Proc. 2001 IEEE-PES Summer Meeting, Vancouver, BC, July 2001.

Effects of Limits in Small Signal Stability

Analysis of Power Systems
André A. P. Lerm Claudio A. Cañizares Nadarajah Mithulananthan
MIEEE SMIEEE SIEEE
E&ARQ School E&CE Department
Universidade Católica de Pelotas University of Waterloo
Pelotas, RS, 96010-000, Brazil Waterloo, ON, N2L-3G1, Canada
alerm@atlas.ucpel.tche.br c.canizares@ece.uwaterloo.ca

Abstract This paper concentrates in demonstrating and evolution of certain system parameters (e.g., load demand
analyzing the effects of limits on power system eigenvalue changes) [5], [6]. Local bifurcations having a significant
computations when system parameters change. A 3-bus system effect on the stability of the system and most studied in the
is used to illustrate and analyze the effect that limits and their literature are the saddle-node (one of the eigenvalues
modeling have on the system eigenvalues. The results obtained becomes zero) and the Hopf bifurcations (a pair of complex
for the simple test system using a classical eigenvalue
computational tool versus those obtained using a tool that
eigenvalues cross the imaginary axes of the complex plane).
adequately models the effects of limits on the eigenvalue The inclusion of limits, however, brings up a new type of
computations are compared and discussed, based on time bifurcation usually referred to as limit induced bifurcation,
domain simulations to confirm the validity of the various results which corresponds to eigenvalues undergoing instantaneous
presented here. changes that may affect the stability of the system, such as
Keywords: Small signal stability analysis, control limit stable eigenvalues turned into unstable ones [4], [7], [8].
modeling, bifurcation theory, power flow. Several methodologies have been discussed in the
literature for the accurate modeling of control limits in power
I. INTRODUCTION system analysis. The authors in [9] study the problem using
the solution of the differential equations representing the
Modern power systems are operating increasingly closer power system as a reference to evaluate inherent
to their control and operational limits, such as those imposed approximations employed in the conventional load flow
by generator Automatic Voltage Regulators (AVR) or other programs. In [10], the problem is analyzed using the actual
devices. In general, this scenario has originated on the equilibrium point of the DAE power system model. A model
increase in demand for electric energy coupled with for generator reactive power and voltage dynamics is
economic and environmental restrictions on power system incorporated in the load flow problem in [11], where the
expansion. Since stressed conditions can lead a system to generator voltage variations are accounted for in the
unstable conditions, it has become necessary to properly calculation of the limits. In [12], the authors present a static
model the effect of such limits on the system. model for the synchronous generators with voltage dependent
It is recognized in the literature that gradual system reactive power limits. This generator model is included in an
parameter variations such as load increase, combined with ordinary power flow program; however, it uses a simplified
contingencies, lead to system instabilities. In large- model of the generator disregarding the voltage regulator. In
disturbance stability analysis, the problem is usually studied [8], a detailed analysis of hard limits in nonlinear dynamic
through time domain simulations. The inclusion and systems is presented in state and parameter space. The static
modeling of limits in software tools used to perform these and dynamic aspects of voltage collapse associated with
types of analyses is well known [1], [2], [3], and is accurately generator reactive power limits are studied using bifurcation
performed by most commercial software packages currently theory in [7]. This work considers that, when a generator
available. On the other hand, from the authors’ experience reaches its field voltage limit, the generator internal voltage
and tests, which form the core of this paper, there is a need to can be treated as a constant value; the dynamics of the
improve the representation and handling of limits on voltage regulator are not modeled in this case. In [13], the
computational packages designed to perform Small Signal limits are represented by hyperbolic functions that allow to
Stability analysis (SSSA). obtain an analytic formulation for the nonlinear equations.
In SSSA, systems are studied using the eigenvalues of the This paper addresses the modeling of generator field
linearization around an operating point of the differential- voltage limits in SSSA. The main aim is to discuss the effect
algebraic equations (DAE) used to model the systems. that these limits have on the system eigenvalues. Thus, a
Although this type of analysis had been widely performed on methodology is first proposed to properly represent these
power systems since the 1960’s, new insight has been gained limits in a conventional power flow, so that initial conditions
into the problem from the application of bifurcation theory to coherent with the modeling of the limits in modal analysis
power system stability analysis since the 1990’s [4]. From and time domain simulations are obtained. Second, a
the point of view of bifurcation theory, local bifurcations and technique to adequately model hard-limits in a linearized
hence system stability are studied through the determination system model is presented and discussed. Finally, the
of a series of system eigenvalues associated with the gradual proposed methodologies are tested in a simple 3-bus system,

1
comparing the results obtained with different eigenvalue Z max
computation programs; time domain simulations of the test
system are used to validate the proposed modeling u K v z
techniques.
This paper is organized as follows: Section II presents a 1+sT
brief overview of eigenvalue computation in power systems Z min
from the point of view of nonlinear system theory, discussing
as well the main issues associated with the proper modeling Fig. 1. Control block with a windup limiter.
of control limits. Section III discusses the modeling of hard- where z∈ℜ represents the variables on which the windup
p
limits in a power flow program for the determination of
proper initial conditions for dynamic analyses, and the proper limiters operate, and h :ℜn×ℜm×ℜl→ℜp is the nonlinear
modeling of these limits in modal analysis. Section IV algebraic function which represents the windup limiters as
presents the numerical results obtained for a simple 3-bus follows:
system, confirming the importance of proper limit modeling  zi for hi ( x, y, µ ) ≤ z i min
in SSSA. Finally, Section V summarizes the main results  min
hi ( x, y, µ ) = hi ( x, y , µ ) for z i min < hi ( x, y, µ ) < z i max (4)
presented in this paper. z for hi ( x, y, µ ) ≥ z i max
 i max i

II. EQUILIBRIA AND EIGENVALUE COMPUTATION where h:ℜn×ℜm×ℜl→ℜp is a smooth algebraic function (Cq,
q≥1).
The stability analysis of power system uses models with a By considering that the inclusion of windup limits does
set of differential and algebraic equations (DAE) of the form not affect the order n of the differential equations, the system
x& = f ( x, y, µ ) (3)-(4) can be simplified and rewritten, without loss of
(1)
0 = g ( x, y , µ ) generality, as
where x∈ℜn typically stands for state variables corresponding x& = f ( x, y, µ )
(5)
to various system elements, such as generators and their 0 = g ( x, y , µ )
controls; µ∈ℜl stands for slow varying parameters over
which operators have no direct control, such as changing where the vector field f and the algebraic function g
loading levels,; f:ℜn×ℜm×ℜl→ℜn corresponds to the include implicitly the actuation variables z.
nonlinear vector field directly associated with the state A. Equilibria and Steady-state Operating Points
variables x; and the vector y∈ℜm represents the set of
algebraic variables defined by the nonlinear algebraic In SSSA, the equilibrium points used as initial conditions
for the linearization process are obtained usually from a
function g:ℜn×ℜm×ℜl→ℜm, which typically correspond to
power flow analysis. Hence, the typical procedure is to first
load bus voltages and angles, depending on the load models
solve the power flow equations and then, based on the
used.
corresponding solutions, find equilibrium points of the
The stability of DAE systems is thoroughly discussed in
dynamic model before proceeding with stability analysis of
[14], where is shown that if Dyg(x,y,µ) can be guaranteed to
the full dynamic system (1). If one is interested in analyzing
be nonsingular along system trajectories of interest, the
the system behavior with respect to changes in a given system
behavior of system (1) along these trajectories is primarily
parameter (e.g., load demand), a set of equilibrium points can
determined by the eigenvalues of the Jacobian matrix
be obtained and depicted in the form of nose curves (e.g., PV
A = D x f |o −( D y f |o [ D y g |o ]−1 D x g |o ) (2) or VQ curves) [4], [15].
The smoothness of the system (1) is destroyed when If there are no limits included in the analysis, the system
limits are incorporated in its model. Although limits can be can be modeled using (1), with the resulting nose curves
classified as windup, nonwindup and relay limits [8], the having a “smooth” shape. With the inclusion of limits, these
present work addresses only the inclusion of windup limits, curves remain continuos but with “sharp” edges at the points
such as those usually imposed on generator field voltages. where limits become active.
A windup limiter applied to a first order control block is B. Eigenvalues
depicted in Fig. 1. Since this type of hard-limit does not
affect directly the associated state variable, there is no change After the computation of the system equilibria and the
in the order n of the system when a limit value is reached. related initial conditions, the local dynamic characteristics of
The inclusion of windup limits in (1) leads to the system can be analyzed using the eigenvalues of the
x& = f ( x, y, z , µ ) Jacobian matrix (2) to assist in its analysis and/or design.
These types of studies are usually performed to improve the
0 = g ( x, y , z , µ ) (3) damping of system oscillations, and more recently to analyze
z = h ( x, y , µ ) voltage collapse problems.
The system dynamic behavior is affected by parameter
changes, resulting in changes to the stability characteristics of
equilibrium points, which can be depicted on the resulting PV

2
curve. Points of interest on these PV curve are those power limits are determined considering only the maximum
associated with changes in the structural stability of the and minimum field voltage Efd.
system, which correspond to bifurcation points [16], [17]. An analytic expression that relates the minimum and
For system (1), instabilities occur generally via saddle- maximum reactive power as a function of the field voltage
node or Hopf bifurcations. Saddle-node bifurcations are limits is difficult to obtain. Therefore a numerical procedure
characterized by a couple of equilibrium points merging at which searches for the reactive power values corresponding
the bifurcation point and then locally disappearing as the to the Efd limits imposed by the voltage regulator is used here.
slow varying parameters µ change. These bifurcations have Thus, assuming that the values of active power dispatched PG
been associated with voltage collapse problems [4], and and terminal voltage Vt are known, the maximum reactive
correspond to an equilibrium point (xo,yo,µo) where the power value QGmax can be determined using the algorithm
system Jacobian A (2) has a unique zero eigenvalue (certain depicted in Fig. 2; this algorithm can be readily adapted in
transversality conditions are also met at this point, order to determine the value of QGmin.
distinguishing it from other types of “singular” bifurcations
[18]). Hopf bifurcations are characterized by a complex 1. Determine Efdmax based on the voltage regulator equations
conjugate pair of eigenvalues crossing the imaginary axes of for given values of PG and Vt.
the complex plane from left to right, or vice versa, as the µ 2. Choose an initial guess for QG (with a value that yields the
parameters slowly change. These types of bifurcations have correct search direction).
been associated with a variety of oscillatory phenomena in 3. Determine Efd, using the generator equations and QG and
power systems [18], [19], and are typical precursors of Vt values.
chaotic motions [5], [6]. 4. If Efd max-Efd < ε, make QGmax = QG and stop the searching
The inclusion of limits leads to the appearance of limit- process. Otherwise, increase QG and go to step 3.
induced bifurcations [8]. These bifurcations correspond to Fig. 2. Algorithm for QG limits determination.
equilibrium points where system limits are reached as the
parameters µ slowly change, with the corresponding The proposed methodology is used to compute the
eigenvalues undergoing instantaneous changes that may reactive power limits at each iteration of the power flow
affect the stability status of the system, such as stable program. This methodology ensures a minimum
eigenvalues turning into unstable ones [7], [8]. A rich set of modification of the conventional power flow program,
new phenomena directly associated with windup limits can be making the generator reactive power limits coherent with the
encountered, ranging from annihilation of equilibria to corresponding AVR field voltage limits used in the modal
emergence of oscillations. analysis and time domain simulations.
D. Inclusion of Windup Limits in a Linearized Model
III. LIMIT MODELING
Since the SSSA is based on the linearization of DAE
The inclusion of limits in the small signal stability system (5), the correct treatment of hard-limits requires not
analysis brings up two important points: only an accurate determination of the equilibrium points
based on solution of the power flow equations, but also the
1. In the determination of an equilibrium point, which proper modeling of these limits in the linearized model.
involves the solution of a power flow and the solution of
(5) for x& = 0 , control limits must be represented in the
power flow and DAE problems in a coherent way.
Zmax z=h(
2. The control limits in (5) must be properly modeled to
ensure that the system Jacobian can be adequately 1
computed. dz/dv

These two issues are addressed in the following sections. v

C. Inclusion of Efd Limits in a Conventional Power Flow Program
Zmin
If hard-limits are present, as in the case of the DAE set Fig. 3. Relationship between variables for the windup limiter.
(5), these must be considered in an adequate manner in the The relationship between the input and output variables v
subset of equations used for power flow analysis. and z, respectively, of the windup limiter of Fig. 1 are
Specifically, generator limits are typically modeled as illustrated in Fig. 3, which is a graphical representation of
constant reactive power limits. However, this is an over equation (4). Observe that within the limits, z = v; hence, the
simplification, as these limits change with the active power linearization procedure yields
dispatch and the generator terminal voltage; a realistic dh(v)
representation of the generator reactive power limits requires ∆z = ∆v (6)
the determination of the capability curve of the machine [2], dv
where
[3], which depend on power factor, mechanical power, and
field and armature current limits. In this paper, the reactive

3
  0 for h(v) ≤ z min equilibrium points, obtained from the power flow program,
dh(v)  consistent with the AVR field voltage limits.
:=  1 for z min < h(v) < z max (7) 140
dv  0 for h(v) ≥ z
 max
138

IV. NUMERICAL RESULTS X

A
136

This section presents the results of applying the

134 B

Qgen 2 (MVAR)
previously discussed concepts to the simple three-bus system
of Fig. 4. In this system, one of the generators is assumed to 132
be an infinite bus, and the other generator is modeled using 5
differential equations (2 for the mechanical dynamics, 3 for 130
the transient dynamics), a simple AVR modeled using the X − gen 2 reaches limits
first order dynamic model depicted in Fig. 1, and no 128 A − limitation with Qgen max fixed
governor. The infinite bus picks up changes in the active B − limitation with Efd max fixed
power demand. The data for this system is shown in Table I. 126

1 2 124
1.65 1.7 1.75 1.8 1.85 1.9 1.95
IB G µ
Fig. 5. Generator reactive power for limits based on (A) constant maximum
reactive power and (B) constant maximum field voltage.

3 2.46

L 2.44
Fig. 4. Three-bus sample system.
2.42 A
TABLE I
P.U. DATA FOR 3-BUS SAMPLE SYSTEM (100.0 MVA BASE). X
Efd gen 2 (p.u.)

2.4
H 10.0 s τ’do 8.5 s Plo 0.8 B
Xd 0.9 τ”do 0.03 s Qlo 0.6 2.38
Xq 0.8 τ”qo 0.9 s Vt1 1.02
2.36
X’d 0.12 Pm 1.0 Vt2 1.0
X − gen 2 reaches limits
X”d 0.08 KAVR 1000.0 Efd2max 2.4 2.34 A − limitation with Qgen max fixed
X”q 0.08 TAVR 2.0 s Xlines 0.2 B − limitation with Efd max fixed
2.32

A. Equilibrium Points
2.3
1.65 1.7 1.75 1.8 1.85 1.9 1.95
The first set of results illustrates different ways to µ
represent generator limits in the determination of initial Fig. 6. Generator field voltage for limits based on (A) constant maximum
reactive power and (B) constant maximum field voltage.
conditions based on a conventional power flow program.
These results were obtained increasing the load on bus 3 by
changing the load parameter µ, as PL = PLo(1+µ) and QL =
QLo(1+µ), where PLo and QLo are the initial load values.
The generator at bus 2 reached its maximum field voltage
limit at µ = 1.772 (point X on Figs. 5, 6 and 7). This
occurred at QG2max = 136.1 MVAR, for Efd2max = 2.4 p.u. and
Vt2 = 1.0 p.u. If the generator limits are treated in the
conventional way, i.e., keeping the reactive power constant
after the maximum limit is reached, curves A on Figs. 5 and 6
are obtained. In this case, the field voltage Efd must increase
in order to keep the generator reactive power constant, since
the terminal voltage decreases as the load increases (Fig. 6).
Hence, the equilibrium points are not coherent with the field
voltage limits represented in (5). Curves B on Figs. 5 and 6
show the generator characteristics when a constant maximum
field voltage is used; observe that the generator reactive
power decreases as the load increases. This would yield Fig. 7. Voltage profile at bus 3, depicting Hopf bifurcation points obtained
from two different eigenvalue programs for constant reactive power limits.

4
 TABLE II
TEST SYSTEM EIGENVALUES
µ Program 1 with limits SSSP
0.0 -39.60; -1.994 -39.532; -1.999
-1.127 ± j 8.365 -1.035 ± j 8.4
-0.543 ± j 5.523 –0.9118 ± j 5.458
1.75 -38.68; -2.545 -39.36 ; -2.149
-1.206 ± j 8.568 -1.141 ± j 8.495
-0.079 ± j 6.060 -0.7293 ± j 5.620
1.772 -37.69; -2.841 -38.39 -2.561
-1.184 ± j 8.735 -1.151 ± j 8.57
-0.5; -0.305 -0.4665 ± j 6.077
1.849 -37.23; -2.862 -37.94; -2.609
-1.166 ± j 8.602 -1.128 ± j 8.388
-0.5; -0.281 -0.4453 ± j 6.290
1.939 -33.11; -2.689 -34.15; -2.717
-1.082 ± j 8.235 -1.519 ± j 7.83
-0.5; 0.000 0.4357 ± j 7.975
B. Stability Analysis
Several bifurcation points for the test system are depicted
on the PV curve of Fig. 7 for the voltage at bus 3. The
bifurcation points were obtained from two different programs
designed for the computation of eigenvalues in power
systems. The first program is based on the well know
EISPACK package, and implements the linearization of both
Fig. 8. Time domain simulation for µ = 1.75 (load increased by 1% at 5 s).
equations (1) and (5) [1], [2], [3]. The second program used
was the software package SSSP [20]. Equilibrium points
were obtained using a standard power flow program that
models generator limits as constant reactive power limits,
defining the limits based on the technique illustrated in Fig. 2.
Three bifurcation points are depicted on the PV curve of
Fig. 7. The first program yields the Hopf HP1 at µ = 1.849
when no limits are neglected, while the second program
MASS yields the Hopf HP2 at µ = 1.933. A limit-induced
bifurcation point X is obtained at µ = 1.75 with the first
program when the AVR limits are properly considered in the
computation process; this program also yields a saddle-node
bifurcation at the maximum loading point µ = 1.939.
Table II shows the eigenvalues obtained with the two
different programs. Observe that when limits are properly
model, the system presents a stable limit-induced bifurcation
when the limits become active at µ = 1.75, as a complex pair
of eigenvalues become real. The second program fails to
detect this structural change, yielding results that indicate that
the effect of limits is not considered in the linearization
process. Also, at the maximum loading point µ = 1.939, the
first program yields a zero eigenvalue, suggesting the
presence of a saddle-node bifurcation, while the second one
implies that the system has gone through a Hopf bifurcation.
To determine which of the two eigenvalue programs is
correct, time domain simulations were carried out using the Fig. 9. Time domain simulation for µ = 1.93 (load increased by 1% at 5 s).
transient stability program ETMSP [21]. A sudden 1 % load bus 2 for µ = 1.75 and µ = 1.93, respectively. Observe the
change was applied to the system at 5 s for different initial small voltage and speed change in the first case. On the other
values of µ. Figures 8 and 9 show the time trajectories for hand, the voltage collapses for the second value of load,
the load voltage at bus 3 as well as the generator frequency at confirming the results obtained with the first program (the
system does not completely collapses, as the load model is

5
automatically switched by the program from a constant power [12] P.A. Löf, G. Andersson, and D.J. Hill, “Voltage dependent
to a constant impedance model when the load voltage is reactive power limits for voltage stability studies,” IEEE
below a certain threshold); no system oscillations are Trans. Power Systems, vol. 10, no. 1, 1995, pp. 220-228.
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V. CONCLUSIONS [15] C. Rajagopalan, B. Lesieutre, P.W. Sauer, and M.A. Pai,
“Dynamic aspects of voltage/power characteristics,” IEEE
Trans. Power Systems, vol. 7, no. 3, 1992, pp. 990-1000.
The paper analyses the effects of control limits on SSSA, [16] B.C. Lesieutre, P.W. Sauer, and M.A. Pai, “Why
demonstrating the importance of correctly modeling these Power/Voltage Curves Are Not Necessarily Bifurcation
limits in the determination of equilibrium points and in the Diagrams,” Proc. North American Power Symposium,
linearization process. A methodology is proposed to properly Washington, Oct. 1993, pp. 30-37.
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flow in order to obtain initial conditions coherent with the “Multi-parameter Bifurcation Analysis of Power Systems,”
dynamic models used in modal analysis and time domain Proc. North American Power Symposium, Cleveland, Oct.
simulations. A thorough discussion on how to model windup 1998.
[18] C.A. Cañizares and S. Hranilovic, “Transcritical and Hopf
limits for inclusion in a linearized model is also presented.
Bifurcations in AC/DC Systems,” Proc. Bulk Power System
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M.Sc. and Ph.D. degrees in Electrical Engineering from Universidade
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[6] R. Seydel, Practical Bifurcation and Stability Analysis-From Federal de Educação Tecnológica-RS, respectively. His main research
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[7] I. Dobson and L. Lu, “Voltage collapse precipitated by the
immediate change in stability when generator reactive power Claudio A. Cañizares received the Electrical Engineer diploma (1984) from
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different positions from 1983 to 1993. His M.Sc. (1988) and Ph.D. (1991)
vol. 39, no. 9, 1992, pp. 762-766. degrees in Electrical Engineering are from the University of Wisconsin-
[8] V. Venkatasubramanian, H. Schättler, J. Zaborsky, “Dynamics Madison. Dr. Cañizares is currently an Associate Professor at the University
of large constrained nonlinear systemsA taxonomy theory,” of Waterloo and his research activities are mostly concentrated on the study
Proceedings of the IEEE, Special Issue on Nonlinear of computational, modeling, and stability issues in ac/dc/FACTS power
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(Eng.) and M.Eng. degrees from the University of Peradeniya, Sri Lanka, and
representation of generator and load dynamics in the study of
the Asian Institute of Technology, Thailand, in May 1993 and August 1997,
voltage limited power system operations,” IEEE Trans. Power respectively. Mr. Mithulananthan has worked as an Electrical Engineer at
Systems, vo. 12, no. 1, Feb. 1997, pp. 304-314. the Generation Planning Branch of the Ceylon Electricity Board, and as a
[10] R.A. Schlueter and I-P. Hu, “Types of voltage instability and Researcher at Chulalongkorn University, Thailand. He is currently a full
the associated modelling for transient/mid-term stability time Ph.D. student at the University of Waterloo working on applications and
simulation,” Electric Power Systems Research, vol. 29, 1994, control design of FACTS controllers.
pp. 131-145.
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