves for Lq(s) magnitude (third order
Table 7
Actual and Per-Unit Values for Equivalent Circuits
Parameter
Actual value
d-axis
Rfd
Lfd
Lc2
Lc1
Rd1
Ld1
Ladu (test)
Ladu (modified)
Ll
Ra
q-axis
Rq3
Lq3
Rq2
Lq2
Rq1
Lq1
Laqu (test)
Laqu (modified)
Ll
Ra
Per-unit value
0.00123
0.0006
0
0
0.0720
0.0009
0.0073
0.0087
0.00039
0.00141
Ohm
Henry
Henry
Henry
Ohm
Henry
Henry
Henry
Henry
Ohm
0.00096
0.1463
0
0
0.0559
0.2194
1.7821
2.1266
0.0950
0.0011
0.05010
0.0019
0.07868
0.00075
0.01669
0.0015
0.0069
0.0082
0.00039
0.00141
Ohm
Henry
Ohm
Henry
Ohm
Henry
Henry
Henry
Henry
Ohm
0.0389
0.4580
0.0610
0.1822
0.0129
0.3798
1.6846
2.0103
0.0950
0.0011
junior power system engineers. All the senior under graduate
students and junior engineers have passed electric machine
courses (I, II, and III). They used the methodology and filled a
questionnaire form. The questionnaire consisted of four questions
shown in Table 9. The students and engineers graded them as
1 (poor), 2 (not much), 3 (good), and 4 (very good). Figure 14
shows the global results obtained from the students’ questionnaire. Figure 15 shows the global results obtained from the
engineers’ questionnaire.
Table 10 shows the average scores for each question out of
students’ feedback.
Table 11 shows the average scores for each question out of
engineers’ feedback.
CONCLUSION
Present paper has outlined and illustrated an approach to derive
dynamic model and equivalent circuit parameters of machine.
The test results are manipulated in MATLAB and a stand-alone
Table 8
Figure 13
model).
The actual and fitted curves for Lq(s) angle (third order
Reactances and Time Constants of Machine
Parameter
Value
Unit
Xd0
Xd00
Xd0
Td0
Xd
Xd000
Tq000
Xq00
Tq00
Xq0
Tq0
Xq
0.1793
0.0157
0.2318
0.7882
2.2216
0.1876
0.0418
0.2110
0.0133
0.4145
0.1156
2.1053
pu
s
pu
s
pu
pu
s
pu
s
pu
s
pu
STANDSTILL FREQUENCY RESPONSE TEST
Table 9 Questionnaire Answered by the Students and Engineers
Questions
Score
1. I had previous knowledge of Microsoft Excel Macro utility
2. The content of this practical is valuable for an engineer
3. Are you understanding the concept of machine dynamic
model better after using this utility
4. Are you more familiar with the test after using this utility
9
utility which is written in Microsoft Excel Macro environment.
By Using this method, with very low cost and effort, the SSFR
test and principles of parameter derivation of dynamic model of
synchronous generator can be taught to senior students. It can also
serve as useful preparatory exercises for power system junior
engineers in this field.
Evaluation of the project involving 20 students and 20 junior
engineers indicates benefits of this project for learning and
mastering the subject.
NOMENCLATURE
Efd
f
fn
G(0)
gd
gq
Id
Ifd
Ifdo
Figure 14
Answers of students to the questionnaire.
Iq
Lad
Laq
Lbase
Ld(0)
Ld(s)
Ll
Lq(0)
Lq(s)
Na
Nfd
Nfda
ra
Rbase
Rd
Figure 15
Answers of engineers’ to the questionnaire.
Table 10 Average Score Obtained From Students’ Answers
Average score
Question
Question
Question
Question
Total
1
2
3
4
3.15
2.85
3.05
2.75
2.95
Table 11 Average Score Obtained From Answers of Engineers
Average score
Question
Question
Question
Question
Total
1
2
3
4
2.85
3.4
2.95
3.05
3.06
rd1
rd2
Rfd
rfd
rfd meas
rfd mod
Rq
rq1
rq2
rq3
s
sG(s)
Sn
Td0
Tq0
Td000
Tq000
Td00
Tq00
T1d, T2d, T3d, T4d
T1q, T2q, T3q, T4q
field voltage
frequency
nominal frequency
current transfer function at zero Hz
d-axis connection configuration factor
q-axis connection configuration factor
current flowing in d-axis
field current
field current to produce nominal voltage at air
gap line of no load curve
current flowing in q-axis
d-axis magnetizing inductance
q-axis magnetizing inductance
armature base inductance
d-axis inductance at zero Hz
d-axis inductance function
armature leakage inductance
q-axis inductance at zero Hz
q-axis inductance function
armature number of turns
field number of turns
field to armature turns ratio
stator resistance
armature base resistance
armature resistance extracted from d-axis
impedance
first damper resistance in d-axis
second damper resistance in d-axis
field resistance calculated from test results
field resistance
measured field resistance
corrected field resistance
armature resistance extracted from q-axis
impedance
first damper resistance in q-axis
second damper resistance in q-axis
third damper resistance in q-axis
Laplace transform parameter
current transfer function of field to armature
nominal apparent power
d-axis transient short circuit time constant
q-axis transient short circuit time constant
d-axis \sub-subtransient short circuit time
constant
q-axis sub-subtransient short circuit time constant
d-axis subtransient short circuit time constant
q-axis subtransient short circuit time constant
time constants in d-axis inductance function
time constants in d-axis inductance function
10
VAHIDI AND TAVAKOLI
Tcoef
Tg1, Tg2, Tg3
Top
Ttest
Vd
Vn
Vq
o
on
Xd0
Xq0
Xd000
Xq000
Xd00
Xq00
Xad
Xadu
Xaq
Xc1, Xc2
Xd
Xd1
Xd2
Xfd
XL
Xq
Xq1
Xq2
Xq3
Zaf0(s)
Zd(s)
Zq(s)
temperature correction factor
time constants in current transfer function
winding temperature at operation
winding temperature at test condition
voltage across d-axis
nominal voltage
voltage across q-axis
angular speed
nominal angular frequency
d-axis transient reactance
q-axis transient reactance
d-axis sub-subtransient reactance
q-axis sub-subtransient reactance
d-axis subtransient \,\,reactance
q-axis subtransient reactance
magnetizing reactance in d-axis
d-axis corrected unsaturated magnetizing
reactance
magnetizing reactance in q-axis
Canay reactances
d-axis reactance
first damper reactance in d-axis
second damper reactance in d-axis
field reactance
armature leakage reactance
q-axis reactance
first damper reactance in q-axis
second damper reactance in q-axis
third damper reactance in q-axis
armature to field transfer impedance function
d-axis impedance function
q-axis impedance function
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BIOGRAPHIES
Behrooz Vahidi (M’ 2000, SM’ 2004) was
born in Abadan, Iran in 1953. He received the
BS in electrical engineering from Sharif
University of Technology, Tehran, Iran in
1980 and the MS degree in electrical engineering from Amirkabir University of Technology,
Tehran, Iran in 1989. He also received his PhD
in electrical engineering from UMIST, Manchester, UK in 1997. From 1980 to 1986 he
worked in the field of high voltage in industry
as chief engineer. From 1989 to the present he has been with the
department of electrical engineering of Amirkabir University of
Technology where he is now a professor. His main fields of research
are high voltage, electrical insulation, power system transient,
lightning protection, and pulse power technology. He has authored
and co-authored 170 papers and five books on high voltage
engineering and power system.
Mohammad Reza Bank Tavakoli was born in
Kerman, Iran, in 1981. He received the BS in
electrical engineering from Tehran University,
Tehran, Iran, in 2003 and MS degree in
electrical engineering from Amirkabir University of Technology, in 2005. Presently, he is a
PhD student at the department of electrical
engineering of Amirkabir University of Technology. His main fields of research are power
system dynamics and component modeling.