IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 51, NO. 4, APRIL 2004
185
A Singular Fixed-Point Homotopy Method to Locate
the Closest Unstable Equilibrium Point for
Transient Stability Region Estimate
Jaewook Lee, Member, IEEE, and Hsiao-Dong Chiang, Fellow, IEEE
Abstract—The closest unstable equilibrium point (UEP) method
is a well-known direct method of the Lyapunov type for optimally
estimating stability regions of nonlinear dynamical systems. One
key step involved in the closest UEP methodology is the computation of the closest unstable equilibrium point that has the lowest
Lyapunov function value on the stability boundary. In this paper,
a new computational algorithm to compute the closest UEP is presented. The proposed algorithm is based on a homotopy-continuation method combined with the singular fixed-point strategy. Numerical simulation results show that the algorithm outperforms
previously reported existing techniques.
Index Terms—Closest unstable equilibrium point (UEP),
nonlinear systems, power system transient stability, stability
boundary.
I. INTRODUCTION
T
HE problem of estimating stability regions (domains of
attraction) of nonlinear dynamical systems is of fundamental importance for many disciplines in engineering and the
sciences. These include economics, ecology, the stabilization of
nonlinear systems, power system transient stability analysis, the
power system voltage collapse problem, schemes for choosing
manipulator specifications, and parameters, etc. (see the references in [3] and [10]).
The closest unstable equilibrium point (UEP) method is a
well-known direct method of the Lyapunov type for optimally
estimating stability regions of nonlinear dynamical systems in
the sense that it gives the largest region within the stability region that can be characterized by the corresponding Lyapunov
function.
One key step involved in the closest UEP method, reviewed
in Section II, is the computation of the closest UEP with the
lowest Lyapunov function value on the stability boundary.
Since the closest UEP normally takes the form of a Type-1 UEP,
the conventional and still widely used method to compute the
closest UEP is zero-finding techniques such as quasi-Newton
methods. These zero-finding techniques, however, suffer from
several disadvantages.
Manuscript received February 14, 2003; revised June 6, 2003. This work
was supported in part by the Korea Research Foundation under Grant KRF2003-041-D00608 and in part by Com MaC-KOSEF. This paper was recommended by Associate Editor A. Ushida.
J. Lee is with the Department of Industrial Engineering, Pohang University
of Science and Technology, Pohang, Kyungbuk 790-784, Korea.
H.-D. Chiang is with the School of Electrical and Computer Engineering,
Cornell University, Ithaca, NY 14853 USA.
Digital Object Identifier 10.1109/TCSII.2004.824058
• First, convergence to a Type-1 UEP is not assured; if they
converge, they only converge to an equilibrium point.
Therefore, a large number of initial points are needed for
these zero-finding techniques to find Type-1 UEPs.
• Associated with the convergence difficulty is the determination of good initial points; in many cases, the size of the
convergence region of a Type-1 UEP is very small compared to that of a stable equilibrium point.
• Repeated calculation of previously known Type-1 UEPs
results in very high-computational complexity.
Recently, a reflected gradient method [13] has emerged as an alternative approach to overcome the difficulties associated with
zero-finding techniques. This method facilitates the search for
a Type-1 UEP by transforming the hard problem of finding
Type-1 UEPs of the original system into the easy problem of
searching stable equilibrium points with respect to the new modified system. This method can, thereby, guarantee the convergence to a Type-1 UEP. Despite some successful applications
of this method to a stability region estimate, this method has a
drawback in effectively computing the closest UEP since it can
normally find only two Type-1 UEPs [11] and so it cannot guarantee the convergence to the closest UEP when there are more
than two Type-1 UEPs on the stability boundary.
In this paper, we propose a new efficient homotopy-based algorithm for computing the closest UEP. The outline of this paper
is as follows. First, the closest UEP method is reviewed. Next,
a novel numerical algorithm is given and shown to converge to
a Type-1 UEP from a stable equilibrium point of our interest.
The algorithm is then combined with the singular fixed-point
strategy utilizing the concept of bifurcation in order to choose a
set of proper initial points that converge to the set of the Type-1
UEPs (including the closest UEP) on the stability boundary and
that avoid or reduce revisits of the same Type-1 UEPs. Several
numerical examples are used to illustrate the efficiency and reliability of the proposed method.
II. MATHEMATICAL PRELIMINARY
In this section, we first introduce some concepts in nonlinear dynamical systems and then review the closest UEP
methodology.
A. Basic Definitions
Consider a nonlinear dynamical system described by
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(1)
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 51, NO. 4, APRIL 2004
where
(1) starting from
is assumed to be smooth. The solution of
at
is called a trajectory, denoted by
. A state vector
is called an equilibrium
point (EP) of system (1) if it satisfies
.
We say that an equilibrium point
of (1) is hyperbolic if
, has no eigenvalues
the Jacobian of at , denoted by
with a zero real part. For a hyperbolic equilibrium point, it is a
Type- equilibrium point if its corresponding Jacobian has exactly eigenvalues with positive real parts. In particular, Type-0
equilibrium point is called a (asymptotically) stable equilibrium
point (SEP); otherwise it is called an unstable equilibrium point
(UEP).
is said to be an energy function for
A function
(i.e.
system (1) if (i) along any nontrivial trajectory
is not an equilibrium point)
Fig. 1. Estimated stability region using the closest UEP method.
(A1). Find all the Type-1 UEPs on the stability
( ).
boundary
(A2). Of these, identify the one, say ^, with the
lowest energy function value. This point
@A x
and the set
has measure zero in ,
is bounded, then
and (ii) if
is bounded. The region of attraction of a SEP
is
defined as
is called the closest UEP and the value of
the energy function at ^ gives the critical
level value of this energy function.
( )
(A3). The connected component of f 2 < :
x
V (^x)g
containing
bility region.
and the (quasi-)stability region of
is defined as
where its boundary is denoted by
. From a
is an open
topological point of view, the stability region
is an
and connected set, and its boundary
dimensional closed, invariant set.
B. The Closest UEP Method
Given an energy function, the central point in the energy function approach to estimating the stability region of a stable equilibrium point is the determination of the critical level value. The
next proposition serves a basis for the closest UEP method.
Proposition 1: [4] Let
be a stable equilibrium point of
system (1). Suppose there exists an energy function
for system (1) and let
be the
be the connected component of
level set of and
containing . Then we have the following:
(i) a point with the minimum value of the energy function
exists and it must be
over the stability boundary
a Type-1 UEP. If we let this one be , then
(ii)
and
number
for any
.
This proposition asserts that the closest UEP method of
as the critical value to estimate the stability
choosing
region
is optimal because the estimated stability region
characterized by the corresponding energy function is the
largest one within the entire stability region. (See Fig. 1.)
x
x
x
Vx <
gives the estimated sta-
III. THE PROPOSED METHOD
Homotopy methods have been developed to overcome the
local convergence nature of many iterative methods including
the Newton method and to compute multiple solutions. Stated
briefly, homotopy methods consist of the following: to solve
, where
, one defines a homotopy
such that
and
function
, where
is a smooth map having
known zero points. Typically, one may choose a convex homotopy function such as
in an attempt to trace an implicitly defined curve (called a homofrom an initial guess
topy path) satisfying
to a solution point
. If this succeeds, then a zero
of
is obtained. Usually it is unclear how
point
should be chosen, but the following two choices for
are
widely used [1], [17].
1) Newton homotopy:
;
.
2) fixed-point homotopy:
Over the last two decades, a significant effort has been spent
in studying theoretical and algorithmic aspects of homotopy
methods and has yielded extremely important contributions toward the numerical solution of nonlinear systems of equations.
In particular, it has been successfully applied to finding dc operating points of transistor or nonlinear circuits in circuit simulations (see [14] and [17] and references therein).
In this section, we propose a homotopy-based method to compute the closest UEP. The next theorem gives a theoretical basis
on the usage of the homotopy approach in effectively finding a
Type-1 UEP.
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LEE AND CHIANG: SINGULAR FIXED-POINT HOMOTOPY METHOD
Theorem 1: Let
(1). Let
given by
187
be a stable equilibrium point of system
be a Newton homotopy function
(2)
If zero is a regular value of the functions
and
, then there exists a parameterized and
,
) in
connected solution curve (or homotopy path) (
containing
. Furthermore, if the homotopy path
and there is exactly one turning point between
contains
and
along this homotopy path, then
is a
Type-1 UEP of system (1).
Proof: The regularity condition implies that: (i)
there exists the solution set of the equation
that is a one-dimensional (1-D) manifold in
and the solution curve (or homotopy path) containing
can be parameterized as (
,
) ([1], [9]);
along this homotopy
(ii) rank
path; and (iii) the derivative of
does not vanish whenever
vanishes on this homotopy path. Hence, by (ii) and (iii),
and the sign of
we get rank
changed only at a turning point of the
,
homotopy path. This implies that the homotopy path (
) does not change its Type with respect to until it passes
a turning point in which case it changes its Type only by one.
(Note, that this property generally holds for any homotopy
satisfying the above conditions. See also ([9],
function
[16], and [17].) Now we know that for the Newton homotopy
. Since is a Type-0
function,
and
SEP and there is exactly one turning point between
along the homotopy path,
is, therefore, a Type-1
UEP of system (1).
Remark:
1) The regularity condition of Theorem 1 is very general in a
sense that it is a “generic” property [9].
2) This theorem shows that by numerically tracing the homoand hoping that it will retopy path starting from
again,
verse the direction and pass through the plane
(see Fig. 2). If the homotopy
we get a Type-1 UEP
path returns to the plane
after more than one turning
is only guaranteed to be a
point, the obtained point
odd Type UEP but not generally a Type-1 UEP. Moreover,
in practice, it has been observed that the point obtained after
an odd number ( 1) of turning points is often located outside the stability boundary of our interest ([11], [12]).
IV. ALGORITHM
One important issue in applying the Newton homotopy
method to optimally estimating the stability region, is how to
choose an initial point in (2) that leads to the convergence to
the closest UEP. Since the closest UEP is the Type-1 UEP with
the lowest energy function value of all the Type-1 UEPs on the
stability boundary, this problem can be transformed into the
problem of how to determine a set of proper initial points ’s
from that the method converges to a set of
near a given SEP
the Type-1 UEPs (including the closest UEP) on
. This
Fig. 2. The shape of the homotopy path (x(s), (s)) with a turning point.
Near a turning point, a pair of critical points dies (or is born) as decreases (or
increases). The Type of x(s) changes by exactly one when passing a turning
point along the homotopy path.
task is, however, extremely difficult unless some knowledge
on the structure of is a priori available. Also, as was shown
in [12], only from a local information near , one cannot
generally determine the number of the Type-1 UEPs on the
stability boundary of . Despite of such theoretical difficulties,
there is a need for finding a set of initial points from which the
Newton homotopy method avoids frequent revisits of the same
Type-1 UEPs. To this end, we adopt a deterministic strategy
inspired by the singular fixed-point strategy [11] for which
is defined by
(3)
The motivation behind this approach is to establish a set of initial
points that does not converge to the same Type-1 UEP utilizing
the concept of bifurcation. Notice that (3) has a bifurcation point
since
at
the Jacobian
has a rank-deficiency (zero-rank) at
. One distinguished feature of this approach is that
several branches from the bifurcation point can be established
and they normally lead to distinct Type-1 UEPs via the Newton
homotopy method, which results in avoiding or reducing the
revisits of the same Type-1 UEPs. (See Fig. 3.)
To obtain the set of initial directions , which correspond
to the set of starting points of each homotopy paths, we make a
second-order approximation for and solve the following:
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(4)
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 51, NO. 4, APRIL 2004
whose energy function is given by
2) Problem [TD]: This example, considered in ([5] and
[10]), represents a tunnel diode circuit whose state-space
model is given by
where
,
,
,
.
The next benchmark problems are gradient systems described by
Fig. 3. Four branches from a bifurcation point x lead to three different Type-1
UEPs.
where
and
is a Hessian matrix of
with
. The computation involves
and several solvings of linear
a decomposition of
systems using this decomposition, all of which can be easily
performed by well-developed numerical linear algebra solvers.
Other ways to compute the bifurcation branches can be found
in [1] or [8].
The results obtained in this section lead to the following conceptual algorithm to compute the closest UEP.
,
1) (Initialization): Find all the initial points
satisfying (4). Set the required accuand
(the variable records all the
racy
Type-1 UEPs).
to do
2) (Find Type-1 UEPs): while
until
Follow the path of (2) starting from
the trajectory reverse the direction and pass through
with the given accuracy
. If the
again
, set
.
obtained point is
Otherwise, this initial point fails to converge to a
solution.
end
3) (Find the closest UEP): Of these, identify the one,
say , with the lowest energy function value.
whose energy function
3) Problem [LE]:([11])
is defined as follows.
4) Problem [DI]:([6])
5) Problem [CA]:([11])
6) Problem [CE]:([2])
7) Problem [FR]:([11])
B. Test Results and Discussion
V. SIMULATION RESULTS AND DISCUSSION
A. Simulated Benchmark Problems
The algorithm described in Section IV has been tested on several test examples we have found in the literature. Here are some
descriptions of the test problems.
1) Problem [CH]: This example, considered in ([13]), represents a reduced gradient system of a three-machine power
system with machine number 3 as the reference machine.
Table I shows the performance of our proposed algorithm
compared to other competing algorithms such as Newton-type
methods with multistarts and reflected gradient methods. Our
performance metrics are based on two factors: (i) robustness
(whether the method can successfully find the closest UEP) and
(ii) efficiency (the ratio of the number of Type-1 UEPs found by
the method and the number of tried initial points). The comparison result demonstrates that the proposed algorithm not only
successfully locates the closest UEPs for all these test problems
but also is efficient than these state-of-art method.
The proposed method may fail to find all of its Type-1 UEPs
on the stability boundary for some problems (problems DI). This
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LEE AND CHIANG: SINGULAR FIXED-POINT HOMOTOPY METHOD
189
TABLE I
SPECIFICATIONS AND RESULTS OF SIMULATED PROBLEMS: RATIO=#(TYPE-1 EP FOUND)/#(INITIAL TRIAL POINTS); (F) FAIL, (S) SUCCESS; (NM-MS):
NEWTON-TYPE METHOD WITH MUTISTARTS, (RGM): REFLECTED GRADIENT METHOD, BOLDFACE: CLOSEST UEPS OF EACH PROB.ID
is inherent for all deterministic methods since they are, in general, forced to deal with somewhat restricted class of systems
in order to exploit mathematically rigorous properties. We remark that, even for the case that the proposed method may fail
to find all the Type-1 UEPs, another choice of a diagonal matrix
in (3) instead of
leads to
obtaining the rest of the Type-1 UEPs, which inspires new research direction in this area.
VI. CONCLUSION
In this paper, we have proposed a homotopy-based algorithm
for the computation of the closest UEP and shown that the
proposed algorithm can serve as an alternative approach. We
have shown from simulation results that the singular fixed-point
strategy utilizing the concept of bifurcation was very effective
and efficient in choosing a set of proper initial points: (i) that
converge to the set of the Type-1 UEPs (including the closest
UEP) on the stability boundary and (ii) that avoid frequent
revisits of the same Type-1 UEPs. The applicability of the
algorithm to large-scale nonlinear systems problems remains to
be investigated.
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