IEEE Transactions on Energy Conversion, Vol. 12, No. 3, September 1997
200
On-Line Estimation of Variable Parameters of Synchronous Machines
sing a Novel Adaptive Algorithm - Estimation and Experimental Verification
*Longya Xu
Senior Member, IEEE
**Zhengming Zhao
Member, TEEE
*Department of Electrical Engineering
The Ohio State University
Columbus, Ohio 43210
**Jianguo Jiang
**Department of Electrical Engineering
Tsinghua University
Beijing 100084, P. R. China
Abstract- On-line estimation of variable parameters of synchronous
machines based on a novel adaptive algorithm is presented. The
estimation process involves instantaneous parameter tracking and
subsequent data processing. The variable parameters are ultimately
expressed as nonlinear functions of operating conditions. On-line
estimation process is applied to a IOOMVA turbogenerator. While
this paper deals with application of the on-line estimation method and
experimental verification, principle discussion and development of
the novel adaptive algorithm are detailed in a companion paper.
1. Introduction
For the last two decades, parameter estimation in frequency or time
domain has improved considerably for calculating nodinear behavior
applications.
It is well known that SM parameters are significant& dependent on
saturation levels, internal magnetic field distributions 'and variations,
and rotor speed changes. For example, synchrono?s reactance is
closely related to the level of airgap flux; leakage dactance can be
heavily influenced by armature current causing local saturation and
distorted flux distribution; and damper resistance can substantially
altered by rotor speed with respect to the rotating flux due to the
induced eddy current in rotor iron. It is for the above keasons the SM
parameters should be more accurately termed variable parameters. It
is apparent that in a SM circuit model, the paramiters should be
continuously adapted to the operating conditions, othkrwise, reliable
t
calculation results can not be guaranteed.
Recently, considerable attention has been given to the possibility of
estimating SM parameters as functions of operating cbnditions [5,6].
Around this issue, there are three major perplexing droblems: 1) to
properly select the model to be identified, that is, td determine the
order of the governing equations of the SM modkl that will be
consistent with the operating conditions; 2) to find an optimal
adaptive algorithm to track the parameter variation, t h h is, to develop
an effective algorithm to track the parameters lpromptly and
be
properly in calculating nonlinear SM performance in a practical
operating situation.
The f i s t problem listed above was well addressed in [7]. This
paper and its companion paper [8] contribute to solving the remaining
two problems. In organizing this paper, initially, a set of discrete,
eighth order incremental equations are derived as the SM model.
Then, a brief description is given of the novel adaptive algorithm and
its application for variable parameter estimation of a SM. The stepby-step implementation of the novel adaptive algorithm is as follows:
* The actual parameters of a lOOMVA turbogenerator are estimated
on line with a perturbation of the field excitation voltage.
A regression method is used to find mathematic expressions for the
variable parameters as functions of ,operating conditions (including
equivalent airgap EMF, armature current, and rotor slip speed).
In order to demonstrate the effective application of the estimated
variable parameters to power systems, the estimated parameters are
used to predict behavior of the lOOMVA generator connected to a
grid the following steps were taken:
Dynamic simulations were done using the estimated parameters and
the designed nominal parameters, respectively, for a new operating
condition.
These simulated responses are compared with the experimental
results.
Finally, the paper ends with a brief summary and conclusions.
2. Identification Model
A systematic approach has been presented in [7] to determine the
order of a SM model, consistent with the operation conditions.
According to this approach, a dynamic model with one damper circuit
in d-axis and two damper circuits in q-axis (in rotor reference frame)
is appropriate as the standard SM identification model. This model is
adopted. The SM equivalent circuits are shown in Fig. 1.
Xdl
xfdl
Xfl
Fig. 1 SM equivalent circuits
for presentation at the 1996 IEEEPES
August 1, 1996, in Denver, Colorado
1995, made available for printing May 10, 1996.
I
I
where Vd and v, are the stator d- and q-axis voltages; x,d and x, are
the d- and q-axis synchronous reactance; xdl,+ X xfl, xldl,
X2,1,
R,, Rfd, Rid, RI,, and R2, are leakage reactance and resistance; and
xfd is the mutual leakage reactance between the field and d-axis
damper.
A. Basic SM State and Incremental Equations
0885-8969/97/$10.00
0 1996 IEEE
Electrical transients of a typical (round or salient rotor) SM can be
described by the following sixth order electrical equation in a matrix
form:
V=ZpI+RI
(1)
where, V is a voltage vector as the input, p the derivative operator, I a
current vector as the output, and Z and R are inductance and
resistance matrices, containing the basic parameters of the SM model.
Explicitly,
v
[ v q , vd. 0, 0, vfd. 0 1
I = t iq. 41.iiq, i2q. ifd, i i d 1 T
(2)
(3)
(4)
B1 =
where v f d is the field voltage; id, ifd and ild are the stator d-axis, the
field and the rotor d-axis damper currents; iq, ilq and i2q are the stator
q-axis, the first q-axis damper and the second q-axis damper
currents. The reactance parameters follow the conventional equations
as: xd = Xad + xdl; xq = xaq + xq1; xllq = xaq + Xlql; X q = xaq + X%1;
xffd = &d xfl; X l l d = %ad + XI&; and Xfld = xad Xfd.
= -xd ido + Xad ifdo
In Eq.(9), AT = T, - T, and in Eq. (12) &x
(id0 and ifdo are the initial values of id and ifd before a transient).
Expressing Eq. (8) in a standard state variable matrix form, it follows
that
P A X = A2AX + B2 AV
(13)
TWOadditional equations are used to describe the SM mechanical
motion as
where A2 = -Aiv1 B1 and B2 = A1-l
B. Discrete Difference Equation
(7)
where Tm is the mechanical torque, Te the electromagnetic torque, w
the angular velocity of the rotor, W O the synchronous velocity, and 6
power angle; H and D are the inertia constant and mechanical
damping coefficient, respectively. Note that o and 00 in the unit of
electrical radians per second; 6 is in electrical radians.
In order to emphasize the parameter variation during transient, a
set of incremental equations are derived based on Eqs. (1) through
(7). Neglecting the high order terms and viewing the parameters as
constants within each incremental interval, a set of incremental
equations are derived
To facilitate adaptive parameter estimation with a digital system, it
is necessary to transform the continuous state variable model of Eq.
(13) into a discrete difference equation form as follows
X (k)= A3 X(k -1) + B 3 V(k)
xi
(14)
m0
where A3 =
(A2 .At)i
i=l
(mo=1,2, ...)
(15)
202
In fiqL(14) and (15), "k" is the instant of concern and At the discrete
time-sthat.the following two equations also hold.
A2=MCM-l
(17)
B2=(Ag-I)-'AzBj
(18)
where M is a matrix consisting of eigenvectors of A3, and C a
diagonal matrix consisting of eigenvalues of A3. Eqs. (17) and (18)
are used when A2 and B2 are computed from matrices A3 and B3
which are made available from estimation process.
3. Adaptive Estimation Algorithm
A block diagram of the proposed adaptive parameter estimation
configuration is shown in Fg. 2. The novel synthesized information
factor (SIF) adaptive estimation algorithm, discussed in detail in the
companion paper, is used.
Fig. 2 Block diagram of adaptive estimation configuration
The basic concept of the SIF adaptive algorithm is that in processing
the data (measured and observed currents) within a sliding window, a
sequence of SIFs are used as the weight factors. Since SIFs are
obtained by multiplying the correlating factors h(k,i) (characterizing
correlation between two data sampled at different instants) by a
proper forgetting factor hk-l (characterizing the degree of forgetting),
the SIFs efficiently synthesize information contained in previous data
with that of the present data. This synthesized information then is
used as fhe driving force for the adaptive mechanism. Essentially,
SIFs attempt to establish an optimal and balanced relationship
between the history and current status of a continuous event.
Mathematical derivation, physics explanation, and application of SIFs
are to be found in the companion paper [SI.
4
4. On-Line Estimation
Following the principles and estimation process based on the novel
adaptive algorithm, presented in [SI, an on-line estimation was
carried out on a lOOMVA turbogenerator. The estimation system
diagram is shown in Fig. 3
1
2
3
4
I
1 -field winding of excirer, 2 - exciter, 3 -field winding of generator,
4 - generator, PT - voltage transformer, CT - current tran.$ormer, L variable inductor, K - switch, AVR - automatic voltage regulator
Fig. 3
System diagram of estimation tests
In order to create a sufficient perturbation so as to excite the inner
modes of the SM and, at the same time, not affect the synchronous
generator normal operation, a field excitation change shown in Fig. 3
is adopted. Before perturbation, switch K is open and the machine is
in steady state. Then, K is closed rapidly to create a sudden change in
excitation voltage. This action produces a reactive power
perturbation. About 30% step change of Vfd is achieved on the
generator by this method. Immediately following the field voltage
perturbation, the transient variables, including vab, vbc, i, ib, ic, ifd,
vfd and the rotor angle 6 are sampled and recorded by a PC data
acquisition system. Then, the adaptive algorithm starts to estimate
the parameter matrices A3 and B3. In the process, two steps are
involved: 1) a self-leaming is carried out to obtain starting values of
the estimated parameters; and 2) the adaptive estimation is executed
to track the trajectories of the variable parameters. Prior to the
system perturbation, the self-learning algorithm uses the design
parameters as the initial conditions to estimate the parameters in
steady state. The results of self-leaming algorithm are then used as
the starting points for the variable parameter estimation. In the
second step, the transient currents, voltages, power angle, and rotor
speed obtained from the test and observer are used to form the
inpuvoutput data. Then, solving related equations defined in the
adaptive algorithm, the trajectories of parameters during the transient
are obtained. The measured transient vfd and 6 are shown in
FigsA(a) and (b).
180
'
0
1
2 3 4 5 6
7 8
(a) Measured field voltage Vfd
9101112
203
1
'
0.15
0.00
0
7 8
(b) Measured power angle 6
1
2
4 5
3
6
1
l
1
1
1
1
l
1
1
1
1
1
2
3
4
5
(d) Estimated variable field parameter R f d
0
9101112
Fig. 4 Transient vfd and power angle 6
. . . . . . . . . .
2*al 7
-1
estimated SM variable parameters corresponding
- to the
transient are shown in Figs. 5(a) irough (g).
h
.- . . . . . . . . . . . -.
q 1.6 '.**--. "d
?
_-_.---------.---------,a 1.2 3
H 0.8 ,"0.4 Time (sec.)
6
208
2.0
1i
'b.
0.52
ld
0.0
1
1
1
1
1
1
1
1
1
~
0.00
0
1
2
3
4
(e) Estimated variable damper parameters
~
.
1.28 a
6 O.% 8 0.64 Fc
0.32 -
5
6
RI,
1.60
?
)
Time(sec.)
0.00
0
1
2
3
4
5
(f) Estimated variable damper parameter R z ~
Time (sec.)
3
4
5
(b) Estimated variable parameters xq and xaq
0
1
2
0.io
6
F
7
.
.
6
. . . . . . .
'1
0.08 -
0.10
?, 0.06
1
0.00
0
1
1
1
2
1
1
3
1
4
1
l
5
~
6
(g) Estimated variable damper parameter R l d
Fig. 5 Estimated variable parameters
0.02
0.00
0
3
4
5
(c) Estimated variable armature parameter R,
1
2
6
As seen from the above figures, in a negative step change of field
voltage, Xd, xq and Xaq show small changes, but %d shows a larger
change. Note that since the armature leakage reactance Q = Q -$d,
the leakage reactance in d-axis has changed substantially. The large
change of the armature leakage reactance can be understood by
observing the intemal flux distribution during the transient. That is,
the transient currents tend to alter the flux path in such a way that
~
~
204
more flux lines go through the air instead of through the iron. This
phenomenon is more apparent along the d-axis since in steady state
nearly all flux lines go through the iron core. As soon as the
transients start, part of flux lines are pushed out of the iron core into
the air, resulting in an increased xs.
Another important feature of the estimation results is that the daxis resistances,Ra, Rfd, and R i d , show large changes, initially
increasing and then decreasing (impulse change). The damper
resistances in q-axis, R I q and R2* are also in a large change initially
and then stay larger (step change). All these changes correspond to
the change in power angle.
The large changes (about eight times the initial value) of the
winding resistances can be explained as follows. When a step change
(30%) of excitation voltage is imposed to the field, substantial
currents are induced in all windings coupled to the field. These
currents are in such a direction and rate that they are firmly against
the attempted change of the magnetic field. Therefore, the di/dt of
the induced currents is very high, proportional to that of the perturbed
field voltage. A detailed spectrum analysis [7] shows that the
equivalent frequency of the rapid perturbation is more than 500 Hz,
starting from 60 Hz. Considering the strong skin effects created by
the high frequency currents through the windings, it is not surprising
to see that resistances of all windings increase dramatically. Simple
calculation also shows that resistance, taking skin-effect into account,
increases approximately proportional to the increase of operation
frequency. This explains well that when the frequency of the
magnetic field and the induced currents vary by 8-10 times, the
resistance correspondingly vary by 8-10 times. It is interesting to
observe that when the transient and sub-transient end, the stator and
field resistances return to their steady-state values. However, the
damper resistances keep their transient values. In fact, when the
transient ends, the damper resistance in steady state is no longer
estimated by this algorithm. Fortunately, it is not significant to know
the damper resistance since before and after any transients (i.e. in
steady state), there is no currents passing through the equivalent
damper winding.
Several similar transient testing and parameter estimations have
been made at different steady-state operation points and similar
results have been obtained. These results all show that the reactances
in the d-axis and the resistances (i.e. R,, Rfd, Rid, RI¶, and R q ) are
very sensitive to the transients of a SM.
functions, accounting for saturation, temperature variation, and eddy
current effects in transient. The parameter functions are
approximated by polynomials:
ml
m2
z =Q+ a0 + C a l j A E +~
j=l
j=l
~
m3
...A i + c a 3 j AJ
j=l
where m i , m2 and m3 are the maximum order of each operating
argument (i.e. AE& AI, and Am), which are automatically determined
in regression procedure and depend on the behavior of each argument
in transient; and (Q, a l j , a2j, a3j) are the coefficients to be
determined by the regression method. Thus the results, a group of
parameter functions (include Xd, Xad, $, X q Xffd, x i i d , Xllq. X22q,
Ra, Rfd, Rid, R I q and Rzq), are found by using the regression
method. For example, Xad and Xaq are found as functions of
operating arguments are:
Xad = XadO + 0.031 - 2.134
- 41.7 &g2+ 549.3
- 14O1AEs4- 5.97AI + 37.9A12 - 103.7 AI3 + 52.6 Aw
(21)
xaq= Xaqo + O.OOO9 - 1.5 AE6 + 5.03 L\E62 + 0.44 AI
- 0.27A12
+ 1.68 Aw
(22)
where XadO and XaqO are the steady-state components. The other
parameter expressions are also obtained in the same manner, but not
listed to save space. Note that the parameter functions may also be
expressed in other formats for convenience, depending on the
application needs. The coefficients in Equation (20) are derived from
three excitation perturbation transients with different steady-state
operating points (1 .OPn, 0.7Pn and OSP,). The perturbation
magnitudes are 25%, 30%. and 40% of Vfd, which covers a large
range of operating conditions. The selection of a step excitation
change as the perturbation for the adaptive estimation is based on the
fact that without disturbing the normal operation of the power
system, this perturbation can stimulate all modes of the dynamic
characteristics of a SM.
B. Experimental Verification
5. Parameters Function and Experimental Verification
The ultimate goal of parameter estimation is to establish an
accurate dynamic model based on estimated variable parameters for
predicting SM behavior in a power system. To this end, parameter
trajectories were estimated in three operating conditions l.OPn, O.7Pn
and 0.5Pn (P, is the rated output power), and are formulated into
parameter functions corresponding to the operating conditions.
A. Parameter Functions
Without loss of generality, the estimated variable parameters are
divided into two parts: the fist part concerns steady state operating
condition and the second transient operating condition. Both parts
are functions of operating conditions (V, I, cos@,Ifd. 6, 0 and their
increments):
Z=Z,+Zt
(19)
For simplicity, only the airgap voltage AEg, armature current AI, and
slip speed A O are employed as the arguments of the parameter
To validate the SIF adaptive algorithm and the parameter
functions, another transient test with a step change in field voltage is
carried out for the same turbogenerator with operating conditions
shown in Table 1.
Table 1: Operating conditions before and after transient
205
The transient outputs of the turbogenerator (id, i ifd, and 6) are
9'
compared with those from computer simulation using the parameter
functions estimated. The results are shown in Fig. 6.
. . . . . . . . . . .
1
o.0517
0.00
1
'?1
-0.10
0
1
2
3
4
5
-0.15
1
0.20
-0.20
0
2
1
c
' 0.15 -
3
5
4
6
-
0.20
6
Simulated
2? 0.10 -
.4
d 0.05
Time (sec.)
-0.05
1
0
2
.
1
1
0
;
1
1
1
2
1
1
3
1
1
1
1
5
4
6
0
?
?
a
v
z
zl
0
0.5
3
2
1
0
5
4
6
I-
4
-
-
a 0.4
E
0.3
24
a3
0.2 -
~
CO
a
0.1
3
4
5
. . . . . . . . . .
6
'1
Time (sec.)
1
0
1
2
-
-
0.0
0.5
1
1
1
1
1
2
1
1
3
1
1
4
1
1
5
,
0
1
2
3
4
5
6
6
Fig. 6 Dynamic responses from experimental testing and from
computer simulation with estimated parameters
It is very encouraging to observe that once the parameter functions
of a SM is obtained, the dynamic behavior of a SM can be predicted
quite accurately by computer simulations. As a c o m P ~ ~ s othe
n~
results from simulations using designed values of the parameters are
presented against the actual responses from the tests in Fig. 7. The
Parmeters to Predict
behavior
~ S ~ Uin rusing
i ~the
of a SM by computer simulation is obvious from these results.
Fig. 7 Dynamic responses from experimental testing and from
computer simulation with designed parameters
6. ~~~~~~~i~~~
The trajectories of the variable parameters of a SM under step
excitation change are first shown. T~~ of the three
problems concerning variable
identification of S M ~ ,
namely, tracking the trajectories of the variable parameters according
to operating conditions and Fmding the proper parameter functions for
applications, are the target of this paper. Based on the concept of SIF
206
(optimally combining forgetting with memorizing effects), a new
adaptive estimation algorithm is developed to obtain the trajectories
of variable parameters. The parameter functions can be found in
terms of the trajectories by a conventional regression method. The
parameters obtained by this new method have been successfully
applied to predict transient behavior of an actual lOOMW
turbogenerator. The output responses of simulation employing
parameter functions are in a very good agreement with the
experimental testing of the actual SM. At present, the on-line
adaptive estimation method is favorably accepted by utility
companies for power system stability calculations and analysis in PR
China. Since the estimated parameters are based on the general SM
model of eighth order, other dynamic behavior of the SM, such as the
transient after a sudden load change, can also be predicted
satisfactorily. It is recommended that the estimated SM parameters
be used as an unseparated group in computer simulation in a properly
selected SM model structure (an eighth order model in this paper) for
accurate prediction. This is because the SIF adaptive estimation
algorithm is conducted at the system level and generates a set of
estimated parameters for a completed description of the SM.
Therefore, a mixture of any individual estimated parameters obtained
by SIF method with those obtained by other methods may not
produce meaningful results if it is used in a computer simulation.
The authors believe that the application of the adaptive estimation
algorithm described in the paper is just a beginning, and more
applications such as real time control of a SM can be processed. The
results of combining the on-line adaptive estimation with real-time
adaptive control will be presented in future papers.
References
P. L. Dandeno, D. H. Baker, et al., "Current usage and
suggested practices in power system stability simulations for
synchronous machines", LEEE Trans. on EC-1, No. 1, pp. 7793,1986
E. Eifelberg and R. G. Harley, "Estimating synchronous
machine electrical parameters from frequency response tests",
IEEE Trans. on EC-2, No. 1, pp. 125-132,1987
Longya Xu was bom in Hunan, China. He graduated from Shangtan
Institute of Electrical Engineering in 1970. He received the B.E.E.
from Hunan University, China, in 1982, and M.S. and Ph.D. from the
University of Wisconsin-Madison, in 1986 and 1990 both in
Electrical Engineering.
From 1982-1984, he worked as a researcher for linear electric
machines in the Institute of Electrical Engineering, Sinica Academia
of China. Since he came to the U.S., he has served as a consultant to
several industry companies including Raytheon Co., US Wind Power
Co., Pacific Scientific Co., and Unique Mobility Inc. for various
industrial concerns. He joined the Department of Electrical
Engineering at the Ohio State University in 1990, where he is
presently an Associate Professor. Dr. Xu received the 1990 First
Prize Paper Award in the Industry Drive Committee, IEEEDAS. In
1991, Dr.Xu won a Research Initiation Award from National Science
Foundation for his research project "A High-efficiency, Low-cost
Flexible Variable Speed Wind Power Generating System." His
research and teaching interests include dynamic modeling and
converter optimized design of electrical machines and power
converters for variable speed generating and drive system. He is an
active IEEE member, currently serving as the Secretary of Electric
Machine Committee of IEEE/IAS and an Associate Editor of IEEE
Transaction on Power Electronics.
Zhengming Zhao received the B.S. and M.S.degrees from Hunan
University,PR China in 1982 and 1985 respectively, the Ph.D. degree
from Tsinghua University, PR China in 1991, all in electrical
engineering.
He worked in the Dept. of Electrical Engineering, Hunan
University during 1985-1987 as a lecturer. He is an Associate
Professor in Tsinghua University since 1993 and currently on leave in
the Department of Electrical Engineering, The Ohio State University
as visiting scholar. His areas of interest include parameter
identification and signal processing, transient analysis of power
system, and design, analysis and control of electric machines.
L. X. Lee and W. J. Wilson, "Synchronous machine parameter
identification: a time domain approach", IEEE Trans. on EC-3,
N0.2, pp. 241-248, 1988
Jianguo Jiang received B.S. and M.S. degrees from Tsinghua
Z. M. Zhao, F. S.Zheng, J. D. Gao and L. Y. Xu, "A dynamic
on-line parameter identification and full-scale system
experimental verification for large synchronous machines",
IEEE Trans. on EC-10, No.3, pp. 392-398, Sept. 1995
C. X. Mao, J. Tan, 0. P. Malik and 6. S . Hope, "Studies of
real-time adaptive optimal excitation control of synchronous
generator and power system stabilizer", LEEE Trans. on EC-7,
No. 3. pp. 598-605, Sept. 1992
From 1965-1988, he worked in the Dept. of Electrical
Engineering of Tsinghua University as a lecturer and Associate
Professor. In 1989, he was a visiting scholar at the Toronto
University in Canada, working on fault diagnosis of electric machines
and digital signal processing. He is currently a Professor in the Dept.
of Electrical Engineering of Tsinghua University. His areas of
interest include machine fault diagnosis and dynamic analysis, signal
processing, design and control of electric machines.
A. M. Farhoud, et al., "Adaptive enhancement of synchronous
generator stabilizer performance using parameter optimization
technique", IEEE IAS, Vol. 1 1993, 147-154
Z. M. Zhao and F. S . Zheng, "Model structure identification
with SVD for synchronous machines", Proceeding of
International conference on EMASM, Zurich, Switzerland, pp.
86-89, August 1991
Z. M. Zhao, L. Y. Xu, and J. G. Jiang, "On-.line estimating
variable parameters of a synchronous machine by a novel
adaptive algorithm -- principles and procedures". IEEE PES
Summer Meeting, July 1996
University in 1961 and 1965, respectively, all in Electrical
Engineering.
207
Discussion
[This is a combined discussion of the companion papers,
“On-Line Estimation of Variable Parameters of Synchronous
Machines Using a Novel Adaptive Algorithm - Principles
and Procedures” (96 SM 356-6) by Z. Zhao, L. Xu, and J.
Jiang and “On-Line Estimation of Variable Parameters of
Synchronous Machines Using a Novel Adaptive Algorithm Estimation and Experimental Verification” (96 SM 355-8 EC)
by L. Xu, Z. Zhao, and J. Jiang.]
I. Kamwa (Hqdo-Qudbec, IREQ, Varennes, Qudbec, Canah J3X ISI).
and A. Keyhani (The Ohio State University, Lkpartment of Electrical
Engineering, Columbus, OH 43210). The authors have used test data for
confirming their proposed parameter estimation approach. Nevertheless,
their two companion papers rise a number of questions both from the
theoretical view point and practical use of their scheme. We will discuss
these issues in the following:
I- Development of the Novel Adaptive Algorithm (SM96 356-6 EC)
El) What the authors have rederived is just yet another form of the
recursive weighted least-squares previously published in well known
scholar books [A-C]. Hence, using a more conventional formulation, let
denote by S, the j-th parameter vector, fi the corresponding measurement
vector, A, the observation matrix and W, a suitable diagonal weighting
matrix.The fundamental equation (9) of SM96 356-6 EC then takes the
form:
This is exactly the equation (4.10) on p. 52 in [A]. Replacing W, in (A)
with the correlation matrix r,given in eq. 13 of SM96 356-6 EC
obviously yields a specific version of the batch weighted least-squares
algorithm, which can be made recursive in a straightforward manner as
outlined in [A] (p. 52-58). In the same reference, the resulting algorithm
was called sequential weighted least-squares (WLS), and the proof for
its convergence was rather complete. The authors should clearly state
what is novel in their algorithm, comparatively to the conventional
sequential WLS with a forgetting factor. If the time-varying weighting
matrix TJ(k) is the only significant difference, then the rather clumsy
demonstration extending from eq. (16) to (30), and including theorems 1
and 2 in appendix was irrelevant for a Transactions paper, just because
more readable proofs are easy to find [A-C].
1-2) The Algorithm development fails to show that the estimated
parameters will always be biased if measurement noise is present, as is
normally the case:
ZJ(k)= H J ( k @ , O
+nJ(k)
where Qp is the true value of the j-th parameter and n, the noise
penetrating in the j-th estimation channel. Refereeing to theorem 1 in
appendix of SM96 356-6 EC, the two criteria ensuring uniform
convergence of the algorithm to the true value Q p are: model noise n, is
(1) a zero mean-value sequence and (2) independent of the data matrix
Hi(k).
While the former condition is already difficult to ensure practically,
due to unavoidable biases in transducers for instance, the latter condition
is simply never verified by the proposed method: since HF) always
involves some terms already present in H,&-I) and hence, dependent upon
n f i - l ) , it follows that the two sequences HF) and nfi) are necessarily
correlated with each other. This result is well known [A,F] for ARMA
models, and has motivated in the past, advanced schemes such as
Generalized Least Squares [A, p.58-611, Instrumental Variables [D] and
Recursive M a x i ” Likelihood @?I,which are statistically much more
efficient than the simplistic sequential weighted least-squares [A] used in
the papers under discussion. These discussers will appreciate if the
authors could outline a rigorous proof of the unbiaseness of their
estimation scheme, if they seriously believe that their scheme is
statistically consistent and unbiased, wen when assuming a perfectly
white measurement noise.
11- Experimental Verification of the Algorithm (SM96 355-8 EC)
11-1) According to our understanding, the authors first estimate the
discrete matrices A3 and B,, then equations (17-18) are used to find the
continuous system matrices AI and BI which contain explicitly the desired
synchronous machine parameters. Equation (17) is certainly income4
since: A3=erp(A2*T),where T is the sampling period [A]. It seems by this
reasoning that the matrix C should involved the logarithm of the
eigenvalues of A3, not these eigenvalues themselves. The authors should
check our claim and provide in their closure, suitable modifications to
~ u a t i o n s(17-18).
11-2) One significant shortcoming of the proposed identification scheme
comes from the necessity to observe indirectly the 1 1 1 state of the system.
If for instance, 6,W ,id, iq and ifd are the only state variables measured
in the field, we need to recollstruct ild, ilq and i2q indirectly. This is a
rather complicated task, which may requires a sophisticated algorithm on
its own [GI.In this context, Figure 2 in SM96 356-6 EC shows a black
box called Observer Algorithm which is not described elsewhere. More
specifically,there is no explanation of the two symbols XI and X2 used to
reconstruct the actual state vector X , which is compared to the predicted
value
in order to generate the error feedback in the adaptive identifier.
2
We feel that there are many findamental problems with the proposed
scheme since the method is based on designing an adaptive observer to
estimate the unmeasurable damper currents that are used to estimate the
damper parameters and the rest of the machine parameters recursively.
We would appreciate the authors comments on the design steps for
construction of their observer which is the key to their parameter
estimationmethcd.
11-3) In SM96 355-8 EC, attempt is made to estimate Xd and Xd,thus
trying to obtain the leakage and magnetbng reactances simultaneously:
this is known, since Canay in 1969 m, to be an ill-posed identification
problem, which violates the parsimony principle and thus giving rise to an
infinite number of admissible solutions. Hence, Xad is reduced
significantlyon Fig. 5, while the machine is evolving towards a much less
saturated state. By contrast, Xd tends to increase (the correct thing),
which means that Xd-Xad is shaped like Ra (Fig. 5). If this is true (as
explained by the authors), therefore, any currently used saturation model,
generally based on the so-called saturation “factors”, is deemed wrong I
We would like the authors to comment these observations.
11-4) It is an established fact that the states of a dynamic system is
function of time and if the system is nonlinear then the Parameters of the
system are also function of states of the system. For a synchronous
machine, the machine parameters are function of excitation, operating
temperature and saturation. Based on these observations, the authors arc
presenting the following time varying parameters from their test data due
to a transient condition given in Table 1:
1. The armature resistance Ra, is changing as a function of time from .O 1
pu at time step 1.5 sec to 0.08 pu at time step 3.0 sec. and back to .01 pu
at time step 5.0 sec. for total transient duration of 3. sec. as shown in
Fig.5 (c).
2. The trajectory of the damper parameters Rlq, R2q, and Rld are
changing in time frame of 3.0 sec. as shown in Fig. 5 (e)-&).
3. The trajectory of Xq and Xd parameters are constant for the same
operating condition.
For the case presented, the authors indicate that the machine k not
saturated and thus the parameters Xd and Xq are essentially constant and
thus the machine core losses are also constant and it will not be a factor in
operating temperature. The authors are suggesting that the change in
machine parameters are due to harmonic frequencies that are present in
the disturbance signal. One would expect that the dynamic change in
system losses (i.e. resistances) will result in corresponding change in
208
operating temperature. It is hard to believe that the machine operating
temperature would change and retum to the same value in 3.0 sec?. We
believe this is not physically possible. The harmonics present in the
perturbation signal are in quasi-steady-state (rectifier operation) and they
should not contribute to large time varying losses as presented
III- General comments (SM 96 356-6 EC & SM96 355-8 EC)
111-1) We wonder if a synchronous machine is a system so stochastic or
uncertain that it requires an adaptive identification algorithm providing
sequences of time-varying inductances, resistances and so forth. To us, the
answer is no: It is just a huge static amount of iron, cupper and other stuff
with an electrical behavior which evolves very slowly through time. The
bottom line here is non-linearity, not time-dependency. Therefore, the
suitable solution is not adaptation through time, but good functional
models of the non linearity. It is quite interesting that after all, the
authors used at various places the concept of functional non linear models
to represent the relationships between the machine parameters and its
state-variables. If equations of the type (35) and (36) in SM96 356-6 EC
or (20-22) in SM96 355-8 EC can describe the machine properly, why not
attempt to determine their static parameters directly for various operating
points, without first using an adaptive estimator to build transient
trajectories and then performing a multivariate polynomial regression? In
this context, we direct the authors' attention to a recent related work
Several questions from the discussion of R.P. Shuh et al. in applied as
well to the papers under discussion, and need therefore, further
consideration by the authors.
m
m.
111-2) Another problem with the SIF is its poor performance when the
excitation is not persistent. In such situations, the prediction residuals
(unforhmatelynot shown in the paper) cannot be white, which violates one
of the basis hypotheses require by the SIF to function properly. In fact,
the damper parameters block at arbitrarily values once the transient
Vanished (section 4, SM96 355-8 EC). The given explanation is that the
steady-state parameters of the dampers are irrelevant. Nevertheless, the
fact that they are undetermined at the end of the estimation process brings
serious concerns about the uniqueness of all the other parameters. To sum
up, the proposed method seems to provide at convergence a black-box, socalled fitting model only; not our favorite model, which uniquely descrik
the machine behavior and must be invariant whatever the test signal is, as
long as the operating domain covered remains unchanged. It will be
helpful if the authors can provide a table showing in terms of standard
dynamic constants (T'd, T"4 ...), the design parameters for the lOOMva
turbogenerator, along with initial and final parameters for one typical run
oftheir algorithm.
111-3) To finish, these discussers want to acknowledge the important
work invested by the authors of in order to perform successfuly the
extensive set of field tests reported in the two papers. Especially, their test
bed which generates an almost purely reactive transient is quite onginal.
However, we feel that a fundamental issue in modeling of dynamical
system has been ignored by the authors: Model Structural Identification,
i.e., both the linear and saturated model structure. Since the authors model
structure is fixed and the saturation is ignored, and the authors have
required match between the test data and the model, they have obtained a
match by allowing the parameters to change as a h c t i o n of time without
regard to physical requirements of the machine. Naturally, with a such
large degree of freedom, match can be obtained between the test data and
t h e -varying model parameters; however, the results are meaningless.
The confirmation of their time-varying model with the test data does not
indicatethe correctness of the proposed procedure.
R~fer~nces
[A] N.K. Sinha, B. Kmzta, Modeling and Identification of Dynamic Systems.
Van Nostrand and Reinhold, New York, 1983.
p]L. Ljung, T . Soderstrom, Theory and Practice of Recursive Identification.
The MIT Press, 1983
[C] T.C. Hsia, System Identification - Least-Squares Methods, Lexington
Books, 1977.
p]P.C. Young, "An instrumental variable method for ml-time identification
of a noisy process", Automatica, 6, 1970, pp.271-287.
[E] R Hastings-James, M.W. Sage, "Recursive Generalized-Least-Squares
Procedures for On-Line Identilication of Process Parameters," Proc. IEE,
116, December 1969, pp. 2057-2062
J. Gertler, C. Banyasz, "A Recursive (on-line) Maximum Likelihood
Identification Method,''IEEE Trans. on Automatic Control, AC-19, 1974,
pp.825-830.
[GI H. Tsai, A. Keyhani, J. Demcko, RG. farmer, "On-Line Synchronous
Machine Parameter Estimation from Small Disturbance Operating Data,"
IEEE Trans. on Energy Conv., EC-IO(I), March 1995,pp.25-36.
NI.M. Canay, "Causes of Discreapancies in calculation of Rotor Quantities
and Exact Equivalent Circuits of the Synchronous Machine," IEEE Trans.
on Power App. and Syst., PAS-88(7) July 1969, pp. 1114-1120.
m A. Keyhani and S. Miri, "Observers for Tracking of Synchronous Machine
Parameters and Detection of Inciepient Faults, " IEEE Trans. on Energy
Conv.,EC-1(2), June 1986, pp. 184-192.
Manuscript received September 3, 1996
Z. Zhao, J. Jiang (Dept. of Elec. Eng, Tsinghua
University, Beijing, 100084, China) and L. Xu (Dept. of
Elec. Eng., The Ohio State University, Columbus, OH
43220) :
The authors wish to thank the discussers for their interests
in the papers since it gives us an opportunity to highlight
our contributions to the subject. To clarify the discusser's
confusion and misunderstanding, the closure follows the
sequence of the discussion.
1-1: With regard to the comments of "rather clumsy
demonstration...", we would like to emphasize that the
form of Eq. (9) is superficially similar to a conventional
recursive weighted least-square (RLS, called WLS by the
discussers) equation, but the concept and behavior of the
weighting factor of Eq. (9) is totally different from those
used in any conventional RLS equations. Upon reviewing
the conventional RLS, we pointed out its serious
drawbacks, including overshoot, one-sided forgetting
factor, poor tracking ability, etc. It is for overcoming the
disadvantages of the conventional RLS method that we
developed the novel synthesized information factor (SIF)
based adaptive algorithm. In effect, the SIF based
algorithm includes many new items, such as the concept of
SIF which establishes an optimal and balanced relationship
between history and current status for a continuous event,
the derivation of the expression of SIF (Eqs. 2 and 3), the
appropriate selection of an sliding time window, the
application of correlation factor in the §IF algorithm, the
proper usage of a forecasting gain matrix K'(k), ... To our
best knowledge, all the aforementioned items have not
appeared on any SM parameter estimation literature. Could
discussers compare our proposed algorithm with the
mentioned books in more detail to justify their comments?
1-2: In Theorem 1 of the appendix, the discussers can find,
if they will, that the uniform convergence proof is provided
with the assumption that the model is subject to a zero
mean-value noise sequence and independent of data Hk, If
other types of noise, not white noise, are present, the SIF
will reject them adaptively because the SIF only relates the
correlated signals to each other, forgetting the others.
Actually, this is one of the salient features exclusively
associated with the SIF based algorithm, not a simplistic
sequential WLS algorithm that the discussers referred to.
209
11-1: In Eq. (17), we stated in the paper that "C" is a
diagonal matrix consisting of the eigenvalues of "A3". This
statement means that "C" is related to the eigenvalues of
of couse,
"A3". The form of the eigenvalues appear in "C",
can be either logarithm or exponential, depending on the
match of the eigenvalues with respect to the corresponding
eigenvectors in matrix "M". These are fundamental
concepts of linear algebra which, we hope, are familar to
the discussers.
11-2: In a previous work, published in 1992, we have
developed an effective adaptive observer algorithm for the
state variables which are not measurable. As we presented
during the oral discussion, the discussers are referred to [4]
for details.
11-3: For a step change of excitation voltage as described in
the paper, the relative positions between the stator MMF
and the rotor field will change (the power angle change
verifies this). Therefore, the flux distribution inside the
machine will change correspondingly. The variations of
Xd-Xad during transients are the terminal reflection of the
flux distribution change inside the machine. It should be
noted that the SIF adaptive parameter estimation is
conducted at system level and generates a set of parameters
for a completed description of the SM machine. We need to
treat the estimated parameters as a group in applications.
11-4: The discussers are obviously confused that the
changes of temperature and resistance are two different
concepts and could be relatively independent. As wellknown, temperature-rise corresponds to energy
accumulation, generally with a large time constant, while
the resistance change could be purely a result of current
distribution change which occurs almost instantaneously in
transient. Note that due to skin effect, the transient currents
flow through a cross-section which is much smaller than
that when the transient is over. Bearing these fundamentals
in mind, the discussers will not have difficulty in
understanding that the resistance could change very rapidly,
de coupled from a slow change of temperature.
111-1: The work of our two papers is an attempt to obtain a
more accurate model accounting for nonlinearities of a SM.
The variable parameters are not, as the discussers claimed,
functions of time, but functions of operating conditions
although we have used the time-domain disturbance and
responses in the process of estimation.
111-2: As discussed in 11-2, we have developed an effective
method to estimate the steady state parameters. The SIF
based algorithm proposed in the two papers is used to
obtain the parameter trajectories in transient. Combining
the transient trajectories with the steady state values, we are
able to obtain satisfactory parameter functions for a wide
range of operating conditions. As to the comparison of
estimated and designed values, we have provided a number
of examples in [3].
111-3: While using a model structure with one damper
circuit in d-axis and two dampers in q-axis, we did not
lower the importance of model structure selection. In fact,
in a related work [4], we have specifically developed a
method for model structure determination and concluded
that for different machines (with salient or round rotor),
different operating modes (three-phase or single-phase
short), and different study objectives (stability analysis or
routine monitor), a SM should be expressed by different
model structures. The model structure selection criterion is
also presented in terms of measured state variables. The
model chosen in the papers is for purpose of illustrating the
principle and effectiveness of the SIF based adaptive
algorithm.
As for saturation effects, at least two methods are
equally well suited: one is from the model structure point of
view, and the other from parameter adaptation point of
view. Parameter estimation belongs to the latter. Since the
estimated parameters are based on the measured inputs and
output variables, saturation and other nonlinearities are
automatically included.
While questioning the
experimental verification for their "meaningful" and
"correct" results [5-161, we feel particularly confident about
the approach of our experimental verification presented in
the paper: a SM at lOOMVA operated in a full-scale large
power system is used and consistent results are obtained.
Closing Remarks
After clearing up the above confusion and
misunderstandings raised by the discussers, we feel that the
rest of the discussion is full of biased and unsubstantiated
comments, much reflecting the discussers' resent upon the
work not fitting to their tastes. Regretfully, we are not
impressed. At this point, we would like to share a point of
view in general with the discussers: any SM models thus
far with lumped circuit parameters is but a subjective
understanding of the physics of an objective SM. Neither
the discussers nor the authors have and will exhaust
abundant research topics and stop enthusiastic exploration
for the SM, such a wonderful piece of art. Comparing with
the work in [5-161, we feel encouraged by the
accomplished results which are not a trivial repetition,
slight variation of one from another, or supported by testing
on toy-size machines. We welcome any serious,
professional discussion. Nevertheless, we will not give up
our efforts to pursue the truth in the area of our interests
because of unwarranted claims.
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