Understanding low frequency oscillation in power systems
*K. Prasertwong, **N. Mithulananthan and *D. Thakur
*Electric Power System Management, Energy Field of Study, Asian Institute of Technology, Klongluang, P.O.
Box 4, Pathumthani, Thailand
**Power and Energy System Group, School of Information Technology and Electrical Engineering, The
University of Queensland, St. Lucia Campus, Brisbane, Qld 4072, Australia
Email: mithulan@itee.uq.edu.au
Abstract: This paper presents a complete overview of low frequency oscillation phenomena in power systems.
Definition of low frequency oscillation and its classification is clearly explained. Some power system blackout
incidents due to low frequency oscillation and the lesson learned from those incidents are given in the paper.
Widely used methodology for studying low frequency oscillation among power utilities is also presented.
Methods for oscillation damping, both at operational and planning stages of power system are briefly discussed.
A comprehensive case study of low frequency oscillation in simple system is presented using eigenvalue analysis.
Time domain results are presented to support the eigenvalue analysis. Conclusions are duly drawn.
Keywords: AVR, eigenvalue analysis, low frequency oscillation, PSS, SSSC, TCSC
Introduction
The main function of power system is to convert energy from one of the naturally available forms to electricity
and transport it through grids to points of consumption. As electrical energy can not be destroyed like any other
forms of energy, whatever produced from generators should be utilized or stored. However, technology for
storing electricity in large quantities has not yet matured. Thus many equipment and controllers have been used to
maintain the balance between what is being demanded and generated in a reliable manner with high degree of
quality.
In modern power systems, apart from a large number of generators and associated controllers, there are many
types of load, ranging from a simple resistive load to more complicated loads with electronic controllers. The
influx of more and more controllers and loads, increase the complexity and nonlinearity of power systems. As a
result power systems are viewed as complex nonlinear dynamical system that shows a number of instability
problems. Instability problems in power systems that can lead to partial or full blackout can be broadly classified
into three main categories, namely voltage, phase angle and frequency related problems [1]-[10]. Though
instability eventually blackout or collapse never happened in a pure form of voltage or phase angle or frequency
problems, the initial part of the incident can be clearly related to one of the categories.
If one looked back the history of power system since its evolution, operation engineers faced with transient
instability problem and researchers struggled to find counter measures to overcome it [4]. Transient instability
problem, considered as part of the phase angle related problem, is defined as the ability of power system to
maintain synchronism when subjected to large disturbances. When the system faces large disturbances such as
large load increase, loss of tie lines, loss of generating units, maintaining constant electrical speed among all the
generators were challenging as some machines speed up while some other slow down to adjust to post
disturbance situation. If there is no control mechanism to keep the speeding up or slowing down generators
within the allowable speed limits, there is a good chance that these generators would fall out of the grid by losing
synchronism. Hence, fast exciter or Automatic Voltage Regulators (AVR) was introduced in the system as one of
the remedial measures to solve the problem. The introduction of fast AVR was able to give the “coarse
adjustment” to keep electrical speed of synchronous generators within the limits and successful in maintaining
synchronism by controlling the first swing. However, the fast AVR could not do the “fine adjustment” to control
oscillation in the speed. Then, Power System Stabilizer (PSS) was introduced in generator to give that fine
adjustment to damp out power oscillations that are referred to as electromechanical or low frequency oscillations
(LFO) [4], [7].
Apart from the fast exciter there are a number of other sources that contribute to oscillation in modern power
system such as frequency load dependency, network characteristics and negative interaction of controllers.
1
Though oscillations are inherent in power system if not properly controlled could lead to partial or full blackout
of the system as it happened in many practical power systems. Hence a complete understanding of the problem
would help in finding effective remedial measures, ways and means to control them.
Rest of the paper is organized as follows. Following introduction, the paper describes what low frequency
oscillation is. Some noteworthy oscillation related incidents and lesson learned is covered next. Then the
methodologies available for power system oscillation studies are described along with general power system
modeling. Most widely used methodology for low frequency oscillation study, eigenvalue analysis is presented in
details in the methodology section. Some remedial measures that have been used all around the world by utilities
for oscillation damping is given in the next section. A simple case study showing the effect of different
controllers on power system oscillation modes in a systematic way is also presented in the paper with some
discussion. Finally, conclusions and contributions of the paper are summarized.
Low Frequency Oscillation
The ability of synchronous machines of an interconnected power system to remain synchronism after being
subjected to a small disturbance is known as small signal stability that is subclass of phase angle related
instability problem. It depends on the ability to maintain equilibrium between electromagnetic and mechanical
torques of each synchronous machine connected to power system. The change in electromagnetic torque of
synchronous machine following a perturbation or disturbance can be resolved into two components – (i) a
synchronizing torque component in phase with rotor angle deviation and (ii) a damping torque component in
phase with speed deviation. Lack of sufficient synchronizing torque results in “aperiodic” or non-oscillatory
instability, whereas lack of damping torque results in low frequency oscillations.
Low frequency oscillations are generator rotor angle oscillations having a frequency between 0.1 -2.0 Hz and are
classified based on the source of the oscillation [4]. The root cause of electrical power oscillations are the
unbalance between power demand and available power at a period of time. In the earliest era of power system
development, the power oscillations are almost non observable because generators are closely connected to loads,
but nowadays, large demand of power to the farthest end of the system that forces to transmit huge power through
a long transmission line, which results an increasing power oscillations.
The phenomenon involves mechanical oscillation of the rotor phase angle with respective to a rotating frame.
Increasing and decreasing phase angle with a low frequency will be reflected in power transferred from a
synchronous machine as phase angle is strong coupled to power transferred. The LFO can be classified as local
and inter-area mode [4][10].

Local modes are associated with the swinging of units at a generating station with respect to the rest of
the power system. Oscillations occurred only to the small part of the power system. Typically, the
frequency range is 1-2 Hz.

Interarea modes are associated with swinging of many machines in one part of the system against
machines in other parts. It generally occurs in weak interconnected power systems through long tie lines.
Typically frequency range is 0.1-1 Hz.
Besides these modes, there can be other modes associated with controllers which happen due to poor design of
controllers [10]. Torsional oscillation is another type of oscillation that happened in series capacitor compensated
system and the frequency of oscillation is typically in sub synchronous frequency range [10].
Oscillatory Instability Incidents and Lesson Learned
Though there have been many incidents related to LFO, not an in-depth study has been performed to see the real
reasons behind many of these incidents. Some of the incidents and the lesson learned are summarized below to
give an understanding of the underlying problem.
Noteworthy incidents related to LFO include [11]:
2







United Kingdom (1980), frequency of oscillation about 0.5 Hz.
Taiwan (1984, 1989, 1990, 1991, 1992), frequency of oscillation around 0.78 – 1.05 Hz.
West USA/Canada, System Separation (1996), frequency of oscillation around 0.224 Hz.
Scandinavia (1997), frequency of oscillation about 0.5 Hz.
China Blackout on 6 March (2003), frequency of oscillation around 0.4 Hz.
US Blackout on 14 August (2003), frequency of oscillation about 0.17 Hz.
Italian Blackout on 28 September (2003), frequency of oscillation about 0.55 Hz.
Most of these incidents involved in a low frequency of oscillation in the range of 0.1 to 0.7 Hz that is considered
as the most serious and could lead to wide spread blackouts [11]. Apart from this, oscillatory incidents in power
systems in Ontario-Canada, Sri Lankan, Malaysia and Bangladesh are also reported in the literatures [11].
Most of the incidents had happened due to faults triggered by some disturbances such as a tree contacting with a
transmission line, some component failure, faults in transmission lines etc. Because of the faults, these lines have
been disconnected from the grid. Then some other lines in the network has been overloaded and sagged on trees
causing more earth faults. Those incidents have been generated sequential line tripping and generator tripping
causing oscillation in power. The tripping of transmission lines significantly modifies the characteristics of the
remaining grid with longer distance (greater equivalent impedance) for the power flow and consequent higher
stability risk. And also the modified grid may have less damping compared with the original grid. The weak tie
lines and the nature of the longitudinal structure are one of the causes for low frequency oscillations.
Concentration of outputs to major power plants with insufficient reserve margins, heavy flow across transmission
interfaces due to seriously imbalanced regional power and pumped storage units were in pumping mode
operation are common causes for low frequency oscillation observed in some of the cases above mentioned [11].
With the heavy tie line power, low frequency electromechanical oscillation modes have been captured the cases
mentioned above and decreasing the tie line power flow made those modes disappeared.
Most of the events happened either during very cold day during winter season or in a very hot day in summer
season [11]. The use of thermostatically controlled loads, such as space heaters, coolers, water heaters are
increasing in these days. One of the properties of these loads is to operate longer period even during low voltage
conditions. As a result, the total number of these devices connected to the system will increase in a few minutes
after a drop in voltage. Therefore there might be some influence on low frequency oscillations from
thermostatically controlled loads [11].
In some cases, during postmortem analysis, it has been identified that the past data of system modeling has
differences with actual values. The analysis with past data has been showed positive damping for power
oscillations but in the actual case it was negatively damped. There are various reasons for this type of
discrepancy. In order to avoid this and to have a complete knowledge of the system, components contribute to
oscillatory problem need to be modeled accurately and good understanding of the phenomena under different
operating conditions are required. With this type of knowledge and understanding a counter measure can be
implemented easily to avoid disastrous consequences.
Methodology
Low frequency oscillation study requires dynamic modeling of most of the power system components. Once the
mathematical model is available different methodologies can be applied to study the system oscillatory behavior
in low frequency range. Eigenvalue analysis, time domain simulation and Prony analysis are used among
researchers. Though each method of these methods has its own merits and demerits, eigenvalues and time domain
simulations are typically used among the utilities to get a complete understanding of system oscillatory
phenomena.
Power system modeling
Dynamic modeling of power system includes a set of differential and algebraic equations (DAE). Low frequency
oscillation studies can be done in two ways depending on the interest. If the interest is to capture the local
behavior related to an area or particular power plant, then that area of power plant can be modeled in details and
3
the rest of the system with simple models. If the interest is to capture both local and global modes such as interareas mode each and every machine in the system and their associated controllers should be modeled in details. It
is important to include loads, controllers and other power system components that would influence the LFO. A
general mathematical model of power system is given by (1).
x  f ( x, y, l , p )
0  g ( x, y , l , p )
(1)
Where x is a vector of state variable; y is a vector of algebraic variables; l and p are uncontrollable and
controllable parameters, respectively. Machine and control dynamics will be included in the differential equations
while basic load flow and other network equations will be included in algebraic equations.
Eigenvalue analysis
The small signal stability or LFO study of the system can be determined by system eigenvalues at an operating
point. The relative participation of state variables and their contribution in certain oscillation mode are given by
the corresponding elements in the right and left eigenvectors. Hence, combination of left and right eigenvectors
yield participation factor matrix. The participation factor matrix can be used to identify the dominant state
variable in a particular mode.
The following steps are followed in studying LFO of power systems.
Step I: Finding equilibrium or operating point
Equilibrium point or operating point of the system can be found by simultaneously solving differential and
algebraic equations given in (1). Assume the equilibrium point is given by (x0, y0, p0) for a fixed value of
uncontrollable parameter l0.
Step II: Linearization DAE model around the equilibrium point
Once the equilibrium point is known DAE model can be linearized around the equilibrium point as given in (2).
Here, the linearized model is considered valid as the disturbances considered is small where nonlinearities can be
ignored.
x   J1 J 2  x 
(2)
 0    J J  y 
3
4 
  



J
Where J1 
f
x
; J2 
( xo , y o , l o , po )
f
y
; J3 
( xo , y o ,l o , po )
g
x
and J 4 
( xo , y o ,l o , po )
g
y
.
( xo , y o ,lo , po )
Step III: Forming the reduced system state matrix
Assuming J4 is nonsingular, (2) can be rewritten by eliminating algebraic variable as shown in (3).
x  ( J1  J 2 J 41J 3 )x  Ax
(3)
That is, the linearized DAE system can be reduced to a set of ODE equations as shown in (3). Matrix A in (3) is
referred to as reduced system state matrix.
Step IV: Finding eigenvalues, eigenvectors and Participation matrix
Small signal stability or steady state stability of the equilibrium point of the system can be analyzed by looking at the
eigenvalues of A or reduced system state matrix. Eigenvalues of A are given by (4) and the number of eigenvalues
depends on the dimension of matrix A or the number of state variables considered in the system.
4
(4)
[ A  I ]  0
Where  represents eigenvalue and  represents right eigenvector. For non-trivial solution determinant of [A-I]
equals to zeros and the eigenvalues can be calculated. Similarly, another equation can be written to find out the left
eigenvector  as given in (5)
 [ A  I ]  0
(5)
In order for the system to be stable or oscillation free, all the eigenvalues should be located in the open left half
plane. This means that real part of the eigenvalues should be negative and damping ratio should be positive with
more than a pre specified value according to utilities‟ practice (typically damping ratio should be higher than
0.05). If at least one of the eigenvalues has positive real part the system is said to be unstable. More specifically,
in oscillatory unstable cases, a pair of complex eigenvaues will appear with positive real part [12].
Given an eigenvalue in complex format, -±j, the initial frequency of oscillation (f) and damping ratio () can
be calculated using expressions given in (6).
f 

2
; 

 2
2
(6)
Participation factor matrix
Once both right and left eigenvectors are known for different eigenvalues, the participation factor matrix can be
calculated by combining the left and right eigenvectors as shown in (7).
P  P1 P2  Pn 
(7)
with
 P1i   1i i1 
 P    
P i   2i    2i i 2 
     
  

 Pni  ni in 
Element Pki  ki ik ; where ki is the kth entry of the right eigenvector with ith mode. ik is the kth entry of the left
eigenvector associated with ith mode.
Time domain analysis
Time domain analysis involves no approximation in the DAE model and considered as the most accurate way to
study LFO problem. However, pertinent information such as various weak modes, the dominant states variable
associated with those modes and sensitivity of those modes to parameter variation and other details can not be
obtained using time domain simulation. Hence, eigenvalue and time domain analyses can be used as
complementary solutions to support each other and verify the results. In time domain analyses, mode is perturbed
and the behavior of state variable is calculated by solving differential equations in (1) using some numerical
integration techniques with the known initial values [13]-[15]. The initial values in this case, are the initial
equilibrium point.
In this paper, both eigenvalue and time domain analyses have been used.
Prony analysis
In Prony analysis, given signal is analyzed to determine modal, damping, phase, and magnitude information
contained in the signal [16]-[18]. It is an extension of Fourier analysis where damping as well as frequency
information is obtained. The Prony analysis gives an optimal fit to a signal Y(t) in the form
5
n
Y (t )   Bi e it
(8)
i 1
The n distinct eigenvalues ( i ‟s) and signal residues (Bi‟s) in (8) are identified by Prony analysis. It is important
to note that Prony analysis results in a residue and eigenvalue decomposition of an output signal; it does not
identify transfer functions directly. Consider the linear system represented in the Laplace domain as shown in (9).
Conventional Prony analysis identifies a modal of Y(s) but it does not use the knowledge of input signal I(s),
therefore it cannot identify the transfer function G(s).
G ( s) 
Y ( s)
I (s)
(9)
But, Prony analysis can give transfer function in the form
n
G ( s) 
 s
i 1
Ri
(10)
i
for a given class of inputs I(s). The distinct eigenvalues ( i ‟s) are associated with the transfer function residues
( Ri ‟s). Obtaining transfer functions in the form as in (10) is valuable to power system analysis in a variety of
areas other than PSS design which is considered as one of the effective ways to damp low frequency oscillation.
The draw back of the method is that the need for an output signal which will capture the modes and related
information.
Damping Low Frequency Oscillation
The traditional approach to address low frequency oscillation problem is to equip PSS in the machines which has
tendency to damp out power oscillations [1]-[8]. However, the present power systems are too complex as many
utilities around the world are interconnected each other to deliver reliable and cheap power from environmentally
clean resources. Moreover, introduction of competition had invited many generating plants to be connected to
power system and started to dispatch power. PSS in some cases founds not sufficient and even detrimental, this
has open the door for a number of FACTS controllers applied to add damping on weak modes. The remedial
measures for oscillation damping can be classified in two broad categories, one at operational level and the other
one is at planning stage.
Operational level approaches for power system oscillation damping include re-tuning excitation control system
and PSS. Re-dispatching of generators and adjusting of load changers can also be considered. At the operational
level, load shedding can also be used as the last line of defense to damp low frequency of oscillation [19],[20].
Planning level: At planning stage a number of damping controllers can be considered for implementation. New
PSS, FACTS controllers [21]-[26], Superconducting Magnetic Energy Storage (SMES) and fly wheel are some of
them [27]-[30].
Case Study and Discussion
A simple test system shown in Fig. 1 is used in the study to show LFO behavior. As this system is similar to
single machine infinite bus system only local modes will be captured in the study. In order to study inter-area
LFO, where generators in one area oscillate against generators in the other area, a larger system should be
considered. The smaller test system used in the study shows the modes from various dynamics of the system and
the impact of various controllers on the critical mode.
PSAT (Power System Analysis Tool) an open source software tool is used to carry out both eigenvalue analysis
and time domain simulations [13],[14].
6
Fig. 1 Simple test system.
In order to study LFO phenomena in the simple system, first eigenvalue analysis is carried out with only machine
in the system. Here, a generator model with six ordinary differential equations (6th order model), including
electromechanical and flux decaying dynamics is considered.
Figure 2 shows the eigenvalue plot of the system with machine only. This case is referred to as the base case and
there are six eigenvalues, including a complex mode related to the electromechanical dynamics of the system.
The participation matrix calculated using (7) reveals and relates different state variables to different eigenvalues
as shown in Table 1. In Table 1, 3 and 4 are complex and its conjugate modes, respectively and the dominant
state variables are states 1 and 2, that are  (rotor angle) and  (rotational speed) of generator. As the complex
mode is closer to the imaginary axis i.e. with the lowest damping ratio, the mode is considered as critical mode of
the system.
8
6
4
Imag
2
0
-2
-4
-6
-8
-40
-35
-30
-25
-20
Real
-15
-10
-5
0
Fig. 2 Eigenvalue plot of the system for base case (machine only).
In a larger system, participation factor analysis can be used in identifying problematic machines for the
placement of PSS.
TABLE 1 Participation Matrix for the Bases Case
1
2
3
4
5
6
State 1
0.00409
0.00484
0.47667
0.47667
0.00234
0.00055
State 2
0.00409
0.00484
0.47667
0.47667
0.00234
0.00055
State 3
0.00771
0.00001
0.01811
0.01811
0.00625
0.99053
7
State 4
0
0.04856
0.0066
0.0066
0.94131
0.00762
State 5
0.98407
0.00025
0.01205
0.01205
0.00008
0.00008
State 6
0.00005
0.94149
0.0099
0.0099
0.04769
0.00066
Now, AVR or excitation system is introduced in the machine to see the impact of it on the critical mode. The
eigenvalue plot of the system with AVR is given in Fig. 3. The installation of AVR has introduced new modes,
including an additional complex mode, as expected. However, the complex mode related to the AVR is far away
from the imaginary axis or damping ratio of the mode is higher. It is interesting to note that the damping ratio of
the critical mode has been reduced by pushing it toward the imaginary axis as AVR is put in place.
20
15
10
Imag
5
0
-5
-10
-15
-20
-80
-70
-60
-50
-40
Real
-30
-20
-10
0
Fig. 3 Eigenvalue plot of the system with machine and AVR.
Figure 4 shows the eigenvalue plot of the system with AVR and PSS installed at machine. As can been clearly
seen, the introduction of PSS has pull out the critical eigenvalue to the open left half plane by adding more
damping on it. Effect of adding Static Synchronous Series Compensator (SSSC) and Thyristor-Controlled Series
Compensator (TCSC) in addition to AVR and PSS are shown in Figs. 5 and 6, respectively.
25
20
15
10
Imag
5
0
-5
-10
-15
-20
-25
-80
-70
-60
-50
-40
Real
-30
-20
-10
0
Fig.4 Eigenvalue plot of the system with machine, AVR and PSS.
However, in a larger system the placement of these FACTS controllers and the best control input signals to add
more damping is critical issues. Controllability and observability indices methods can be used in identifying
locations and best control input signals respectively [26]. Extended eigenvector method also proposed in the
literature for placement of FACTS controllers for oscillation damping. In this simple system, SSSC and TCSC
are placed in transmission line between buses 2 and 3.
8
Introduction of SSSC and TCSC has introduced new eigenvalues as shown in Figs. 5 and 6, respectively and also
added damping on the critical mode as well as other complex mode. Comparison of critical mode with different
controllers is presented in Table 2.
20
15
10
Imag
5
0
-5
-10
-15
-20
-80
-70
-60
-50
-40
Real
-30
-20
-10
0
Fig. 5 Eigenvalue plot of the system with AVR, PSS and SSSC.
20
15
10
Imag
5
0
-5
-10
-15
-20
-80
-70
-60
-50
-40
Real
-30
-20
-10
0
Fig. 6 Eigenvalue plot of the system with AVR, PSS and TCSC.
Table 2 shows the comparison of critical eigenvalue, associated frequency of oscillation and damping ratio for
various cases. Introduction of AVR has reduced the damping ratio on the critical mode and the corresponding
initial frequency of oscillation has been increased, slightly. However, the combination AVR and PSS is the best
among all the cases and gives maximum damping ratio on the critical mode.
Introduction of SSSC and TCSC, in addition to PSS controller has not improved the damping ratio on the critical
mode. Comparison between SSSC and TCSC cases along with AVR show the damping ratio on the critical mode
is higher in SSSC along with AVR case. But when PSS is introduced along with SSSC and TCSC, TSCS is
showing a better performance. All these results have been verified with time domain simulation as shown in
Figs. 7 to 9. SSSC and TCSC give almost a comparable performance under different combinations.
9
TABLE 2 Comparison of Critical Eigenvalue, Frequency and Damping ratio
Case
Critical Eigenvalue
Base
AVR
AVR+PSS
AVR+SSSC
AVR+TCSC
AVR+PSS+SSSC
AVR+PSS+TCSC
Frequency
(Hz)
1.1738
1.2205
1.0417
1.0873
1.1096
0.9626
0.9795
-0.34310±7.3752
-0.19071±7.6685
-2.1570±6.54490
-0.31507±6.8315
-0.29064±6.9716
-1.8910±6.04850
-1.93660±6.1544
Damping
Ratio
0.0465
0.0249
0.3130
0.0461
0.0417
0.2984
0.3002
1.0004
1.0003
Rotor speed (pu)
1.0002
1.0001
1
0.9999
AVR
AVR+PSS
AVR+TCSC
AVR+SSSC
0.9998
0.9997
0.9996
0
0.5
1
1.5
2
Time (s)
2.5
3
3.5
4
Fig. 7 Comparison of rotor speed for different cases.
1.0004
AVR+PSS
AVR+SSSC
AVR+PSS+SSSC
1.0003
Rotor speed (pu)
1.0002
1.0001
1
0.9999
0.9998
0.9997
0.9996
0
0.5
1
1.5
2
Time (s)
2.5
3
3.5
4
Fig. 8 Comparison of rotor speed with PSS, SSSC and PSS and SSSC together.
Time domain simulation is performed for a small perturbation in mechanical power to the machine in all the
cases.
10
1.0004
AVR+PSS
AVR+TCSC
AVR+PSS+TCSC
1.0003
Rotor speed (pu)
1.0002
1.0001
1
0.9999
0.9998
0.9997
0.9996
0
0.5
1
1.5
2
Time (s)
2.5
3
3.5
4
Fig. 9 Comparison of rotor speed with PSS, TCSC and PSS and TCSC together.
Conclusions
The paper presents a comprehensive low frequency oscillation study in a simple test system to capture local
electromechanical mode in a step by step manner. Base case system shows an electromechanical mode with low
damping ratio and the introduction of AVR push the critical mode towards the imaginary axis by adding negative
damping on it. Adding PSS on the machine with AVR is pulling the critical eigenvalue by adding more damping.
Similarly, TCSC and SSSC, two of the series FACTS controllers also help system improving the damping which
got reduced due to AVR. However, adding TCSC or SSSC to the system with machine, AVR and PSS not
improving the damping on the critical mode, in this case. Overall, the PSS is found to be very effective in adding
damping on critical mode.
All the eigenvalue results have been corroborated with the helping of time domain simulations.
References
F. P. deMello and C. Concordia, „Concept of Synchronous Machines Stability as Affected by Excitation
Control‟, IEEE Trans. Power App. Syst., PAS-88 (April 1969), 189-202.
2. E. V. Larsen and D. A. Swann, „Applying Power System Stabilizers, Parts I-III‟, IEEE Trans. Power
App. Syst., PAS-100, (June 1981), 3017-3046.
3. P. Kundur, M. Klein, G. J. Rogers, and M. S. Zywno, „Application of Power System Stabilizers for
Enhancement of Overall System Stability‟, IEEE Trans. Power Syst., PWRS-4, (May 1989), 614-626.
4. P. Kundur, Power System Stability and Control, (McGraw-Hill: New York, 1994).
5. G. Rogers, „Demystifying Power System Oscillation‟, IEEE Computer Applications in Power, 9(3),
(July 1996), 30-35.
6. J. Machowski, J. W. Bialek and J. R. Bumby, Power System Dynamics and Stability, (John Wiley &
Sons: New York, 1997).
7. G. Rogers, Power System Oscillations, (Kluwer: Massachusetts, 2000).
8. J. H. Chow, J. J. Sanchez-Gasca, H. Ren, and S. Wang, „Power System Damping Controller Design‟,
IEEE Control Systems Magazine, (August 2000), 82-90.
9. IEEE/CIGRE, „Definition and Classification of Power System Stability‟, IEEE Transactions on Power
Systems, 19(2), (May 2004), 1387-1401.
10. B. Pal and B. Chaudhuri, Robust Control in Power Systems (Springer Inc. New York, 2005).
11. M. C. J. Angammana, Analytical study of factors affecting to electromechanical oscillations in power
systems, Master thesis, Asian Institute of Technology, Thailand, (May 2006).
1.
11
12. C. A. Cañizares, N. Mithulananthan, F. Milano, and J. Reeve, „Linear Performance Indices to Predict
Oscillatory Stability Problems in Power Systems‟, IEEE Trans. on Power Systems, 19(2), (May 2004),
1104-1114.
13. F. Milano, „An Open Source Power System Analysis Toolbox‟, IEEE Transactions on Power Systems,
20(3), (August 2005), 1199-1206.
14. F. Milano, PSAT: Power System Analysis Toolbox, version 2.0.0-beta, (March 2007).
15. IEEE Std 421.5-2005 „IEEE Recommended Practice for Excitation System Models for Power System
Stability Studies‟, IEEE, NY, USA, (April 2006).
16. J. F. Hauer, C. J. Demeure and L.L. Scharf, „Initial Results in Prony Analysis of Power System
Response Signals‟, IEEE Transactions on Power Systems, 5(1), (February 1990), 118-126.
17. J. F. Hauer, „Application of Prony Analysis to the Determination of Modal Content and Equivalent
Models for Measured Power System Response‟, IEEE Transactions on Power Systems, 6(3), (August
1991), 1062-1068.
18. D. J. Trudnowski, J. R. Smith, T. A. Short, and D. A. Pierre, „An Application of Prony Methods in PSS
Design for Multimachine Systems‟, IEEE Transactions on Power Systems, 6(1), (February 1991), 118126.
19. R. Wang and R. H. Lasseter, „Re-dispatching generation to increase power system security margin and
support low voltage bus‟, IEEE Transactions on Power Systems, 15(2), (May 2000), 118-126.
20. C. Y. Chung, L. Wang, F. Howell, and P. Kundur, „Generation rescheduling methods to improve power
transfer capability constrained by small-signal stability‟, IEEE Trans. Power Syst., 19(1) (February
2004), 524-530.
21. N. G. Hingorani, „Flexible AC Transmission‟, IEEE Spectrum, 30(4), (April 1993), 40-45.
22. J. J. Paserba, N. W. Miller, E. V. Larsen, and R. J. Piwko, „A Thyristor Controlled Series Compensation
Model for Power System Stability Analysis‟, IEEE Trans. on Power Delivery, 10(3), (July 1995), 14711478.
23. H. F. Wang, and F. J. Swift, „Capability of the Static VAr Compensator in Damping Power System
Oscillations‟, IEE Gene. Transm. Didtrib., 143(4), (July 1996), 353-358.
24. N. G. Hingorani, and L. Gyugyi, Understanding FACTS, (IEEE Press, 2000).
25. 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. on Power Systems,
18(2), (May 2003), 786-792.
26. B. K. Kumar, S. N. Singh and S. C. Srivastava, „Placement of FACTS controllers using modal
controllability indices to damp out power system oscillations‟, IET Gener. Transm. Distrib., 1(2),
(March 2007), 209-217.
27. Y. Mitani, K. Tsuji, and Y. Murakami, „Application of superconducting magnet energy storage to
improve power system dynamic performance‟, IEEE Trans. Power Syst., 3(4) (1988), 1418-1425.
28. M. G. Rabbani, J. B. X. Devotta, and S. Elangovan, „Application of Simultaneous Active and Reactive
Power Modulation of SMES Unit under unequal α-mode for Power System Stabilization‟, IEEE Trans.
on Power Systems, 14(2), (May 1999), 547-552.
29. Y. Mitani, K. Tsuji, and Y. Murakami, „Fundamental analysis of dynamic stability in superconductive
power systems‟, IEEE Trans. on Magnetics, 27(2), (March 1991), 2349-2352.
30. A. Soni and C. S. Ozveren „Improved Control of Isolated Power System by the Use of Feeding‟, IEEE
UPEC '06. Proceedings of the 41st International, 3(Sept. 2006), 974-977
12