1
Bifurcation Analysis on a Detailed Multimachine
Power System Model
Gustavo Revel, Andrés E. León, Diego M. Alonso and Jorge L. Moiola
Abstract—In this article the dynamics of a multimachine power
system model is studied using bifurcation theory. The classical 9
bus WSCC power system model is considered. It is represented
using a full differential ODE set, including the corresponding
dynamics of the control loops and the transmission lines. The
analysis is carried out using the load demands as bifurcation
parameters, in order to reveal possible dynamical scenarios for
different loading conditions. For variations of one of the loads,
Hopf and saddle-node bifurcations were detected. Furthermore,
the bifurcation analysis varying two loads simultaneously reveals
the existence of a pair of double-Hopf and a zero-Hopf bifurcations, acting as organizing centers of the dynamics.
Index Terms—Multimachine power systems, bifurcation analysis, oscillations, double-Hopf and zero-Hopf bifurcations.
I. I NTRODUCTION
A
CTUAL power systems are continuously approaching
their stability limits due to an increasing power demand
and other factors such as quality regulations, environmental
and economic restrictions. In addition, the operating point of a
power system network is continuously suffering perturbations
and depending on their severance the system may become
unstable. Instability might be caused by many different phenomena (or due to a combination of them), the most common
are the changes in the load consumption that reaches the limits
of the transmission network or the generation capacity, action
of badly tuned controls such as automatic voltage regulators
(AVR) and power system stabilizers (PSS), line and generator
tripping, etc. Some countermeasures to improve the system
robustness are the reinforcement of the transmission lines
between the generation and load centers, improvement in
control devices and real-time communications, decisions based
on reliable information and system contingency plans such as
emergency load shedding schemes [1].
Since a power system is essentially a non-linear system with
a set of parameters, like controller gains, load parameters, etc.,
bifurcation theory seems to be a suitable approach to study its
dynamics. There are several articles that introduce this topic
[2]–[6], where a description of the fundamental implications of
bifurcations like saddle-node and Hopf in the system stability
are presented. One of the most popular benchmark to study
bifurcations in power systems was introduced in [7], where the
authors presented a simple 3-bus model including a variable
load. This particular example has been widely studied by many
The authors are with Instituto de Investigaciones en Ingenierı́a Eléctrica IIIE (UNS-CONICET) and Departamento de Ingenierı́a Eléctrica y de Computadoras, Universidad Nacional del Sur, Av. Alem 1253, (8000) Bahı́a
Blanca, Argentina. (E-mail: grevel@uns.edu.ar).
The authors appreciate the financial support of UNS (PGI 24/K041),
ANPCyT (PICT 06-00828) and CONICET (PIP 5032).
researchers using different sets of parameter values (see [8]
and references therein). The dynamics of this simple model has
proved to be complex including oscillations, voltage collapse
and chaos.
In this work, one and two parameter bifurcation analyses are
carried out for a multimachine detailed model of the common
9 bus Western Systems Coordinating Council (WSCC). The
system is formulated as an entirely differential ODE model
composed by three generators with their corresponding control
loops, and three aggregated loads formed by induction motors
and static components. The dynamics of the transmission lines
and stators are included to perform a bifurcation analysis using
standard continuation packages like MATCONT. The underlying goal is to analyze the system behavior for different loading
scenarios and to show that useful information, related to
instabilities and oscillations can be extracted using bifurcation
theory. Additional information, not evident from a classical
linear analysis, can be obtained performing a bifurcation study
varying two parameters simultaneously. For example, it is possible to determine regions of qualitative different dynamics and
find organizing centers where bifurcation branches emerge.
There are few papers that perform a multiparameter bifurcation
analysis in multimachine power systems, some examples are
[9] and [10], but they are oriented to the use of bifurcation
diagrams to find appropriate values of control parameters, like
the AVR gains.
This work is organized as follows. In Section II the multimachine power system model is described. In Section III the
dynamical behavior of the system varying one of the loads is
studied using the bifurcation diagram. Then, in Section IV, a
two parameter bifurcation study varying two loads simultaneously is performed. A description of some organizing centers
of the dynamics is also presented in this section. Finally,
concluding remarks are given in Section V.
II. M ULTIMACHINE P OWER S YSTEM M ODEL
The analysis presented in this work is based on the 9-bus
model of the WSCC power system represented schematically
in Fig. 1 [11], [12]. It is composed by three generators, one
hydro unit (#1) and two steam units (#2 and #3) with their
respective voltage controls (Automatic Voltage Regulator AVR) and a frequency regulator (Governor - GOV) placed
only on generator #1. There are three composite loads with
static and dynamic characteristics at buses 5, 6 and 8. The set
of equations used in this article are described in the following
subsections. All the variables are in per unit. The definition
of the state variables and the parameter values are listed
respectively in Tables II and III of Appendix A.
2
Fig. 1. Schematic representation of the 9-bus WSCC system, including the voltage and frequency control loops. The values of the parameters of the
transmission lines are indicated on each one. The nominal operating point is also indicated.
A. Synchronous Generators
The mathematical model of the generators is a 6-th order
ODE set that considers the mechanical and electrical behavior of the rotor and the stator dynamics. The later ensures
compatibility between the generator model and the network
dynamics, presented in subsection II-D. The model is valid for
both hydraulic and steam-turbine generators, and the equations
for the i-th (i = 1, 2, 3) generator are,
δ̇i = ΩB (ωi − ωs ) ,
2Hi ω̇i = Pmi − Pei − Kdi (ωi − ωs ) − Dbi ωi ,
Td0i
ėqi = −eqi + (Xdi − Xdi
) idi + ef di ,
Tq0i ėdi = −edi − Xqi − Xqi iqi ,
Xqi
Tq0i
i̇qi
=
−
Xdi
Td0i
i̇di =
−
−edi − iqi (Xqi − Xqi
)
Tq0i ΩB (vqi − idi Xdi − eqi + iqi Rsi ),
eqi − ef di − idi (Xdi − Xdi
)
Td0i ΩB (vdi + iqi Xqi − edi + idi Rsi ),
(1)
(2)
(3)
(4)
(5)
(6)
iqi idi .
with Pei = eqi iqi + edi idi + Xdi
− Xqi
B. Excitation Systems
Two types of normalized excitation systems and AVR will
be considered for the generators. The first is the IEEEDC1A DC Excitation System, for generator #1. This is a
slow response DC excitation/AVR system. The second one is
the IEEE-ST1A+PI Static Excitation System, for generators
#2 and #3. This excitation/AVR model is based on static
elements and it has a fast dynamic response [13]. The block
diagrams of these control systems are shown in Fig. 1 and the
corresponding equations are not included here since they are
simple to extract from the diagrams.
C. Prime-Mover Turbine and Governor
The dynamical behavior of the hydraulic turbine attached
to generator #1 is represented by a nonlinear model used
for transient stability studies [14]. The linear dynamic of the
servomotor driving the gate of the turbine is also considered.
The governor unit used for primary and secondary frequency
regulation is a PI controller, and a transient term is added to
compensate the non-minimum phase effect introduced by the
turbine. The equations of the turbine-governor can be obtained
from the block diagram included in Fig. 1.
D. Transmission Network
In the traditional DAE model of a multimachine power
system, the dynamics of the transmission network is considered so fast that it is neglected [15]. This model is difficult
to use in bifurcation studies, since the most common continuation packages, such as AUTO [16] or MATCONT [17],
do not support this type of representations. Although there
are algorithms based on continuation methods developed for
3
transformed to the synchronous DQ frame (as shown in Fig. 3)
before being considered in in .
E. Composite Load Model
Fig. 2.
The load modeling is a critical issue in power system
analysis, since the stability properties of a given network
is strongly dependent on the load considered. This is an
extremely complex subject that is still under investigation [21].
In this paper, following the recommendation of [22]–[24],
composite loads consisting of static and dynamic elements
are used. The static portion is a constant ZIP (impedance (Z),
current (I) and power (P)) load. Then, the active and reactive
power drawn by the j-th (j = 5, 6, 8) load are,
vj2
vj
+ PP j ,
(12)
Pj = αj PZj 2 + PIj
V0j
V0j
vj2
vj
Qj = βj QZj 2 + QIj
+ QQj ,
(13)
V0j
V0j
A single transmission line π equivalent circuit.
1
Fig. 3.
Reference frame transformation.
tracking equilibria bifurcations in DAE systems [18], [19],
they do not offer additional features such as continuation of
periodic solutions and their bifurcations. Therefore to make the
power system model compatible with the classical continuation
packages, a fully differential ODE model of the multimachine
power system must be derived. Toward this end the fast
dynamics of the currents and voltages of the transmission
network are considered [4], [20]. It is important to notice
that all the transmission system variables are referred to a
synchronous rotating DQ frame.
The lines are represented by the lumped parameter π
equivalent circuit shown in Fig. 2. The values of the network
components are indicated in Fig. 1. Then, for each transmission line there are two differential equations describing the
current
Ω−1
B Lnm i̇Qnm =vQn −vQm −Rnm iQnm +ωs Lnm iDnm ,(7)
−1
ΩB Lnm i̇Dnm =vDn −vDm −Rnm iDnm −ωs Lnm iQnm .(8)
In addition, there are two differential equations for the dynamics of the bus voltages (excluding generation buses)
Ω−1
B Cn v̇Qn = iQCn + ωs Cn vDn ,
−1
ΩB Cn v̇Dn = iDCn − ωs Cn vQn ,
(9)
(10)
where iQCn and iDCn are the DQ components of the current
through the capacitor Cn , obtained from the node equation
iCn = in − inm − iL ,
(11)
where iL is the current in the load (if any), inm is the current
between the nodes n and m and in represents all other currents
involved in node n. Notice that, since the currents of the
generators are expressed in their own frame, they must be
2
2
+ vDj
) 2 , is the voltage magnitude of
where, vj = (vQj
the corresponding load bus (referred to the synchronous DQ
frame). The DQ components of the current drained by the ZIP
load should be obtained from (12) and (13) to calculate the
corresponding part of the load current iL in (11).
The dynamic portion of the load is given by induction
motors with a two axis transient model including the stator
dynamics. The differential equations are
2Hj ω̇j = αj (Aωj2 + Bωj + C)Pmj
− Pej − Dbj ωj ,
Tdq0j
ėqj = −eqj + (Xdqj − Xdqj
)idj
− Tdq0j ΩB (ωj − ωs )edj ,
Tdq0j ėdj = −edj − (Xdqj − Xdqj
)iqj
+
Tdq0j
i̇qj
Xdqj
=
+
Xdqj
Tdq0j
i̇dj =
+
(14)
(15)
(16)
Tdq0j
ΩB (ωj − ωs )eqj ,
Tdq0j
ΩB (Xdqj
idj + eqj − vQj )
(17)
iqj (Xdqj − Xdqj − Rsj Tdq0j
ΩB ) − edj ,
Tdq0j
ΩB (−Xdqj
iqj + edj − vDj )
(18)
idj (Xdqj − Xdqj − Rsj Tdq0j ΩB ) + eqj ,
with Pej = eqj iqj + edj idj . The parameter αj is introduced
to vary in equal proportions the power consumption of the
static and the dynamic parts of the loads [see Eqs. (12) and
(14)]. This parameter indicates the level of power demand,
and αj = 1 corresponds to the nominal operation point. The
parameter βj has a similar function, but for the reactive power
demand. Notice that the induction motor do not have angle
dynamics and thus the variables are in the synchronous DQ
frame and are directly added in the total load current iL .
III. O NE PARAMETER B IFURCATION A NALYSIS
The nominal operating point of the WSCC model, i.e. the
power transmitted by the generators and the power drained
by the loads is obtained from [11], and indicated in Fig. 1.
This point is locally stable and it should be robust enough to
4
(a)
(b)
Fig. 4. Bifurcation diagrams varying the power consumed by the load connected to bus 5 (α5 ). (a) Bifurcation diagram varying α5 with Pm2 = 1.90 p.u.
(b) Expanded view of the rectangle in Fig. 4a.
TABLE I
G ENERATOR POWERS : NOMINAL AND RE - DISPATCHED .
Gen.
#1
#2
#3
Nominal
P
Q
0.716
0.270
1.630
0.067
0.850
-0.109
Re-dispatched
P
Q
0.482
0.314
1.900
0.129
0.850
-0.088
overcome typical perturbations, such as small variations in the
demand of the loads.
In this work the generation units are re-dispatched in order
to analyze a stressed operating condition. This is a common
practice on a real system due to changes in the load demand,
security reasons, economic considerations, physical restrictions or emergency conditions. It is important to mention that
the re-dispatch can lead to oscillatory instability [25], [26]. In
the following, the mechanical power of generator #2 is set near
the maximum value allowed (Pm2 = 1.90 p.u.), the generation
unit #1 will diminish the power generated due to the governor
action, and the set point of generator #3 is left unchanged.
The re-dispatch of generators is indicated in Table I.
The dynamical behavior of the power system is analyzed
in terms of variations of the active power of the load at bus
5, i.e. considering variations of parameter α5 . The behavior
of the magnitude of the voltage at the load bus 5 (|V5 |) is
depicted in the bifurcation diagram of Fig. 4. Solid lines
represent stable operation conditions, while the unstable points
are shown with dashed lines. The filled dots represent the
amplitude of stable oscillations (limit cycles) and unfilled dots
are unstable periodic solutions.
Let us now describe the bifurcation diagram. Decreasing
the load on bus 5 below the nominal value (α5 = 1) the
equilibrium point becomes unstable due to a Hopf bifurcation,
namely H1 in Fig. 4a. Consequently, a pair of complex
eigenvalues of the linearization crosses the imaginary axis
(and the non-degeneracy conditions are satisfied [27]) for
α5 = 0.9050. Hence, the equilibrium point becomes unstable
and a stable limit cycle (sustained oscillation) emerges towards
the left of H1 , i.e. for decreasing values of α5 . Notice that even
though the linearization around the equilibrium has a pair of
unstable complex eigenvalues, there is a stable oscillation that
may prevent the system to diverge.
When the load on bus 5 is increased from its nominal value,
the bus voltage begins to decrease. Then, after a significant
increment of α5 , the equilibrium point becomes unstable at
α5 = 2.1282 due the Hopf bifurcation H2 shown in Fig. 4b
(blow up of the rectangle in Fig. 4a). The equilibrium recovers
stability after the Hopf bifurcation H3 located at α5 = 2.1289.
Afterwards it collides with the saddle point (denoted with
dashed lines in Fig. 4a) and both disappear in a saddle-node
bifurcation LP . Beyond this value of α5 there are no possible
operating points.
The unstable limit cycle born at H2 collides with the stable
cycle created at H3 in a cyclic fold bifurcation CF for
α5 = 2.1280. Hence, for values of 2.1280 < α5 < 2.1282
there are two nested limit cycles and, then, a bistability
competition between a stable equilibrium and a stable periodic
solution arises depending on the initial conditions. This special
structure of cycles (Fig. 4b) is confined to the neighborhood
of LP , where the dynamics is very influenced by the saddle
equilibrium, restricting the basin of attraction of the stable
equilibrium or even the stable cycles. From a one parameter
analysis it is not evident that for variations of other parameters
this oscillatory behavior could not appear in regions of practical importance. Therefore, it is important to recognize the
differences between bifurcations like H1 or H3 (supercritical)
and those like H2 (subcritical), since it might be useful in
cases where a similar structure of nested cycles like the one
shown in Fig. 4b appears. For example, if the operating point
is on the left of H2 and the load is increased until H2 is
crossed, the system will exhibit a “large” amplitude stable
5
becomes stable again. These phenomena are related to voltage
stability problem. Notice that this mode describes the ellipsoid
shown in Fig. 5.
Then, a real eigenvalue becomes unstable for a higher value
of α5 , leading to the saddle-node bifurcation LP in Fig. 4a.
The participation factors reveal that the state variable of the
swing equation of the motor at bus 5 is dominant in this
bifurcation.
IV. T WO PARAMETER BIFURCATION ANALYSIS
Fig. 5. Movement of the relevant eigenvalues corresponding to the equilibria
shown in Fig. 4a. The arrows indicate the movement direction as α5 is
increased from α5 = 0.8. Only those with positive imaginary part are shown.
oscillation. Then, if the load is decreased beyond H2 the
oscillation persists until CF is reached (hysteresis). This is
not the case in oscillations like H1 or H3 . This information
can not be deduced from a linear stability analysis.
A. Eigenvalue Movement
Since the bifurcations of equilibria seen in the preceding
section (Hopf and saddle-node) are local phenomena, it is
interesting to study the eigenvalues associated to the equilibrium and their movement when the parameter α5 is varied
in the range used in Fig. 4a. The movement of the relevant
eigenvalues as α5 is increased is shown in Fig. 5 (only the
positive imaginary parts are shown and the arrows indicate
the movement direction). The analysis can be improved calculating the participation factors [12], [15], to determine which
state variables are the main responsible for the occurrence of
bifurcations. Then starting from α5 = 0.8 there is a pair
of unstable complex eigenvalues at 0.016 ± i8.563, while
the rest of the eigenvalues are in the left half plane. When
α5 is increased the unstable pair moves toward the left and
crosses the imaginary axis at 0 ± i8.547, developing the Hopf
bifurcation H1 (see Fig. 5). The participation factors near this
singular point reveal a strong influence on the state variables
δ2 and ω2 , and thus the instability of the equilibria and the
resulting oscillation due to the Hopf bifurcation H1 can be
associated to an angle instability.
Increasing the load on bus 5 further more, the previous angle
oscillatory mode does not become unstable again, but a pair
of eigenvalues related to the excitation system of generator
#1 moves toward the imaginary axis. More precisely, the
dominant state variables are eq1 and rF 1 . The first crossing of
these eigenvalues is at 0 ± i0.894 and determines the Hopf
bifurcation H2 . The equilibrium remains unstable until the
same pair crosses the imaginary axis again at 0 ± i0.729
determining the Hopf bifurcation H3 . Then the equilibrium
There are several advantages in performing a bifurcation
analysis varying two parameters. For example, regions of
qualitative different behavior can be found by tracing the
codimension one bifurcation curves. In addition, it brings the
possibility to study codimension two bifurcations that in a
two parameter plane might be seen as organizing centers of
the dynamics. Non-trivial phenomena may also be revealed
studying the normal form of such codimension two singularities. In this section the dynamical aspects of the system will
be studied when two loads are varied simultaneously.
As seen before, the Hopf bifurcation H1 is caused by an
angle instability associated to the generator #2 swing equation
due to a variation in the load on bus 5. Then, it is interesting
to study how this oscillation mode behaves when two loads
are varied simultaneously. Similar arguments are valid for the
other bifurcations shown in Fig. 4. To accomplish this task,
parameter α8 , i.e. the power drained by the load on bus 8 (see
Fig. 1), is varied (decreased). At the same time α5 is varied
satisfying the Hopf bifurcation condition and thus curve H1 in
Fig. 6 is obtained. Analogously, the bifurcation curves for H2 ,
H3 and LP are obtained by continuation of the corresponding
bifurcation conditions. The curves have been computed using
the continuation package MATCONT [17].
Some conclusions about the system behavior can be drawn
from the Hopf bifurcation curves provided by the two parameter bifurcation scheme shown in Fig. 6a. One of the most
important facts to be remarked is that the angle instability
is the main problem, since the bifurcation curve H1 exists
for a wide range of values of α5 and α8 , and the remaining
bifurcation curves are confined at values of α5 ≈ 2.1,
regardless the value of α8 . Notice that the relative distances of
the phenomena shown in Fig. 4b (curves H2 , H3 and LP in
Fig. 6) are practically not affected by the variation of α8 . This
means that the limit imposed by the saddle-node bifurcation
LP is virtually unaffected by the consumption of the load on
bus 8 (the saddle-node is associated to a mechanical state of
the load motor in bus 5).
The grey area enclosed by the curves H1 and H2 in Fig. 6
denotes a region where stable equilibria exist. Caution should
be taken regarding the robustness of the equilibrium as the
parameter is close to any of the bifurcation curves. Also notice
that this is not the only area with stable equilibria since there is
another one enclosed by the curves H3 and LP . This area has
no practical meaning because it is very narrow and therefore
is not colored for the sake of simplicity.
Suppose that the system is in a feasible operating condition
(grey area), and the load on bus 5 is constant (α5 fixed).
6
(a)
(b)
Fig. 6. Two-parameter bifurcation diagram. (a) Bifurcation diagram varying α5 and α8 . (b) Expanded view of the rectangle in Fig 6a. The gray area indicates
the existence of stable operating conditions in terms of equilibrium points.
Then, if the load on bus 8 (parameter α8 ) is decreased and
crosses the curve H1 , the system will experience an angle
oscillation. Also, setting α8 constant and increasing α5 , the
instability will be caused by a voltage phenomenon such as
the Hopf bifurcation H2 . In addition, keeping α8 constant and
decreasing α5 it also leads to the angle instability due to H1 .
A. Organizing centers of the dynamics
Figure 6b offers an expanded view of the rectangle shown
in Fig. 6a. Even though the parameter range is very small, this
area contains a collection of codimension two singularities and
concentrates the organizing centers of the dynamics.
1) Double Hopf Bifurcation: The intersection of the curves
H1 and H2 defines the double Hopf (or Hopf-Hopf) bifurcation HH1 in Fig. 6b and establishes the point in the parameter
plane where both Hopf singularities occur simultaneously. In
addition, the intersection of the curves H1 and H3 denotes
another double Hopf point, namely HH2 in Fig. 6b. Double
Hopf bifurcations were studied in several engineering applications, such as electrical circuits [28]–[30], mechanical systems
[31], [32], and neural systems [33]. The special resonant cases
are studied in [34], [35].
The double Hopf singularity is a highly complex codimension two bifurcation with many possible scenarios. The normal
form predicts eleven different unfoldings (six of them include
non-trivial global phenomena), i.e. with different local dynamical phenomena [36]. These unfoldings are distinguished by
the values of the normal form coefficients. A procedure for
obtaining these coefficients (and those of the normal form
for all the bifurcations shown in this paper), is explained in
detail in [36] and it is implemented in the continuation package
MATCONT.
A common factor in all the double Hopf scenarios is the
presence of quasi-periodic oscillations (2D tori), involving two
frequencies or modes. In the neighborhood of the Hopf-Hopf
Fig. 7. Schematic representation of the unfolding of the Hopf-Hopf HH1 .
The tori (2D and 3D) are enclosed by the curves T R1 and T R2 .
point these frequencies correspond to that of the limit cycles
associated to the Hopf bifurcations, since the torus is born at
local bifurcations (Neimark-Sacker) of the limit cycles. The
case denoted as HH1 has additional non-trivial phenomena
such as heteroclinic obits and a 3D torus with three main
frequency components. The unfolding is depicted schematically in Fig. 7, where the Neimark-Sacker bifurcations are
denoted as T R1 and T R2 . The continuation software brings
the possibility to continue the Neimark-Sacker curves varying
two parameters, but in this case it demands a big computational
burden due to the high dimension of the model (70 state
equations). The quasi-periodic solutions are within the area
enclosed by T R1 and T R2 . In addition, the 3D torus can be
found between the curve C that represents a bifurcation of
the 2D torus [37], and the curve Y which is a heteroclinic
connection.
Analyzing the normal form it can be determined that the
region with tori for the HH1 singularity is located to the left
of HH1 (locally), between H1 and H2 curves, enclosed by
two (not shown) Neimark-Sacker bifurcation curves. In the
HH2 case, the unfolding corresponds to one of the simple
cases that includes a single 2D torus, besides the limit cycles
7
emanated from the Hopf bifurcations.
2) Zero Hopf Bifurcation: The singularity ZH shown in
Fig. 6b is a zero-Hopf (Gavrilov-Guckenheimer) bifurcation.
At this point, the Hopf bifurcation curve H1 becomes tangent
to the saddle-node curve LP . Therefore, the linearization of
the equilibrium has a pair of complex conjugated eigenvalues
and a single eigenvalue with zero real part, denoting the
simultaneous occurrence of the Hopf bifurcation H1 and the
saddle-node LP . This bifurcation can be interpreted as the
generator of the saddle-node bifurcation shown in Fig. 4, since
it corresponds to a point of the saddle-node curve emanating
from the ZH singularity. The normal form of this bifurcation
is also described in [36], and in this case the corresponding
unfolding has a quasi-periodic orbit in a neighborhood of the
singularity. There is a high chance that this quasi-periodic orbit
is in direct relation with the tori emanated from the Hopf-Hopf
singularities. This fact might be confirmed by continuation of
the Neimark-Sacker curve associated to ZH.
V. C ONCLUSIONS
In this paper the dynamics of a detailed multimachine power
system was studied using bifurcation theory and classical
tools, such as the participation factors. The combination of
these techniques gives an improved knowledge of the system
behavior and the possible problems coming from the variation
of system parameters. For example, using the bifurcation
analysis it is possible to determine the specific values of the
parameters where the system equilibrium becomes unstable
due to a Hopf bifurcation, and also find out the stability of
the associated oscillation. While using the participation factor
it is possible to get information about the problem nature, i.e.
the state variables responsible for such instability, in order to
take the appropriated countermeasures.
The bifurcation analysis varying two parameters brings
additional information such as the location of organizing
centres of the dynamics like the double Hopf and zero-Hopf
bifurcations. Then, studying the normal form of these codimension two points it is possible to explain some non-trivial
phenomena like quasi-periodic motions. In addition, feasibility
regions in the parameters space can be determined. Among the
drawbacks found using continuation methods in power systems
is the computational burden demanded to trace the bifurcation
curves (specially the torus continuation), mostly due to the
high dimension of the system. Even though, the continuation
algorithms are very robust if their parameters are set correctly.
A PPENDIX A
The definition of the variables and parameters of the system
are listed in Table II and Table III, respectively. All values are
in per-unit, using a 100 MVA base, otherwise the corresponding unit is indicated in parenthesis.
R EFERENCES
[1] M. D. Ilic, H. Allen, J. W. Chapman, C. A. King, J. H. Lang, and
E. Litvinov, “Preventing future blackouts by means of enhanced electric
power systems control: from complexity to order,” Proceedings of the
IEEE, vol. 93, no. 11, pp. 1920–1941, Nov. 2005.
TABLE II
D EFINITION OF STATE VARIABLES .
Description
Synchronous Generators
Generator angle
Rotor speed
d, q axis transient EMF
d, q axis stator current
Induction Motors
Angular speed
d, q axis transient EMF
d, q axis stator current
Transmission Lines
D, Q axis bus voltages
D, Q axis line currents
IEEE-DC1A Excitation System
Exciter output voltage
Exciter input voltage
Feedback rate (see [12])
Transductor output voltage
Turbine-Governor
Turbine flow rate
Gate opening
Regulator internal state
Transient droop internal state
IEEE-ST1A+PI Excitation System
Exciter output voltage
PI regulation state
Transductor output voltage
Symbol
i = 1, 2, 3
δi
ωi
edi , eqi
idi , iqi
j = 5, 6, 8
ωj
edj , eqj
idj , iqj
(n, m) = 4, ..., 9
vDn , vQn
iDnm , iQnm
Gen. #1
ef d1
vR1
rF 1
vm1
Gen. #1
q1
gc1
x1
x2
Gen. #2 & #3 (i = 2, 3)
ef di
xRi
vmi
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TABLE III
D ESCRIPTION OF SYSTEM PARAMETERS AND THEIR VALUES .
Description
Parameter
Value
Synchronous Generators
Generation type
Rated power
Base angular frequency
Synchronous angular frequency
Stator winding resistance
Inertia constant
Mechanical power
d axis reactance
d axis transient reactance
q axis reactance
q axis transient reactance
d axis transient open circuit time const.
q axis transient open circuit time const.
Amortisseur damping coefficient
Mechanical losses damping coefficient
Excitation System
Exciter field proportional constant
Exciter field time constant
Amplifier gain
Amplifier time constant
Feedback gain
Feedback time constant
Regulator proportional gain
Regulator integral gain
Transducer gain
Transducer time constant
Gen. terminal & regulator ref. voltage
Turbine-Governor
Proportionality power factor
Water time constant
Main servo gain
Main servo time constant
Permanent droop (primary regulation)
Transient droop
Dashpot time constant
Integral gain (secondary regulation)
Composite Loads
Stator resistance
Motor + driven load inertia constant
Mechanical power
Induction motor reactance
Induction motor transient reactance
Transient open circuit time constant
Mechanical losses damping coefficient
Motor torque model
p.u. constant impedance load (active)
p.u. constant current load (active)
p.u. constant power load (active)
p.u. constant impedance load (reactive)
p.u. constant current load (reactive)
p.u. constant power load (reactive)
Initial voltage magnitude
Bifurcation Parameters
Load active power (nominal value)
Load reactive power (nominal value)
#1
Hydro
SN (MVA) 247.50
ΩB (r/s) 2π60
ωs
1.0000
0.0000
Rs
H (s)
23.640
Pm
Xd
0.1460
Xd
0.0608
Xq
0.0969
Xq
0.0969
(s)
Td0
8.9600
(s)
Tq0
0.3100
0.0000
Kd
Db
0.0000
#2
Steam
192.00
2π60
1.0000
0.0000
6.4000
1.6300
0.8958
0.1198
0.8645
0.1969
6.0000
0.5350
0.0000
0.0000
#3
Steam
128.00
2π60
1.0000
0.0000
3.0100
0.8500
1.3125
0.1813
1.2578
0.2500
5.8900
0.6000
0.0000
0.0000
KE
TE (s)
KA
TA (s)
KF
TF (s)
KP R
KIR
KR
TR (s)
vt , vt
1.0000
0.3140
20.000
0.2000
0.0630
0.3500
1.0000
0.0010
1.0400
200.00
0.0200
1.0000
0.2500
1.0000
0.0200
1.0250
200.00
0.0200
1.0000
0.2500
1.0000
0.0200
1.0250
2.4750
1.0000
1.0000
0.4000
0.0500
0.0486
5.0000
0.7500
#5
Rs
0.0010
H (s)
0.5269
Pm
0.8658
Xdq
3.8943
Xdq
0.1683
Tdq0
(s)
0.7826
Db
0.0000
{A, B, C} {0,1,0}
PZ
0.3750
PI
0.0000
PP
0.0000
QZ
0.1500
QI
0.0000
QP
0.0000
V0
0.9956
#5
α
1.0000
β
1.0000
#6
0.0010
0.5269
0.6251
3.8943
0.1683
0.7826
0.0000
{0,1,0}
0.2700
0.0000
0.0000
0.0900
0.0000
0.0000
1.0126
#6
1.0000
1.0000
#8
0.0010
0.5269
0.6941
3.8943
0.1683
0.7826
0.0000
{0,1,0}
0.3000
0.0000
0.0000
0.1050
0.0000
0.0000
1.0159
#8
1.0000
1.0000
At
Tw (s)
KG
TG (s)
RDp
RDt
TRt (s)
KID