Energy and Power Engineering, 2011, 3, 1-8
doi:10.4236/epe.2011.31001 Published Online February 2011 (http://www.SciRP.org/journal/epe)
Singular Hopf Bifurcations in DAE Models of
Power Systems
Wieslaw Marszalek1, Zdzislaw Trzaska2
1College of Engineering & Information Sciences, DeVry University, North Brunswick, USA
2Department of Management and Production Engineering, Warsaw University of Ecology and Management,
Warsaw, Poland
Received August 25, 2010; revised October 10, 2010; accepted October 15, 2010
Abstract
We investigate an important relationship that exists between the Hopf bifurcation in the singularly perturbed
nonlinear power systems and the singularity induced bifurcations (SIBs) in the corresponding different-
tial-algebraic equations (DAEs). In a generic case, the SIB phenomenon in a system of DAEs signals Hopf
bifurcation in the singularly perturbed systems of ODEs. The analysis is based on the linear matrix pencil
theory and polynomials with parameter dependent coefficients. A few numerical examples are included.
Keywords: Power Systems, Singularly Perturbed Systems, DAEs, Bifurcations, Matrix Pencils
1. Introduction
In an effort to better understand dynamical properties of
power systems, their stability features and the impact of
various parameters, the DAEs approach seems to be very
important as was shown in a number of recent papers
(see for example [1-10]).
Some of these papers deal with the existence of Hopf
bifurcations in the singularly perturbed systems of non-
linear ODEs. DAEs are closely related to singularly per-
turbed ODEs, therefore it is natural to expect similar
types of behavior. However, there are also obvious dif-
-ferences between the qualitative properties of DAEs and
singularly perturbed ODEs. One can mention, for exam-
ple, the SIB phenomenon, which is present in DAEs and
not in ODEs. Also, singularities in DAEs add another-
layer of difficulty in the bifurcation analysis, in particular
in the Hopf bifurcation. Because of a local nature of
these phenomena, it is feasible to consider the singular
Hopf bifurcation and the SIBs by analyzing certain
properties of linear matrix pencils and their characteristic
polynomials [9,11-14].
The matrix pencil and characteristic polynomial ap-
proach used in the analysis of the singularly perturbed
systems and DAEs models has one obvious drawback: the
available results give sufficient conditions only and, as of
today, no significant results are available for multi-pa-
rameter bifurcations of the underlying DAEs. The fun-
damental SIB theorem for DAEs reported in [9] has been
slightly improved in [12,13,15] to weaken some of the
sufficient conditions and to include the case of second-
order slow subsystems (typical in power systems). In [16]
the sufficient conditions of the SIB phenomenon were
given for quasi-linear DAEs and other interesting appli-
cations of DAEs have been reported in [17,18].
In this paper we extend some of the results presented
in [1]. In particular, we present further studies of the
singularly perturbed power systems, their SIB, singular
Hopf bifurcations and interactions between them. The
topic of bifurcations in power systems (in particular the
Hopf bifurcation) is one of the most important and inter-
esting ones, and, at the same time quite difficult, as any
realistic power grids have thousands of buses, generators
and other devices. Even small failures of various devices
of secondary importance may yield catastrophic conse-
quences, power outages, or unstable systems’ behavior
[7,8].
Section 2 provides a short introduction into the DAEs
and singularly perturbed ODEs as models of power sys-
tems. Section 3 constitutes the main part of this paper.
First, we discuss the SIBs and Hopf bifurcations in the
context of singularities of DAEs. Next, several results
related to these bifurcations in power systems are given
and illustrated by several numerical examples. Section 4
provides a summary of the results and a few suggestions
for possible extensions. The three part appendix is an

W. MARSZALEK ET AL. 2
important addition as it provides several properties of
linear matrix pencils and polynomials with parameter
dependent coefficients that are frequently used in our
analysis in Section 3.
2. DAEs as Models of Power Systems
The DAE model of a power system is as follows [1,2,
19,20]
2
2,1,,
,1, ,
,,1, ,
i
iiimi G
idiGG PV
idii diGPVG PVPQ
dd
i
MDfP i N
dt dt
fP iNNN
f
P gQ iNNNNN


 
 



(1)
where fi and gi form the load flow equations and are de-
fined as (n ≡ NG + NPV + NPQ)
 
 
1
1
sin cos
1,,
sin cos
1,,
n
iijijijijiji
j
n
iijijijijiji
j
GV
fVVBVVG
i n
gVVG VVB
iNN n
 
 















j
j






(2)
with the following notation:
Mi is the rotor inertia of the ith generator,
Di is the damping coefficient of the ith generator,
Vi is the voltage magnitude of the ith bus,
αi is the angle of the ith bus,
Bij, Gij are the transfer susceptances and conductances,
Pmi is the turbine mechanical power injection of the ith
generator, and
Pdi, Qdi are the real and reactive power loads at the ith
bus.
The above system can easily be written in the following
DAE form
 
2
2,, ,0,,
dx dx
MDfxy gxy
dt dt


  (3)
with x ≡ [α1,, αNG]T , y ≡ [αNG+1,, αn, V1,, Vn]T , λ
is the vector of k parameters, and the matrices M, D are
positive and semi-positive definite, respectively.
One of the generator buses is usually considered as a
swing bus. This allows us to introduce a set of new vari-
ables (relative angles) and reduce the number of differ-
ential equations in the first subsystem in (3) by one [3].
The DAE model (3) and its bifurcations are closely
related to the behavior of the singularly perturbed ODE
model of the form
 
2
2,, ,,,
dx dxdy
MDfxy gxy
dt dt dt
when ε ≠ 0. We shall examine the singular Hopf bifurca-
tion of (4) for small 0 < ε << 1.
The singularly perturbed ODEs (4), when linearized,
yield linear matrix pencils. Either first or second order
matrix pencils can be used depending on an individual
preference (see Appendices 1.1 and 1.2 for more details).
A second order matrix pencil would normally be preferred
for power systems as it may directly lead to double SIB
points [1,21].
3. Bifurcations of Power Systems at
Singularities
3.1. The Singularity Induced Bifurcation
The SIB is a phenomenon attributed exclusively to DAEs.
Since DAEs are used quite often as models of electrical
power systems, it is important to study the changes of
stability of those systems to better understand their dy-
namical properties. This may help to prevent future fail-
ures and catastrophic events such as, for example, mas-
sive power outages. It is also known that DAEs may ex-
hibit other types of bifurcations that are widely found in
nonlinear ODEs. One of such bifurcations is the well-
known Hopf bifurcation when a linearized system has a
pair of complex conjugate eigenvalues on the imaginary
axis with all other eigenvalues lying off the axis. The
bifurcating parameter may cause the subcritical or super-
critical Hopf bifurcations. The bifurcation is called sub-
critical if the periodic solutions are unstable and super-
critical if the periodic solutions are stable. The Hopf bifur-
cation in power systems has been analyzed in [2,22].
The singularity of nonlinear semi-explicit parameter
dependent DAEs = f(x, y, λ), 0 = g(x, y, λ) occurs when
det(gy) = 0 for some (x∗, y∗, λ0) Rn+m+1, x(t)
x



Rn, y(t)

Rm, λ

R, dtdxx 
and dydgg y. The λ is a
slowly varying parameter. Equivalently, the linear matrix
pencil
0,
00
I
AB
CD






(5)
where A(λ) = fx, B(λ) = fy, C(λ) = gx, D(λ) = gy, has index
greater than 1 at (x∗, y∗, λ0) (see Appendices 1.1 and
1.2).
For the singularly perturbed ODEs we have= f(x, y,
λ),
x 
y 

= g(x, y, λ) where 0 < ε << 1 is small and constant.
A natural extension to second-order DAEs and singularly
perturbed ODEs is to have),,,,(

yxxfx y 

= g(x, y,
λ).


  (4)
It turns out that under the assumption of the SIB theo-
rem (with an algebraically simple zero of gy(x
∗
, y
∗
, λ0)
[9]), the SIB phenomenon for semi-explicit parameter

W. MARSZALEK ET AL.3
dependent DAEs may be equivalent to Hopf bifurcation
of the singularly perturbed ODEs. The following example
illustrates such a case.
Example 1: Consider the following singularly per-
turbed parameter dependent DAEs
2
1
, 2
x
yyx y   

(6)
Two solutions of (6) for λ = 0 are shown in Figures 1-2.
If ε = 0, then the DAEs (6) undergoes the SIB as the pa-
rameter λ → 0. The equilibrium locus is (λ2/2, λ), i.e. x =
λ2/2, y = λ, and an eigenvalue −1/λ diverges through in-
finity as λ → λ0 = 0. The stability of the system is changed
at λ0. The point λ0 is called a single SIB point (see [11]).
The system is asymptotically stable for λ > λ0 and unstable
for λ < λ0. This relatively simple case with scalar f(x, y, λ)
and g(x, y, λ) is covered by Lemma 1 in [15] which, under
the conditions gy(λ0) = 0, g′y(λ0) ≠ 0 along the equilibrium
locus, and gx(λ0) fy(λ0) = −ω2 < 0, predicts a pair of com-
plex eigenvalues
 


2
0
2
,1
2
x
f
i -


 


 


1 O (7)
as ε → 0. The ′ = d/dλ and 1i.
For (6) we have fx(λ0) = 0, gy(λ) = −λ, g′y (λ) = −1 and
ω2= 1. Thus, the eigenvalues (7) are: α(ε, λ(ε)) ~ ±ω √ ε +
Figure 1. Trajectories for ε = 0.01.
Figure 2. Trajectories for ε = 0.09.
O(1). The linearization of these DAEs along the equi-
librium locus (λ2/2, λ) yields the matrix pencil (see Ap-
pendix A)
100 1
,
001








(8)
which has index 1 for λ ≠ 0 and index 2 for λ = 0 (Ap-
pendix A). The CB = −1, v = 1, and CBv Im(D) at λ =
0. Therefore, at λ = 0, the index jumps by one, the well
established fact for DAEs undergoing the SIB. Also, the
characteristic equation of the pencil (8) is (see Appendix
1.1, case k = 1) −λs + 1 = 0 with the root s = 1/λ diverging
through ±∞ as λ → 0.

Example 2: Now, suppose that d2x/dt2 +γ dx/dt =λ−y,
εdy/dt = x −12y2, with γ > 0. The eigenvalues are given
by

2
221

  , and the system is asymp-
totically stable for λ > 0 and unstable for λ < 0. It can be
checked that the linear system has index 1 if λ ≠ 0, and the
index jumps to 3 at λ = 0. If λ ≠ 0, then along the equilib-
rium locus (λ2/2, λ, λ) we have (see Appendix 1.2):
det(E(0)s − L(λ)) = s2λ + sγλ + 1, while at the SIB point λ
= 0 we have det(E(0)s − L(0)) = 1. This drop by 2 in the
degree of det(E(0)s − L(λ)) is equivalent to the index
increase from 1 (at λ ≠ 0) to 3 (at λ = 0). If λ < 0 one of the
eigenvalues is unstable while the other remains stable. At
λ = 0 the eigenvalues diverge through ±∞. The point λ = 0
is called the double SIB point, the concept introduced by
Beardmore in [11].
For λ ≠ 0 the Kronecker Normal Form of the pencil
(E(0), L(λ)) is
 
10001 0
0010,1 0
00000 1
E L 
 
 

 
 
 








(9)
and for λ = 0 the structure of the pencil changes to

010 100
001,0010
000 001
E L
 
 

 
 
 
(10)
3.2. Properties of DAEs with SIB Phenomenon
The main theorem indicating Hopf bifurcation in the
singularly perturbed ODEs is due to Beardmore [15], as
follows.
THEOREM: Suppose that the singularly perturbed
ODEs

x
f(x, y, λ), y 

=g(x, y, λ) have a trivial equi-
librium for λ = λ0 and
1)




0y
ker gv   with



0
Im v vg

,
2)






00
Im
yy
gvg


,

W. MARSZALEK ET AL. 4
3) det 0
xy
xy
ff
gg




4)
 
0
T
y
ker gu   and

00 20
T
xy
T
ugf v
uv
 
 .
Then, there exists ε1 > 0 such that ε1 > ε > 0 there is a
λ0(ε) such that the singularly perturbed linearized ODEs
have a pair of purely imaginary eigenvalues at λ = λ0(ε)
and λ0(ε) is continuous in ε at 0 with λ0(0) = λ0.

In the above, the ker, Im and det denote the kernel,
image and determinant of a matrix, respectively, the < > is
a column vector and T denotes the transposition. It is easy
to check that the above sufficient conditions are satisfied
for (6) with v = u = 1, g′y (0) = −1, fxgy − fygx = 1, gx(0)fy(0)
= −1, yielding ω = 1.
LEMMA 1: Suppose that in (18) in Appendices 1.2 we
have for ε = 0: ind(E(0), L(λ)) = 1 for λ ≠ λ0, detL(λ0) ≠ 0,
kerD(λ0) ≡ vRm, D′(λ0)vImD(λ0) and C(λ0)B(λ0)v

ImD(λ0). Then,
    


10
det 0,
nm
nm
EsLasa sa




  
(11)







2
010
det 0,
nm
EsLsd sd
00



 
(12)
and the two eigenvalues of the matrix pencil (E(0), L(λ))
diverging through ±∞ are the zeros of the second-degree
polynomial
 





0 2
2
0
nm
nm
nm
as a
asbsc sd


 




 

(13)
with a(λ) = an−m(λ), a(λ0) = b(λ0) = 0, and c(λ0) ≠ 0. The
index of (E(0), L(λ0)) is 3.
Lemma 1 is a consequence of the above theorem. In
particular, from the Kronecker Normal Form of (E(0),
L(λ)) and the assumption that ind(E(0), L(λ)) = 1 for λ ≠ λ0,
we obtain (11). The other assumptions are needed to have
the two eigenvalues divergent through ±∞ [11], and as a
consequence, for λ = λ0 we have (12).
Notice that in example 2 det(E(0), L(λ)) = s2λ + sγλ + 1
and det(E(0), L(0)) = −1, therefore a(λ) = λ, b(λ) = γλ, c(λ)
= 1 and the two diverging eigenvalues are the roots of s2λ
+ sγλ + 1 = 0. As a direct consequence of the above lemma
we have the following result.
LEMMA 2: If the assumptions of lemma 1 are satis-
fied and if Γ(λ) ≡ 0 in (18) (see Appendix 1.2), then b(λ) ≡
0 and the two diverging eigenvalues are purely imaginary.
If c(λ)/a(λ) changes its sign from positive to negative at λ
= λ0, then the system undergoes transition from a center to
a saddle, or vice-versa, from a saddle to a center if
c(λ)/a(λ) changes sign from negative to positive.
The proof of the above lemma follows from the fact that
the two diverging eigenvalues are
 
1,2
sca

  .
No damping term exists in the system. A similar behavior
in active RLC circuits has been reported in [23].
3.3. An Illustrative Example
The following example illustrates the above described
features of a power system undergoing the singularity
induced bifurcation.
Example 3: Consider the 4-bus, 3 generator power
system shown in Figure 3. If we assume that the bifurca-
tion parameter λ defines the real and reactive powers at
bus 4 with P = P0 (1 + λ), Q = Q0 (1 + λ), (P0 = const, Q0 =
const), then the DAE model (3) of this power system
undergoes the SIB at λ0 = 0.537305.
Assuming that the rotor inertia of all generators equals
1, the line transfer conductances Gij are all zero, the
damping generator coefficients to be γ, and considering
bus 1 as a swing bus, the DAE system is as follows






 

2
11
21 21121
2
1313 22424 31
2
22
31 13132
2
12 1 2134 3432
4224 314334 32
4242 3143343
2sin
sin sin
2sin
sin sin
0sin sin
0sin sin
mm
mm
dd
PP BVV
dt dt
BVV BVV
dd
PP BVV
dt dt
BVV BVV
BVV BVVP
BVV BVV






 
 







2
2444
BV Q
(14)
with the relative angles θi ≡ αi+1 − α1 for i = 1, 2, 3.
Suppose that the parameters are: B12 = B13 = 1, B24 = B34
= 2, B44 = −4, Bij = Bji, Pm2 − Pm1 = 1, Pm3 − Pm1 = 2, P0 = 1,
Q0 = 0.3, γ = 1.5 and Vi = 1 for i = 1, 2, 3. Then, the DAEs
(14) have an equilibrium placed at the singularity for
Figure 3. A simple 4-bus, 3-generator power system.

W. MARSZALEK ET AL. 5




1234
,,,,0.8496, 2.3761, 2.4940, 0.6334, 0.5373V   
For the singularly perturbed DAEs with P = P0 (1 + λ)
and Q = Q0 (1 + λ) + εdθ4/dt we get the same equilibrium.
Solutions θi(t), i = 1, 2, 3, for ε = 0.002 and various values
of γ are shown in Figure 4. Notice that V4(t) can be
eliminated from (14) and the system reduces to two sec-
ond-order differential equations and one first-order dif-
ferential equation containing εdθ4/dt. It turns out that the
above equilibrium is unstable and at λ = 0.537305, ε =
0.002, γ = 0 the pencil (18) has the following eigenvalues
−5.6245 ± 9.7933i, ±0.1342i, 11.2487.
The solutions in Figure 4 escape the above equilibrium
and oscillate around another equilibrium (not on the sin-
gularity)




1234
,,,,0.6812, 1.0816, 1.2770, 1.1018, 0.5373V   
with the eigenvalues ±1.3238i, ±1.5720i, −4742.9260 for
γ = 0 and ε = 0.002.
For ε = 0 and γ ≠ 0 the characteristic equation of the
(a) (b)
(c) (d)
(e) (f)
Figure 4. (a, c, e) The θi(t), i = 1, 2, 3, for γ = 0; (b, d, f) the θi(t), i = 1, 2, 3, for γ = 0.05.

W. MARSZALEK ET AL. 6
linear matrix pencil (18) is (see (19)-(21) in Appendix
1.3): a4(λ)s4 + a3(λ)s3 + a2(λ)s2 + a1(λ)s + a0(λ) = 0 and for
λ = λ0 = 0.537305 the equation reduces to 2.869398s2 +
4.304097s + 0.051705.
Clearly, we have the case k = 2 (see Appendix 1.3) and
in (21) we have c2(λ0) = c1(λ0) = 0, c0(λ0) = 2.869398,
b1(λ0) = γ = 1.5000, and b0(λ0) = 0.018019. For λ → λ0
the DAEs have two eigenvalues (divergent through in-
finity)






2
110
1,22
22 2
24
ccc
scc
c





 (15)
whose real parts are such that
 
 
lim 2cc
012


 
 and the imaginary
parts diverge to infinity as λ → λ0. The point λ0 =
0.537305 is called a double SIB point and the linear ma-
trix pencil has index 3 at λ = λ0 and 1 if λ ≠ λ0.
One can also examine matrix R(λ) in Appendix 1.4 to
check that detR(λ) = 0 if the characteristic polynomial of
the matrix pencil (18) admits a pair of purely imaginary
roots. Figure 5 shows detR(λ) for system (14). Clearly,
this determinant is 0 for λ = 0.537305 because of the two
complex eigenvalues ±0.1342i of the matrix pencil.
4. Conclusions
In the paper we have examined sufficient conditions un-
der which the single and double SIB points in systems of
DAEs are responsible for the Hopf bifurcation in singu-
larly perturbed systems of ODEs. Both, the DAEs and
singularly perturbed ODEs are widely used as dynamical
models of power systems making their analysis very
relevant in the studies of stability, reliability and sensi-
tivity to various parameters. The analysis of power system
models around their singularities is important from both
the qualitative and numerical points of view. The quali-
tative analysis presented in this paper should be supple-
mented by a numerical one [17], as a typical power system
Figure 5. The det R(λ) as a function of λ (see Appendices
1.4).
model includes hundreds or even thousands of equations
[20].
It is also our opinion that the future work in the quali-
tative analysis of the SIB and singular Hopf phenomena
via matrix pencils and parameter dependent polynomials
should focus on the formulation of the necessary and
sufficient conditions and the analysis of multi-parameter
bifurcations at singularities. To the authors’ knowledge
no formulation of the necessary conditions of the SIB
phenomenon exists at the present time.
5. References
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

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Proceedings of 32nd Conference on Decision and Con-
trol, San Antonio, December 1993, pp. 3727-3733.
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36142993253461
1. Appendix
1.1. Matrix Pencils for First Order Systems
For a pair of constant square n × n matrices, say E and L,
with detE = 0, if det(sE − L) ≠ 0, then there exist non-
singular matrices U and V, such that
-
0
,
00
mm
nm nm
IJ
UEV ULV
N




 (16)
where N n−m is a nilpotent matrix of size (n − m) × (n − m)
and index ν ≤ n − m. That is, ν is a positive integer such
that Nν = 0, Nν−1 ≠ 0. The Ik is the unit matrix of size k × k.
The pencil (UEV, ULV) is said to be in Kronecker Normal
(Canonical) Form and σ(E, L) = σ(Jm), i.e. the finite
spectrum of the pencil (E, L) is the same as the spectrum
of matrix Jm. It is also well-known that for DAEs with the
matrix pencil
0,
00
I
AB
UEV ULVCD
 

 
 



(17)
and detL ≠ 0, the index(UEV, ULV)=1 if and only if detD
≠ 0. If ker(D) = <v > ≠ 0 and CBv Im(D), then the
index (UEV, ULV) = 2.

1.2. Matrix Pencils for Second Order Systems
The linearization of ),,,,(

yxxfx 

y 

= g(x, y, λ)
yields the matrix pencil (E(ε), L(λ)) with

W. MARSZALEK ET AL. 8
000 0
() 00,()()()()
00 ()0D()
n
n
m
II
n
EI LAΓB
IC





 

 

 






(18)
and A(λ) = fx, B(λ) = fy, C(λ) = gx, D(λ) = gy, Γ(λ) =xand
E(ε), L(λ)R2n+m and det(E(ε)s − L(λ)) = det(E(ε)s − L(λ))
= an+m sn+m + an+m−1sn+m−1 + …+ a1s + a0 with ai = ai(ε, λ).
f
1.3. Parameter Dependent Polynomials
The change of the index of DAEs is related to certain
properties of the characteristic polynomial of the pencil
(UEV, ULV). Let the polynomial be
 


1
11
,rr
rr
psasasa sa
0




  

, 
(19)
with ar(λ0) = ar−1(λ0) = = ar−k+1(λ0) = 0, ar−k(λ0) ≠ 0.
Polynomial (19) admits a real decomposition [13]

 
12
,,psp sps   (20)
with
 




 
1
11
1
21
,
,
kk
kk
rk rk
rk
psc scsc
pss bsb
0
0

 






 
 

 
(21)
where ck, , c0 and br−k−1, , b0 are real and smooth,
and ck(λ) = ar(λ), ci(λ0) = 0 for 1 ≤ i ≤ k, c0(λ0) = ar−k(λ0) ≠
 
0.
If k = 1, then one root of p1(λ, s) diverges through ∞ as λ
→ λ0. This is easily seen by observing that with k = 1 we
have c1(λ) = ar(λ) = (−1)detD, and the root −c0(λ)/c1(λ) →
∞ as λ → λ0. This divergence is equivalent to the index
jump of the matrix pencil from 1 at λ ≠ λ0 to 2 at λ = λ0. If k
= 2, then two roots of p1(λ, s) diverge through ∞ and the
index of the corresponding matrix pencil jumps from 1 at
λ ≠ λ0 to 3 at λ = λ0. The two diverging eigenvalues, say
s1,2, are such that limλ→λ0s2
1,2(λ)(λ − λ0) = μ with μ =
−(uTC(λ0)B(λ0)v)/(uTD′(λ0)v), u ≡ kerD(λ0)T , v ≡ kerD(λ0),
and D′ ≡ dD(λ)/dλ (see [1] for more details).
1.4. Crossing the Imaginary Axis: Loss of
Stability
Applying the Liénard-Chipart splitting [24], polynomial
(19) with continuous real coefficients ai(λ) can be written
as p(λ, s) = peven(λ, s2) + spodd(λ, s2), where



 

 
224
02 4
224
13 5
,|
,|
evens j
odds j
paasas
paasas


 




 


(22)
Polynomial p(λ, s) has purely imaginary roots ±jω if
and only if peven(λ,−ω2) = 0 and podd(λ,−ω2) = 0. These
conditions imply that the resultant matrix is singular [25].
Determinant detR(λ) = ±detHr(λ), where Hr is the Hurwitz

135 1
13 31
135 31
24 20
2420
24 420
000 0
00
00
000
00
00
rr r
rr
rr r
rr r
rr
rr r
aaa a
aa aa
aaa aa
Raa aa a
aaa aa
a aaaaa




























00
0
0

matrix of (19). Note that detHr(λ) = a0detHr−1 and



12 1
1,1
det 1,
rr r
rr
iki k
r

ik
H
as


 
s (23)
This implies that detHr−1 is zero not only for the purely
imaginary roots ±jω, but also for a pair of real symmetric
roots s1,2 = ±σ and for two complex and symmetric pairs
of roots s1,2 = σ ± jω and s3,4 = −σ ± jω. If, during the
bifurcation process with λ → λ0, a power system oper-
ates initially in a stable mode (all eigenvalues are in the
open left half plane), then, providing that none of the
eigenvalues diverges through ∞, the eigenvalues have to
cross the imaginary axis first (at λ0) before reaching
unstable locations in either of the symmetric patterns
with respect to the origin. Thus, in a typical situation, the
increased ill-conditioning of the matrix Hr(λ) with the
changing parameter λ will indicate an imminent possi-
bility of the Hopf bifurcation as some of the eigenvalues
approach the imaginary axis and become ±jω at λ = λ0 for
finite ω.