1
Damping Torque Estimation and Oscillatory
Stability Margin Prediction
Hassan Ghasemi and Claudio Cañizares
Department of Electrical and Computer Engineering
University of Watreloo, Waterloo, ON N2L 3G1, Canada
Emails: hghasemi@uwaterloo.ca and ccanizar@uwaterloo.ca
Abstract— The damping torque of linearized models of power
systems is studied here as a possible on-line security index,
based on system identification techniques applied to realistic
measurements. First, the theoretical values of damping and synchronizing coefficients of the electromagnetic torque are discussed
in detail. These values are then used to investigate the accuracy of
damping coefficient identified from on-line measurements using
the ordinary least square (OLS) method. It is demonstrated that
OLS may not be able to correctly estimate the coefficients due
to the nonlinear nature of power system oscillations. Hence,
generalized least square (GLS) and robust fitting with bisquare
weights (RFBW) are applied to this system identification problem,
showing to be better alternatives. Based on these results, the
damping coefficient is proposed and studied as an index to
calculate the distance to the closest oscillatory instability point.
The results obtained from 3 test cases show that the index is
an effective tool, and can be of significant help to operators for
on-line security monitoring of power systems.
Index Terms— Damping torque, system identification, oscillatory stability, stability indices, bifurcations.
I. I NTRODUCTION
On-line stability monitoring and system surveillance are
considered useful tools in the operation of power systems, as
these allow operators to guarantee system security by ensuring
that there is enough stability margin even in the case of severe
contingencies. These stability margins are typically associated
with voltage and/or oscillatory instabilities which have been
the cause for recent major blackouts (e.g. August 2003 NorthEast blackout [1], August 1996 WSCC blackout [2]).
Some research has been carried on detection of oscillatory
instability points as well as their prediction and control [3],
[4], [5], [6]. They may be categorized as off-line and online methods. Off-line methods are basically system studies
based on certain models and their corresponding differential
algebraic equations (DAE’s), and hence are highly dependent
on modeling assumptions as well as availability of accurate
system data. On the other hand, on-line methods, if properly
designed and implemented, can properly capture the actual
dynamic behavior of the system without the need for modeling
assumptions from on-line measurements.
In [3] and [4], the authors propose indices to predict the
closest oscillatory instability or Hopf bifurcation (HB) point.
These indices are based on the critical electromechanical
mode of the system that eventually crosses the imaginary
This research was partially supported by NSERC, Canada.
axis, as system loading changes, assuming that the critical
mode is known. When the system is close to HB point, these
indices present a fairly linear profile allowing for estimation of
stability/security margins. However, electromechanical modes
may present a highly nonlinear behavior, making it difficult
to determine the critical mode, unless the system is very
“close” to the instability point; in this case, these indices fail
to produce accurate estimates of the stability/security margins.
The authors in [7] and [8] investigate the behavior of
the damping and synchronizing torque of the system in a
single-machine-infinite-bus (SMIB), using the time domain
simulation results of the system and utilizing an ordinary
least square (OLS) method. This is then applied to a multimachine power system in [9], using classical model of generators without modeling exciters and governors. Other system
identification techniques such as Kalman filtering and genetic
algorithm have also been utilized to estimate damping and
synchronizing coefficients to achieve less computational time
as well as robustness to noisy measurements [10], [11]. Since
the authors in [9] demonstrate that damping torque variations
are closely related to the trend of electromechanical modes,
this torque could be in principle used as another index to
calculate the stability/security margins. However, the accuracy
of these estimated values have not been studied.
In this paper, the theoretical values of the damping and
synchronizing coefficients of the electromechanical torque
of linearized models (LM) of power systems are discussed,
in order to investigate the accuracy of parameter estimates
obtained using OLS techniques. The LM used is based on subtransient models of generators as well as a full representation
of exciters and governors. Alternative methods known as
generalized least square (GLS) and robust fitting with bisquare
weights (RFBW) are proposed to handle the cases when OLS
does not provide accurate estimates. Furthermore, the use of
the estimated damping torque coefficient as an index is studied.
The paper is structured as follows: In Section II, the
theoretical background on damping and synchronizing torques
is presented, and the procedure to derive the damping and
synchronizing coefficients from an LM is described. The
identification techniques used to estimate damping coefficient
from the measured response of the system are explained in
Section III. The results of applying the proposed identification
methods to calculate the damping coefficient are presented
and discussed in Section IV for a variety of test cases,
discussing the behavior of the damping torque coefficient as
2
a stability/security index. Finally, Section V summarizes the
main contributions of this paper.
∆ Tm j (s )
+
1
2 Hs
∆ ω j (s )
-
∆ Te j (s )
II. BACKGROUND
The concept of damping and synchronizing torques for
a single-machine-infinite-bus (SMIB) system was first introduced in [12]. Thus, electromechanical torque T e deviations of
a machine can be expressed in terms of its speed ω and angle
δ deviations, known as damping and synchronizing torque,
respectively. The damping and synchronizing torques are the
in time phase components proportional to the speed and angle
deviations [7], and are defined as:
∆Te (t) = Kd ω0 ∆ω(t) + Ks ∆δ(t)
damping
synchronizing
(1)
or in the frequency domain:
Ks 

T j ( s ) =  K d j + j  ω0
s 

Fig. 1.
Torque-speed block diagram.
A. Damping & Synchronizing Coefficient From Linearized
Models
The state-space model of a power system, when linearized
about an operating point, can be represented as:
∆ẋ(t) =
A ∆x(t)
∆y(t) =
C ∆x(t)
(5)
where ∆x ∈ is a vector of state variables that represents
the state variables of generators, loads and other system
controllers; ∆y ∈ m is the vector of output variables; and A
and C are constant matrices resulting from the linearization
process. Let assume that only the i th mode of the system
λi = αi + jβi is excited by an initial condition x 0 in the
direction of right eigenvector U i associated with λi . Hence,
any output y k (t) can be expressed as y k (t) = Ck Ui eλi t , or
yk (s) = Ck Ui /(s − λi ) where Ck is the k th row of C. In
this case, one may write the following relationship between
the speed and electrical torque of machine j, defined as D j :
n
Ks
) ω0 ∆ω(s)
∆Te (s) = (Kd +
s
(2)
where Kd (p.u./rad/sec) and K s (p.u.) are damping and
synchronizing coefficients, respectively; and ω 0 (rad/sec) is
the system angular frequency. The generator’s angle and speed
deviations in (1) are typically measured with respect to a center
of inertia (COI), i.e. δ i = δ̂i − δCOI and ωi = ω̂i − ωCOI ,
where
δCOI
ωCOI
g
=
=
g
1 Mi δ̂i
MT i=1
(3)
Dj =
1 dδCOI
ω0 dt
where MT = i=1 Mi is the total inertia of the g generators.
Since ω0 ∆ω(t) = d∆δ(t)/dt or in discrete form
ω0 ∆ω[k] = (∆δ[k] − ∆δ[k − 1])/Ts , where Ts is the sampling time, one can rewrite (1) as:
∆Te [k] = A ∆δ[k − 1] + B ∆δ[k]
(4)
where A = −Kd /Ts and B = Kd /Ts + Ks . Equation (4) can
be used to calculate A and B from sampled signals, yielding
Kd and Ks . Using (4), which only requires two signals T e and
δ, as opposed to (1), which requires three signals T e , ω and δ,
reduces the errors due to mean value and low frequency trends
in the measured signals, due to the effect of the governor’s
response.
For a single-machine-infinite-bus (SMIB) system, the synchronizing and damping coefficients correctly define the frequency and damping of the electromechanical mode. However,
for a multi-machine power system, the electromechanical
oscillations contain different modes, and hence a single mode
cannot simply be assigned to a single machine [9], since the
oscillations of each machine are a linear combination of all
the modes. Hence, in this case, one must be aware that the
damping coefficient of a given machine includes the effect of
several modes.
CTej Ui
∆Tej (s)
∆Tej (t)
=
=
∆ωj (t)
∆ωj (s)
Cωj Ui
(6)
This ratio is a complex number, i.e. D j = DRj + jDIj and
can be calculated using state space model. Thus, from (2) and
(6),
α D
Ij
+ DRj
β
Kdj =
(7)
ω0
2
−DIj |λi |
Ksj =
ω0 β
The Kdj and Ksj coefficients may be used in the torquespeed block diagram as depicted in Fig. 1. This system is stable
if both Kdj and Ksj are positive, and becomes oscillatory
instable at an operating condition where K d = 0 (HB point).
Hence, Kd may be used as an index to predict the closest
oscillatory instability.
It is worth mentioning that the K d and Ks coefficients presented here are different from the damping and synchronizing
coefficients discussed in [13], [14], since in the latter, these
coefficients are defined by disabling the shaft dynamics in all
generators, which is not the case here.
III. I DENTIFICATION T ECHNIQUES
A. Ordinary Least Square (OLS):
Damping and synchronizing coefficients in (1) can be estimated using an ordinary least square (OLS) method [7]. It is
3
basically a fitting problem which can be described as:
Y
=
[Ω ∆] θ + =
Xθ + C. Robust Fitting with Bisquare Weights (RFBW)
(8)
where Y
=[Te [1]
Te [2]
...
Te [N ]]T ∈
N ;
T
Ω = ω0 [∆ω[1] ∆ω[2] ... ∆ω[N ] ] ∈ N ;
∆ =[∆δ[1] ∆δ[2] ... ∆δ[N ]]T ∈ N ; ∈ N represents
“residuals” introduced to account for fitting errors and
measurement noise; θ = [K d , Ks ]T ; and N is the number of
samples. The ordinary least square estimate can be obtained
as:
(9)
θ̂OLS = (X T X)−1 X T Y.
Here, θ̂OLS is unbiased, efficient (or Markov estimate), and
consistent when is white noise with zero mean [15]. Furthermore, it is also identical to the maximum likelihood estimator
when is normally distributed. However, the condition of
whiteness for the current application is not guaranteed, thus θ̂
might be biased. This will be clearly shown in a number of
case studies in Section IV.
B. Generalized Least Square (GLS)
From the system identification point of view, (8) can be
represented as follows:
y[k] = x[k] θ + H(q −1 ) e[k]
−1
(10)
−1
where e[k] is white noise; H (q ) is a whitening filter;
and q is the backward shifting operator defined by q −1 y[k] =
y[k − 1].
Selecting a proper candidate model set in this case,
which includes the true model, is an essential aspect to
obtain accurate parameter estimates. On the other hand, overparameterization can lead to high computational costs as well
as numerical problems (e.g. convergence and local minima).
Therefore, one should search for a model set that includes
the true model with as low number of parameters as possible.
It is also important to consider that choosing a rather simple
model, i.e. under-parameterization, might lead to inaccurate
parameter estimates [16].
The candidate model (10) with H(q −1 ) = 1, which leads to
(9), is not a general model and may not correctly represent the
power system dynamics. For example, when the perturbations
are large and hence the linearity assumption does not hold,
the OLS estimates can lead to erroneous results. Experience
shows that by adding a transfer function such as H(q −1 ) =
1/(1 + h1 q −1 + h2 q −2 + ... + hp q −p ) with p = 1 or 2, good
results can be obtained [15]. This method is also known as
generalized least square (GLS) and can be used to correctly
model the residuals or filter the nonlinear effects. In this case,
θ along with the parameters of the whitening filter have to
be estimated using prediction error methods (PEM). These
methods are basically based on an optimization problem such
as:
M in
s.t.
V
e[k]
=
=
N
1 2
e [k]
N
H
k=1
−1 −1
(q
) {y[k] − x[k] θ}
(11)
Robust fitting may also be employed as another remedial
method for handling the nonlinear effects, which can occur
due to large deviations from an operating point. This method
is basically an iterative weighted least square (WLS) method,
and is able to give different weights to each residues during
the fitting process; data with lower quality are given less
weight, since the part of the signal that is distorted due to
the nonlinear effects is hard to fit, and hence should be less
important in the estimation process. One may consider using
small perturbations as a way to avoid nonlinearities; however,
in the current application, small disturbances may not be
feasible, since the perturbations have to be large enough to
be distinguished from measurement noise.
Robust fitting employs the following objective function,
which assigns weights to different predicted errors:
M in
V
=
N
1 ωk 2 [k]
N
(12)
k=1
The estimated damping and synchronizing coefficients in this
case are:
θ̂W LS
=
(X T W X)−1 X T W Y.
(13)
where W = diag{ω1 , ω2 , ... , ωN } ∈ N ×N . There are
different weights which can be utilized in the objective function (12) [17]; in this work, bisquare weights are used. This
method has been previously used in economics to decrease the
sensitivity of least square to “extreme” values called outliers.
IV. T EST CASES
Several test cases based on 2 simple test systems and
their associated transient stability models, and the Power
System Toolbox (PST) [18] were used to generate the required
signals, i.e. electrical torque deviations, generator angle and
speed deviations. White Gaussian noise was added to the
mentioned signals as measurement noise, such that the signalto-noise ratio (SNR) would be 40 db. A Monte-Carlo type of
simulation was used to test the feasibility and accuracy of the
identified damping coefficient by simulating 20 independent
cases at each operating condition. Assuming a direction for
load change and generation dispatch, the damping coefficients
using (7) for the electromechanical mode of interest were
computed at different operating conditions using the small
signal stability analysis functions of PST. These values were
then compared to the mean value of the results obtained
from the identification methods OLS, GLS and RFBW based
on the time domain simulation results for the system. The
damping coefficient behavior with respect to load increase was
studied as a possible index to predict the margin to the closest
oscillatory instability.
It is important to mention that, in order to achieve highly
accurate parameter estimates, a proper time window for the
measurement that does not contain nonlinear and saturation
effects should be selected. However, it is not practical to delay
the measurements until the oscillations are small enough and
nonlinear effects have completely disappeared, because this
can result in low SNR and consequently erroneous estimates.
4
1
G1
Fig. 2.
5
6
7
8
9
10
11
3
G1
G3
G2
Single-Machine-Infinite-Bus (SMIB).
G4
2
LM
OLS
GLS
RFBW
0.07
4
Area 1
Fig. 4.
Area 2
Two-area benchmark system.
0.06
(a): Inter−area mode
5.5
0.04
Imag (rad/s)
Kd (p.u./rad/s)
0.05
0.03
0.02
*
5
: Schedule 1
o : Schedule 2
4.5
0.01
4
0
−0.3
−0.2
−0.01
−0.1
0
(b): Local mode − Area 1
0.1
0.2
*o
: Schedule 1
500
600
700
Fig. 3.
Damping coefficient Kd for SMIB.
In all test cases, in order to excite the system, a three phase
fault was applied and quickly removed. A sampling time of
0.1 sec was used so that the measured signals contain at least
three oscillations periods. Choosing a longer window did not
provide satisfactory results, since once the oscillations are
well damped out, measured signals would not be informative
enough. This is especially important for the RFBW method,
as this would try to assign larger weights to the tale of the
signal, which has an almost negligible magnitude, leading to
erroneous results.
A. Single-Machine-Infinite-Bus (SMIB)
The generator in the simple test system depicted in Fig.
2 was modeled using sub-transient model. There is an electromechanical mode which moves toward the right half plane
(RHP) as the load increases from 100 MW to 700 MW,
resulting in an increase in tie-line power as well as a gradual
decrease in the generator’s damping coefficient K d . The results
obtained for “theoretical” and estimated values of K d are
shown in Fig. 3; observe that all the identification methods
were able to correctly calculate K d . However, for heavy loads,
the different values for K d obtained from the various methods
used coincide, which is to be expected, since at light loading
conditions the mode is well-damped. This is akin to the case
when eigenvalues are to be identified using the time domain
response of the system, as reported in [3], [19].
B. Two-area Benchmark System
The single line diagram of the two-area benchmark system
used here is shown in Fig. 4 [20]. There is an inter-area mode
and two local modes that change as the loads are increased.
Depending on the load and dispatch scenarios, one could
7.4
7.2
: Schedule 2
7
−1.1
−1
−0.9
−0.8
−0.7
(c): Local mode − Area 2
−0.6
−0.5
* : Schedule 1
8.6
Imag (rad/s)
400
Loading (MW)
o : Schedule 2
8.4
8.2
8
−1.2
−1
−0.8
−0.6
−0.4
−0.2
−1
Real (sec )
0
0.2
Fig. 5. Eigenvalue profiles for different dispatch scenarios; Schedules 1 and
2.
1
Schedule 1
Schedule 2
HB
0.99
*
0.98
*
0.97
HB
0.96
7
300
V (p.u.)
200
Imag (rad/s)
7.6
−0.02
100
0.95
0.94
0.93
0.92
2800
Fig. 6.
2900
3000
3100
3200
3300
Loading (MW)
3400
3500
3600
PV curve at Bus 7 for the 2-area benchmark system.
3700
3800
5
G
G
0.01
0
0.02
0.01
2500
3000
Loading (MW)
3500
−0.01
2000
G
0.02
0
0.015
0.04
0.06
0.04
0.02
−0.02
−0.02
2500
3000
Loading (MW)
3500
2400
4
0.05
2600 2800 3000
Loading (MW)
G
3200
3400
2400
3
0.005
0
−0.005
0.03
0.02
0.01
0
−0.01
2500
3000
Loading (MW)
3500
−0.02
2000
3200
3400
3200
3400
4
0.08
0.06
Kd (p.u./rad/s)
0.01
2600 2800 3000
Loading (MW)
G
0.04
Kd (p.u./rad/s)
Kd (p.u./rad/s)
3
−0.01
2000
0.04
G
0.02
0.08
0
0
Kd (p.u./rad/s)
−0.01
2000
0.06
0.03
2
LM:Inter−area mode
LM:Local mode Area−2
OLS
GLS
RFBW
0.08
Kd (p.u./rad/s)
0.02
d
K (p.u./rad/s)
0.03
1
0.04
Kd (p.u./rad/s)
LM: Inter−area mode
OLS
GLS
RFBW
G
G
2
1
Kd (p.u./rad/s)
0.04
0.02
0
−0.02
2500
3000
Loading (MW)
3500
0.04
0.02
0
−0.02
2400
2600 2800 3000
Loading (MW)
3200
3400
2400
2600
2800 3000
Loading (MW)
Fig. 7. Damping coefficients of all generators for the 2-area benchmark
system for Schedule 1.
Fig. 8. Damping coefficients of all generators for the 2-area benchmark
system for Schedule 2.
expect to have different eigenvalue profiles. For instance, in
this system, two dispatch scenarios, Schedules 1 and 2, result
in fairly different eigenvalue profiles, as depicted in Fig. 5.
There is an HB point before the nose point, as shown in the
corresponding PV curves depicted in Fig. 6.
1) Schedule 1: This dispatch results in the inter-area mode
becoming unstable as loading increases. In this case, generators G2 and G3 are dispatched proportionally to their base
power as the loads at Bus 7 and 9 increase. Loads are modeled
as constant PQ loads and are increased with a constant power
factor. The inter-area mode becomes unstable at a loading level
of about 3050 MW. This can also be observed clearly in Fig.
7, which depicts the theoretical and estimated values of K d
for this case study. Observe that all the calculated K d ’s tend
to zero as the load increases; however, the GLS method offers
superior performance in terms of predicting the HB point,
since the estimates are fairly close to the theoretical values.
2) Schedule 2: Schedule 2 is an interesting dispatch scenario where the inter-area mode is primarily the critical one
until at certain loading level (3030 MW), the local mode
in Area 2 becomes critical, crossing the imaginary axis and
hence leading the system to oscillatory instability conditions
at a loading level of about 3150 MW. In this case, the loads
are increased as in Schedule 1; however, only Generator G 3
is dispatched to respond to the load change. This certainly
stresses G3 more, and thus the local mode in Area 2 becomes
critical, as shown in Fig. 5.
For this scenario, it is possible to compute two values of
Kd using (7) which are associated with the inter-area mode
and the local mode in Area 2, as depicted in Fig. 8. Since both
inter-area and local mode in Area 2 are dominant modes, time
domain simulation results, in this case, would be influenced
by both of them, and so would be the identified K d ’s. Notice
that in this case, the values obtained for K d from OLS are
not accurate. Neither are the predicted stability margins. On
the other hand, the K d ’s obtained using GLS present the best
results in terms of accuracy in estimation as well as prediction.
The RFBW’s accuracy is between the OLS and GLS, and from
computational point of view, it is slower than OLS and faster
than the GLS.
The advantage of using K d , when compared to eigenvalue based indices, is that, in order to monitor system
stability/security margin, the damping coefficient is able to
provide accurate margins without the need for monitoring
particular modes. This can be an issue for eigenvalue-based
indices. For instance, the damping torque is able to capture
the oscillatory phenomenon for both schedules; on the other
hand, for Schedule 2, if only the inter-area mode were used
as an index, it would yield inaccurate margins.
V. C ONCLUSION
The procedure to derive the theoretical values of damping
and synchronizing coefficients obtained using linearized models of power system was demonstrated. The accuracy of the
estimates based on system identification and measured time
domain response of the system was investigated by comparing
them to theoretical values.
The GLS and RFBW identification methods are better alternatives to the OLS method. In particular, it is shown that OLS
may not be able to accurately estimate the coefficients due to
the nonlinear effects that appear in the measured response of
a power system, especially when the system is well-damped.
These effects can be reduced using smaller perturbations.
However, there is a trade off between the magnitude of the
perturbation and SNR of the measured signals, i.e. the lower
the SNR, the less accurate the estimates.
The damping coefficient is shown to be a useful index
to predict the distance to instability points when system
load changes. It is basically a close duplicate of the electromechanical modes’ behavior; however, it does not require
the monitoring of certain modes which can be an issue in
large systems. Hence, this index may help system operators
6
to determine proximity to an oscillatory instability, without
requiring modeling assumptions and simulations and thus
allowing to take proper action in the case of contingencies
and/for congestion problems.
ACKNOWLEDGMENT
Hassan Ghasemi (S’01) received his B.Sc. and M.Sc. degree from the
University of Tehran, Iran, in 1999 and 2001, respectively. He did research
on analysis and design of machine drives during his master program. He also
worked in Jovain Electrical Machines Co. (JEMCO) 2000-2001 as a parttime electrical engineer. He has been in the Ph.D. program at the Electrical
and Computer Engineering Department, University of Waterloo, Canada, since
2002 working in the field of power system modeling and application of system
identification to stability analysis of power systems.
The authors would like to express their gratitude to Powertech Labs Inc. personnel and Hamidreza Zareipour from
University of Waterloo for their valuable help, comments and
discussions.
R EFERENCES
[1] “Interim Report: Causes of the August 14th Blackout in the United
States and Canada,” Tech. Rep., November 2003, available at
http://www.nrcan-rncan.gc.ca/media/docs/814BlackoutReport.pdf.
[2] “System Disturbance Stability Studies for Western System Coordinating
Council (WSCC),” EPRI Final Report prepared by Powertech Labs Inc.,
Surrey, BC, Canada, Tech. Rep. TR-108256, September 1997.
[3] H. Ghasemi, C. A. Cañizares, and A. Moshred, “Oscillatory stability
limit prediction using stochastic subspace identification,” accepted to
IEEE Trans. Power Systems, 10 Pages, September, 2005.
[4] C. A. Cañizares, N. Mithulananthan, F. Milano, and J. Reeve, “Linear
performance indices to predict oscillatory stability problems in power
systems,” IEEE Trans. Power Systems, vol. 19, no. 2, pp. 1104–1114,
May 2004.
[5] T. Kim and E. H. Abed, “Closed-loop monitoring system for detecting
impending instability,” IEEE Trans. Circuits and Systems, vol. 47, no. 10,
pp. 1479–1493, October 2000.
[6] N. Mithulananthan, C. A. Cañizares, J. Reeve, and G. J. Rogers,
“Comparison of PSS, SVC and STATCOM controllers for damping
power system oscillations,” IEEE Trans. Power Systems, vol. 18, no. 2,
pp. 786–792, May 2003.
[7] R. T. H. Alden and A. A. Shaltout, “Analysis of damping and synchronous torques: Part I - A general calculation method,” IEEE Trans.
Power Systems, vol. PAS-98, no. 5, pp. 1696–1700, Sept/Oct 1979.
[8] ——, “Analysis of damping and synchronous torques: Part II - Effect
of operation conditions and machine parameters,” IEEE Trans. Power
Systems, vol. PAS-98, no. 5, pp. 1701–1707, Sept/Oct 1979.
[9] A. A. Shaltout and B. A. Abu Al-Feilat, “Damping and synchronous
torque computation in multimachine power systems,” IEEE Trans. Power
Systems, vol. 7, no. 1, pp. 280–286, February 1992.
[10] E. A. Feilat, “Performance estimation techniques for power system
dynamic stability using least squares, Kalman filtering and genetic
algorithms,” in Southeastcon 2000, Proceedings of the IEEE, April 2000,
pp. 489 – 492.
[11] E. A. Feilat, “On-line adaptive assessment of the synchronizing and
damping torque coefficients using Kalman filtering,” in Southeastcon
1999, Proceedings of the IEEE, March 1999, pp. 145 – 148.
[12] F. P. Demello and C. Concordia, “Concepts of synchronous machine
stability as affected by excitation control,” IEEE Trans. Power Apparatus
and Systems, vol. PAS-88, no. 4, pp. 316 – 329, April 1969.
[13] M. J. Gibbard, “Co-ordinated design of multimachine power system
stabilisers based on damping torque concepts,” in IEE Proceedings, Pt.
C, vol. 135, no. 4, July 1988, pp. 276–284.
[14] P. Pourbeik and M. J. Gibbard, “Damping and synchronizing torques induced on generators by facts stabilizers in multimachine power systems,”
IEEE Trans. Power Systems, vol. 11, no. 4, pp. 1920–1925, November
1996.
[15] T. C. Hsia, System Identification: Least-Square Methods. Lexington,
MA: Lexington Books, D.C. Health and Company, 1977.
[16] R. Söderström, System Identification. Cambridge, UK: Prentice Hall
International (UK) Ltd., 1989.
[17] Statistic Toolbox 1.1 in MATLAB, The Mathworks, 2000.
[18] Power System Toolbox (PST) Version 2.0: Dynamics Tutorial and
Functions, Cherry Tree Scientific Software, 2002.
[19] R. W. Wies, J. W. Pierre, and D. J. Trudnowski, “Use of ARMA block
processing for estimating stationary low-frequency electromechanical
modes of power systems,” IEEE Trans. Power Systems, vol. 18, no. 1,
pp. 167–173, 2003.
[20] P. Kundur, Power System Stability and Control. New York: McGrawHill, 1994.
Claudio A. Cañizares (SM’00) received in April 1984 the Electrical Engineer
diploma from the Escuela Politécnica Nacional (EPN), Quito-Ecuador, where
he held different teaching and administrative positions from 1983 to 1993. His
MS (1988) and PhD (1991) degrees in Electrical Engineering are from the
University of Wisconsin-Madison. Dr. Cañizares has held various academic
and administrative positions at the E&CE Department of the University of
Waterloo since 1993 and is currently a full professor. His research activities
concentrate in the study of stability, modeling, simulation, control and
computational issues in power systems in the context of electricity markets.