![Bifurcation Analysis to Power Systems 3
Consequently, the power system dynamics starting from the old stable equi-
librium point will converge to the new stable equilibrium point and will make
the system state track its new stable equilibrium point, whose position is
changed continuously but slowly, and yet the system remains stable under
this load disturbance. The typical ways in which a study power system may
lose stability, under the influence of load variations are through the following:
• the stable equilibrium point and another equilibrium point coalesce and
disappear in a saddle-node bifurcation as parameter varies, or
• the stable equilibrium point and another equilibrium point coalesce and
exchange stability in a limit-induced bifurcation (a type of transcritical
bifurcation) as parameter varies,
• the stable equilibrium point and an unstable limit cycle coalesce and
disappear and an unstable equilibrium point emerges in a subcritical
Hopf bifurcation as parameter varies,
• the stable equilibrium point bifurcates into an unstable equilibrium point
surrounded by a stable limit cycle in a supercritical Hopf bifurcation as
parameter varies.
It is now well recognized that bifurcations are rather mundane phenomena
that can occur in many physical and man-made systems where nonlinearity is
present. The pioneer work on the local bifurcation analysis of power systems
can be dated back to the 1970’s and earlier [62,63,41]. Within the last 20
years or so nonlinear dynamical theory has become a subject of great inter-
est to researchers and engineers in the power system community. Powerful
computational tools for bifurcation analysis have been applied during this
period to study important nonlinear problems arising in power systems, and
in some cases, to relate this study to observed nonlinear phenomena in power
systems [45,46,59,53]. In addition, some counter-measures to avoid bifurca-
tions have been developed to design control schemes for prevention of power
system instabilities [23,24,52,73,34,35].
From the engineering viewpoint, one important task in performing bifur-
cation analysis to nonlinear systems, such as electric power systems, is the
analysis of both the mechanism leading to disappearance of stable equilib-
rium points due to a bifurcation and the system dynamical behaviors after
the bifurcation. After a bifurcation occurs, the system state will evolve ac-
cording to the system dynamics. The dynamics after bifurcation determine
whether the system remains stable or become unstable; and what is the type
of system instability. Local bifurcation theory does not describe the dynam-
ical behaviors after a bifurcation. We will review some work on the analysis
of the system dynamics after typical local bifurcations in this chapter.
Electric power systems comprise a large number of components interact-
ing with each other in nonlinear manners. The dynamical response of these
components extends over a wide range of time scales. The different time-
scale components of power systems all have their corresponding influences
on power system dynamical responses. It has become convenient to divide](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-17-320.jpg)
![4 H.-D. Chiang
the time span of dynamic response simulations into short-term (transient),
mid-term and long-term, covering the post-disturbance times of up to a few
seconds, 5 minutes and 20 minutes or so, respectively. Up to present, most
power system models used for bifurcation analysis involve only short-term
dynamical models (transient stability models). It raises the concern about
the validity of short-term dynamical models, which have disregarded slow
dynamics, for local bifurcation analysis. The effects of un-modelled dynamics
on the local bifurcation analysis of a power system model is also discussed in
this chapter.
P-V and Q-V curves have been widely used by power system analysis
engineers to study voltage stability [60]. These curves represent one impor-
tant aspect of the saddle-node bifurcation occurring in power systems due
to variations of loads and generations. While global analysis tools based on
continuation methods developed in the last decade can generate P-V and
Q-V curves in a reliable manner, these tools may be too slow for certain
power system on-line applications. To overcome this difficulty, a number of
performance indices intended to measure the severity of the voltage stability
problem have been proposed in the literature. We will examine several exist-
ing performance indices and discuss a performance index which has rendered
practical applications. This performance index is based on the normal form
theory of saddle-no de bifurcation point.
In this chapter, we will present an overview on the application of local bi-
furcation analysis and theory to (i) develop models explaining power system
nonlinear behaviors and various power system instabilities such as voltage col-
lapse and low-frequency oscillations, to (ii) develop a powerful global analysis
tool based on continuation methods to trace power system quasi-steady-state
behaviors due to load and generation variations in realistic power system
models, and to (iii) develop powerful computational tools for detecting and
estimating local bifurcations of power systems. An overview on the extension
of saddle-node bifurcation, Hopf bifurcation and limit-induced bifurcation to
include the analysis of the system dynamics after the bifurcation will be pre-
sented. In addition, the effects of un-modelled dynamics due to fast and slow
variables on local bifurcations will be analyzed.
2 Local Bifurcations and Power System Behaviors
Recently, local bifurcation theory has being applied to interpret observed non-
linear dynamical behaviors in power systems. In some cases, local bifurcation
theory has been extended to include the analysis of dynamics after a local
bifurcation. A comprehensive bifurcation and chaos analysis of a 3-bus power
system was carried out in [11]. Numerical bifurcation analysis of a simplified
model of a 9-bus power system and a 39-bus power system were conducted
in [12]. Other numerical bifurcation analysis of simple power systems can be
found in, for example, [1,4]. It has been found that the bifurcation phenomena](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-18-320.jpg)
![Bifurcation Analysis to Power Systems 5
observed in the 3-bus and 9-bus power system are similar. These bifurcation
phenomena have been observed in the 39-bus power system as well, includ-
ing Hopf, period-doubling, and cyclic fold bifurcations. Furthermore, some
bifurcation phenomena not appearing in the 3-bus and 9-bus systems have
surfaced in the 39-bus system. These numerical studies favor the claim that
various types of bifurcations can occur in real power systems.
Local bifurcation theory has been applied to provide an explanation for
various observed power system nonlinear behaviors and power system insta-
bilities such as voltage collapse and low-frequency electro-mechanical oscil-
lations that occur in electric power networks. Abed and Varaiya [2] were
probably the first to suggest a possible role for Hopf bifurcations in explain-
ing the low-frequency electro-mechanical oscillation phenomena. Later Chen
and Varaiya [10] numerically demonstrate that degenerate Hopf bifurcation
can occur in a simple power system model. In [20], Dobson and Chiang inves-
tigated a generic mechanism leading to disappearance of stable equilibrium
points due to a saddle-node bifurcation and the subsequent system dynam-
ics after the bifurcation for one-parameter dynamical systems. A saddle-node
voltage instability model to analyze the process of voltage instability was then
proposed to explain voltage stability/instability due to slow load variations
in three stages. Iravani and Semlyen investigate the transition of growing
torsional oscillations into limit cycles occurring in torsional dynamics based
on Hopf bifurcation [28]. Their studies indicate that the range of instability
(growing oscillations) based on Hopf bifurcation is noticeably narrower than
the one predicted by an eigen-analysis method.
Another dynamic phenomenon of concern in power systems that can be
explained via Hopf bifurcation is subsynchronous resonance (SSR). This is a
condition where the electric network exchanges energy with a turbine genera-
tor at one or more of the natural frequencies of the combined system below the
synchronous frequency of the system. The IEEE SSR working group suggests
that if the locus of a particular eigenvalue approaches or crosses the imaginary
axis, then a critical condition is identified that will require the application
of one or more SSR counter measures. The critical condition is closely re-
lated to the Hopf condition. To provide high-quality electricity, utilities must
endeavor to minimize the effects of large fluctuating loads associated with
large motors and furnaces on the transmission network. One of the effects,
characterized by visible fluctuations in other customers’ electric lighting, is
called voltage flicker. This phenomenon is often categorized as cyclic and
non-cyclic. From a dynamic system viewpoint, cyclic flickers may relate to
limit cycles or quasi-periodic motion in power systems and non-cyclic flickers
to quasi-periodic or chaotic trajectories.
Another advance of applying bifurcation analysis to power systems can be
manifested in the development of several bifurcation-based models to explain
several instances of recent power system voltage instability and/or collapse.
This kind of blackout has occurred in several countries such as Belgium,](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-19-320.jpg)
![6 H.-D. Chiang
Canada, France, Japan, Sweden and the United States [49,60]. Voltage in-
stability and/or collapse, a frequent concern on modern power systems, are
generally caused by either of two types of system disturbances: load distur-
bances or contingencies, i.e., event disturbances. Among several examples of
voltage collapse, the 1987 occurrence in Japan [43] was due to large load vari-
ations, while the collapse in Sweden in 1982 [71] was caused by a contingency.
The dynamic process of voltage instability or collapse usually starts with a
power system weakened by a contingency due to a transmission line or gen-
erator outage, or by an unusually high peak load (a high load variation), or
by a combination of such events. The system may be further weakened due
to an inappropriate transmission under-load tap-changer (ULTC) setting,
or insufficient reactive power supports, or load restorations that have been
temporary reduced because of low voltage. Three bifurcation-based voltage
collapse models will be discussed in some details in this chapter.
3 Local Bifurcations in Power Systems
A power system model relative to a disturbance comprises a set of first-order
differential equations and a set of algebraic equations
ẋ = f(x, y, u, λ) (1)
0 = g(x, y, λ)
where λ ∈ R1
is a parameter, x is a dynamic state variable and y a static
“state” variable, such as the load variables of voltage magnitude and angle.
The vector field depends on the value of parameter and will change its dimen-
sion accordingly. It describes the internal dynamics of devices such as gener-
ators, their associated control systems, certain loads, and other dynamically
modelled components. The set of algebraic equations describe the electrical
transmission system (the interconnections between the dynamic devices) and
internal static behaviors of passive devices (such as static loads, shunt ca-
pacitors, fixed transformers and phase shifters). The differential equations
(1) can describe as broad a range of behaviors as the dynamics of the speed
and angle of generator rotors, flux behaviors in generators, the response of
generator control systems such as excitation systems, voltage regulators, tur-
bines, governors and boilers, the dynamics of equipments such as synchronous
VAR compensators (SVCs), DC lines and their control systems, and the dy-
namics of dynamically modelled loads such as induction motors. The state
variables x typically include generator rotor angles, generator velocity de-
viations (speeds), mechanical powers, field voltages, power system stabilizer
signals, various control system internal variables, and voltages and angles
at load buses (if dynamical load models are employed at these buses). The
forcing functions u acting on the differential equations are terminal voltage
magnitudes, generator electrical powers, and signals from boilers and au-
tomatic generation control systems. Some control system internal variables](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-20-320.jpg)
![Bifurcation Analysis to Power Systems 7
have upper bounds on their values due to their physical saturation effects.
Let z be the vector of these constrained state variables; then the saturation
effects can be expressed as
0 ≤ z(t) ≤ ẑ (2)
We term the above model as a set of parameter-dependent differential and
algebraic equations with hard constraints. A detailed description of equations
(1) - (2) for each component can be found, for example, in [36,42,55]. For a
500-generator power system, the number of differential equations can easily
reach as many as 10,000.
From a nonlinear dynamical system viewpoint, (1-2) is an one-parameter
dynamical system while, in power system applications, it can represent a
power system that operates with one of the following conditions:
1. the real (or reactive) power demand at one load bus varies while the
others remain fixed,
2. both the real and reactive power demand at a load bus vary and the
variations can be parameterized. Again the others remain fixed,
3. the real and/or reactive power demand at some collection of load buses
varies and the variations can be parameterized while the others are fixed,
4. the real power transfer at one transmission corridor (e.g. interface transfer
and import/export) varies while the others remain fixed,
5. the real power transfer at some collection of transmission corridors (e.g.
interface transfer and import/export) varies while the others remain fixed.
The only typical ways in which a power system may lose stability (un-
der the influence of one parameter variation) are through the saddle-node
bifurcation, or limit-induced bifurcations or the Hopf bifurcation. In [21], it
has been shown that for generic one-parameter dynamical systems, before a
saddle-node bifurcation the equilibrium point x1(λ) is type-one. By type-one,
we mean that the corresponding Jacobian matrix has exactly one eigenvalue
with a positive real part and the rest of the eigenvalues have negative real
parts. Furthermore, x1(λ) lies on the stability boundary of xs(λ). The Ja-
cobian matrix, when evaluated at xs(λ), has all of its eigenvalues with only
negative real parts and among them, one of the eigenvalues is close to zero. At
the bifurcation occurring at say, the bifurcation value λ = λ∗
, equilibrium
points xs(λ) and x1(λ) coalesce to form an equilibrium point x∗
(= xs(λ∗
)
= x1(λ∗
)). The Jacobian matrix evaluated at x∗
has one zero eigenvalue and
the real parts of all t he other eigenvalues are negative. If the parameter λ
increases beyond the bifurcation value λ∗
, then x∗
disappears and there are
no other equilibrium points nearby.
Another local bifurcation peculiar to power systems is the so-called limit-
induced bifurcation. Physically, the generation reactive power capability is
limited. The reactive power capability of a generator can reach a limit due
to the excitation current limit or the stator thermal limit. Power systems](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-21-320.jpg)
![8 H.-D. Chiang
are vulnerable to voltage collapse when generation reactive power limits are
reached. Given a load/generation variation pattern, the effect of reaching a
generator reactive power limit is to immediately change the system equation.
From a static analysis viewpoint the generator whose reactive power limit is
reached may be simply modelled by replacing the equation describing a P-V
bus by the equation describing a P-Q bus. In [21], numerical examples and
general arguments were developed to show that a sufficiently heavily loaded
but stable power systems can become immediately unstable via a transcritical
bifurcation when a reactive power limit is encountered. We term this type of
bifurcation as limit-induced bifurcation. We note that when a transcritical
bifurcation occurs at say, the bifurcation value λ = λ∗
, the stable equilibrium
point xs(λ) and a type-one unstable equilibrium point xu(λ) coalesce to form
an equilibrium point x∗
(= xs(λ∗
)). The Jacobian matrix evaluated at x∗
has
a single, simple eigenvalue and the real parts of all the other eigenvalues are
negative. After the bifurcation, the two equilibrium points change stability
to become a type-one equilibrium point and a stable equilibrium point.
Hopf bifurcation can occur on generic one-parameter dynamical systems.
Before a subcritical Hopf bifurcation, the unstable limit cycle xl
1(λ, t) lies
on the stability boundary of xs(λ). The Jacobian matrix, when evaluated
at xs(λ), has all of its eigenvalues with only negative real parts and among
them, a pair of complex eigenvalues are close to zero. At the bifurcation
occurring at say, the bifurcation value λ = λ∗
, the equilibrium point xs(λ)
and the unstable limit cycle xl
1(λ, t) coalesce to form an equilibrium point
x∗
(= xs(λ∗
) = xl
1(λ∗
)). The Jacobian matrix evaluated at x∗
has two zero
eigenvalues and the real parts of all the other eigenvalues are negative. If the
parameter λ increases beyond the bifurcation value λ∗
, then x∗
becomes a
type-two equilibrium point and there are no other equilibrium points or limit
cycles nearby. As for the supercritical Hopf bifurcation, we make the following
remarks. Before the bifurcation. The Jacobian matrix, when evaluated at
xs(λ), has all of its eigenvalues with only negative real parts and among
them, a pair of complex eigenvalues are close to zero. A t the bifurcation
occurring at say, the bifurcation value λ = λ∗
, the equilibrium point xs(λ)
is an equilibrium point x∗
(= xs(λ∗
). The Jacobian matrix evaluated at x∗
has two zero eigenvalues and the real parts of all the other eigenvalues are
negative. If the parameter λ increases beyond the bifurcation value λ∗
, then
x∗
becomes a unstable equilibrium point x∗
(λ) surrounded by a stable limit
cycle.
4 Dynamics after Local Bifurcations
Local bifurcation theory has been developed to describe mechanisms leading
to disappearance of stable equilibrium points due to a local bifurcation. Re-
cently, local bifurcation theory has being extended to include the analysis of
system dynamics after a local bifurcation. The knowledge of subsequent sys-](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-22-320.jpg)
![Bifurcation Analysis to Power Systems 9
tem dynamics is essential to determine whether the system remains stable or
becomes unstable, and the type of system instability if the system is deemed
unstable. This section briefly summarizes the model for analyzing the system
dynamics after a saddle-node bifurcation. Models for analyzing the system
dynamics after the limit-induced bifurcation and Hopf bifurcations are also
described.
4.1 Saddle-node bifurcation
In [20], Dobson and Chiang investigated a generic mechanism leading to
disappearance of stable equilibrium points due to a saddle-node bifurcation
and the subsequent system dynamics for one-parameter dynamical systems.
When a saddle-node bifurcation occurs at say, the bifurcation value λ = λ∗
,
equilibrium points xs(λ) and x1(λ) coalesce to form an equilibrium point
x∗
(= xs(λ∗
) = x1(λ∗
)). The Jacobian matrix evaluated at x∗
has one zero
eigenvalue and the real parts of all the other eigenvalues are negative. The
eigenvector p that corresponds to the zero eigenvalue points in the direction
along which the two vectors xs(λ) and x1(λ) approached each other. There
is a curve made up of system trajectories which is tangent to eigenvector
p at x∗
. This curve is called the center manifold of x∗
and is the union of
a system trajectory Wc
− converging to x∗
, the equilibrium point x∗
and a
system trajectory Wc
+ diverging from x∗
. Next, we consider the case that λ
remains fixed at bifurcation value λ∗
. When the system trajectory is near
Wc
+ at the moment that the bifurcation occurs and if λ remains fixed at its
bifurcation value λ∗
, then the system trajectory after the bifurcation moves
near Wc
+. The system dynamics due to the bifurcation are then determined
by the position of Wc
+ in state space. If Wc
+ is positioned so that some of
the voltage magnitudes decrease along Wc
+, then we associate the movement
along Wc
+ with voltage collapse. This is the center manifold voltage collapse
model due to saddle-node bifurcation. This model has two advantages from
a computational point of view.
1. Since p is tangent to Wc
+ at x∗
, the initial direction of Wc
+ near x∗
is
determined by p which can be computed f rom the Jacobian matrix at
x∗
.
2. Since Wc
+ is a system trajectory, the dynamics of voltage collapse can be
predicted by integrating system equations (1) starting on Wc
+ near x∗
.
4.2 Limit-induced bifurcation
In [21], Dobson and Lu studied a mechanism leading to disappearance of sta-
ble equilibrium points due to a limit-induced bifurcation and the subsequent
system dynamics. Before the bifurcation, the system is operated around the
stable equilibrium point xs(λ). At the limit-induced bifurcation, the stable](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-23-320.jpg)
![12 H.-D. Chiang
point or the limit-induced bifurcation point. Stage 3 describes the system
dynamical behavior after bifurcation to assess whether the system, after bi-
furcation, remains stable or becomes unstable; and determines the types of
system instability (voltage collapse and/or angle instability). Hence, the volt-
age collapse model describes both the static aspect (stages 1 and 2) and the
dynamic aspect (stage 3) of the problem.
4.6 Hopf voltage collapse model
Stage 1: the system is in quasi-steady state and is tracking a stable
equilibrium point.
Stage 2: the system reaches its steady-state stability limit when the
stable equilibrium point undergoes a subcritical Hopf bifurcation.
Stage 3: the system dynamics after bifurcation are captured by a two-
dimensional center manifold.
The Hopf voltage collapse model also describes both the static aspect (stages
1 and 2) and the dynamic aspect (stage 3) of the problem. This model dic-
tates that mathematically speaking, the steady-state stability limit may be
determined by the subcritical Hopf bifurcation, instead of the Saddle-node
bifurcation. One implication is that the load margins will be less that one
might expect if the nose point was taken as the point of voltage collapse.
Since detecting Hopf bifurcation requires the knowledge of the eigenvalues
of the system Jacobian, the traditional repetitive power flow approach cannot
detect Hopf bifurcation.
The saddle-node bifurcation has been a widely accepted model for voltage
collapse analysis. Most computational tools developed so far have been con-
centrated on the identification of saddle-node bifurcation point, also termed
point of collapse [6]. The SAD voltage collapse model includes the saddle-
node bifurcation point as a point of collapse. In fact, it can be shown that
the SAD voltage collapse model encompasses many existing models used to
explain voltage collapse such as the multiple power flow model, the power flow
feasibility model, the static bifurcation model, the singular Jacobian model
and the system sensitivity model. Indeed, Stage 1 is related to the feasibility
of the power flow solution [37,40]. It has been shown that stage 2 itself is a
generalization of many existing models used to explain voltage collapse [13].
From an algebraic point of view, the point (x(λ0), λ0) abbreviated by
(x0, λ0) is a saddle node of (1) if the following conditions hold:
1. f(x0, λ0) = 0.
2. fx(x0, λ0) has a simple eigenvalue 0.
3. fλ(x0, λ0) ;∈ Range space of (fx(x0, λ0)).
4. there is a parameterization (x(t), λ(t)) with x(t0 ) = x0, λ(t0) = λ0 and
d2
λ(t0)
dt2 ;= 0.](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-26-320.jpg)
![Bifurcation Analysis to Power Systems 13
We now use the above algebraic characterizations to examine several models
proposed by several researchers for predicting voltage collapse. Note that
the voltage collapse models based on the determinant of system Jacobian
in [67,56] can be viewed as providing necessary conditions for the first two
conditions for saddle nodes. We show in the following that the model based on
the sensitivity of system Jacobian in [19] also provides a necessary condition
for the first two conditions of saddle nodes. More specifically, we show that
the sensitivity of state vector with respect to its parameter at a saddle node
is infinity. Suppose that (x∗
, λ∗
) is a solution of (1) and fx(x∗
, λ∗
) is non-
singular, the implicit function theorem guarantees the existence of a unique
solution curve (x(λ), λ) passing through (x∗
, λ∗
), i.e . x∗
= x(λ∗
). And we
have the following identity:
dx(λ)
dλ
= − fx(x(λ), λ)− 1
fλ(x(λ), λ) (3)
Although the matrix fx(x(λ), λ)− 1
does not exist at the saddle node (x(λ0
),
λ0
), the property that fλ(x0, λ0) ;∈ Range space of (fx(x0, λ0)) ensures
that there is a unique solution curve passing through (x(λ0
), λ0
) and dx(λ)
dλ
becomes infinite there and a small change in λ yields a large change in +x+.
This result also explains the “knee” phenomenon in the P-V curve and Q-V
curve.
The voltage collapse model based on multiple power flow solutions [61,64,
57] uses the presumption that the existence of a pair of very close power flow
solutions indicates that the system is about to undergo a voltage collapse. We
note that before the bifurcation, there are two equilibrium points (power flow
solutions) close to each other. One is stable and the other is type-one. As these
two points approach each other an annihilation occurs at the saddle node
bifurcation, while at the same time the system Jacobian becomes singular.
The above two voltage collapse models can be extended to general slow
time-varying one-parameter dynamical systems. The assumption that the
parameter λ freezes at a bifurcation point may be inadequate to reflect real
power system behavior. A more realistic model is to allow a slowly time-
varying parameter both before and after the bifurcation. In particular, the
assumption that the system parameter “freezes” at the bifurcation point of
interest can be removed. The system dynamics after the bifurcation are then
captured by the center manifold of the bifurcation point x∗
with respect to
(2).
ẋ = f(x, λ) (4)
λ̇ = Mg(x, λ)
where M is a small number and g is a locally Lipschitz function.
In order to extend the voltage collapse model to system (4), it will be
necessary to examine the adequacy of the center manifold of x∗
with respect
to system (1) in capturing the dynamics after a bifurcation. In this regard,](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-27-320.jpg)
![14 H.-D. Chiang
we examine the relationship between the trajectories of (1) and (4) which
start at the same point near the center manifold of x∗
. It can be shown that
system trajectories of (4) follow the system trajectories of (1) during each
choice of time interval provided the rate of parameter variation is sufficiently
small as shown in the following:
Proposition 1
Let x1(t), x2(t) be the system trajectories of (1) and of (4) starting from the
initial condition (x0
1, t0) and (x0
2, t0) respectively. Let U be a compact set of
the state space containing the center manifold as far as it is of interest, and
let K be the Lipschitz constant of (4) on the set U. If x0
1 = x0
2 ∈ U, then
|x1(t) - x2(t)| ≤ CM
K [eK(t−t0)
- 1], where M is a constant.
The above result shows that we may approximate the system dynamics
after a local bifurcation point by applying (1) instead of (4). Hence, if the
parameter changes slowly enough, then the solutions of (4) which lie near the
center manifold of (1) at the time of bifurcation will subsequently track the
center manifold trajectory of (1) in the state space. This analysis validates
the voltage collapse model for power system with slowly variation loads.
5 Computational Tools
Power systems are subject to parameter variations. It is important to study
the impacts of parameter variations on power system behaviors by tracing
the quasi-steady-state of realistic power system models subject to parameter
variations. A powerful global analysis tool based on continuation methods
can meet this requirement. Continuation methods, sometimes called curve
tracing or path following, are useful tools to generate solution curves for
general nonlinear algebraic equations with a varying parameter. The theory
of continuation methods has been studied extensively and has its roots in
algebraic topology and differential topology. Continuation methods have four
basic elements: parameterization, predictor, corrector and step-size control.
The application of continuation methods to power system analysis has
been very actively investigated in recent decades, see for example [38,14,48,3,
7,15,32]. The most attractive feature of a continuation method is that it al-
lows users to globally analyze a given power system relative to parameter
variations in a reliable and efficient manner. In [12], a survey of existing and
pioneering continuation methods applied to power system analysis, which
may contain thousands of nonlinear algebraic equations with some limits on
some of the state variables, is presented. This survey also includes a com-
parison among different implementations of continuation methods for power
system applications according to predictor type, corrector type, step-size con-
trol strategy, parameterization schemes and modelling capability.
A widely used approach in the power industry to investigate potential
voltage stability problems, with respect to a given parameter increase pattern,
is the use of repetitive power flow calculations. The main advantages of a](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-28-320.jpg)
![16 H.-D. Chiang
The step-size along the arc-length s yields the following constraint:
n
V
i=1
{(xi − xi(s))2
} + (λ − λ(s))2
− (∆s)2
= 0
Third, solve the following n + 1 equations for the n + 1 unknowns x and λ
f(x, λ) = 0 (7)
n
V
i=1
{(xi − xi(s))2
} + (λ − λ(s))2
− (∆s)2
= 0 (8)
It can be shown that the above set of augmented power flow equations is well-
conditioned, even at the “nose” point. These augmented power flow equations
can be solved to obtain the solution curve passing through the “nose” point
without encountering the numerical difficulty of ill-conditioning.
The task of computing maximum loading points (it saddle-node points or
limit-induced bifurcation points) relative to a given load/generation variation
pattern has important applications in power system operations and planning.
The maximum loading points have a strong relationship with the operating
points where voltage collapse may occur. A widely used approach in power
industry to determine the maximum loading point, with respect to a given
load/generation increase pattern, is the repetitive power flow calculations to
generate the so-called P-V or Q-V curve relative to the variation pattern. An
operating point on the curve is said to be the maximum loading point of the
system if the point is the first point on the curve where power flow calculation
does not converge. Note that, due to the its shape in the bifurcation diagram,
the saddle-node bifurcation point is termed nose point in power engineering
community. Depending on the physical meaning of the underlying parameter
and the power network conditions, nose points have been physically related
to maximum loading points, or to maximum transfer capability points, or to
voltage collapse points.
Several issues arise regarding this approach [44]. These issues however can
be resolved by applying the local bifurcation theory. First, the point where
the power flow diverges (which is a numerical failure caused by a numerical
method) does not necessarily represent the maximum loading points (which
is a physical limitation). Second, the point where power flow calculations fail
to converge may vary, depending on which numerical method was used in
the power flow calculation. In other words, based on the criterion of power
flow divergence, different numerical methods may come up with different
calculated maximum loading points of the system while the maximum loading
point physically is unique. We note that the set of power flow equations is ill-
conditioned near nose points making Jacobian-based numerical methods such
as the Newton method diverge in the neighborhood of nose points. It is well
recognized that the Jacobian at a nose point has one zero eigenvalue, causing
the set of power flow equations ill-conditioned near nose points. Recently,](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-30-320.jpg)
![Bifurcation Analysis to Power Systems 17
considerable progress has been made in calculating nose points in a reliable
and exact way by using continuation methods and the characteristic equations
of nose points. Continuation methods are reliable to overcome the singularity
of the Jacobian near nose points and can provide partial initial conditions for
solving the characteristic equations.
The standard formulation for the characteristic equations of nose points
for a set of n-dimensional power flow equations is a set of (2n+1)-dimensional
nonlinear equations. Solutions to the characteristic equations give the nose
point (n-dimensional), the bifurcation value and the left or right eigenvector
(n-dimensional corresponding to the zero eigenvalue. To solve the characteris-
tic equations, continuation methods can only provide good initial conditions
for an estimated nose point and an estimated bifurcation value. What is
missing is a good initial guess for the eigenvector which is an additional fac-
tor affecting convergence to the solution. Another method which solves an
extended (2n+1)-dimensional system of equations characterizing the saddle-
node bifurcation point was proposed in [6] and more recently in [7]. The
methods attempt to compute the saddle-node bifurcation point directly. The
success of the above two methods depends greatly on a good initial guess
of the desired saddle-node bifurcation point. Otherwise, the methods may
diverge or converge to another saddle-node bifurcation point. This is because
these methods are static in nature, they do not make use of any information
on the particular branch of solutions and they do not confine their iterative
process to the desired branch of solutions.
A simpler set of characteristic equations for nose points of power flow
equations can be developed by exploring a decoupled parameter-dependent
property of power flow equations. In [44], a test function was developed to
characterize nose points of power flow equations. The test function possesses
a monotonic property in the neighborhood of nose points that it is positive
on one side of the bifurcation value while it is negative one the other side.
Hence, it offers an effective way to bracket the parameter value during a
search procedure of bifurcation values to guarantee a solution exists inside the
bracket. This test function in conjunction with the set of power flow equations
constitute a set of (n+1)-dimensional characteristic equations for saddle-node
bifurcation points of general nonlinear equations with decoupled parameter
[39]. Distinguishing features of the new set of characteristic equations are
that they are of dimension n+1, instead of 2n+1, for n-dimensional power
flow equations and that the required initial conditions (bifurcations point and
bifurcation value) can be completely provided by the continuation method.
The task of computing Hopf bifurcations points has physical importance
in power system analysis and control. This task, though more involved in com-
putation, receives less attention than the task of computing nose points. For
n-dimensional power flow equations, the standard formulation for the char-
acteristic equations of Hopf bifurcation points is a set of (2n+2)-dimensional
nonlinear equations. A simpler set of characteristic equations for Hopf bi-](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-31-320.jpg)
![18 H.-D. Chiang
furcation points of power flow equations can be developed by exploring a
decoupled parameter-dependent property of power flow equations. The new
set of characteristic equations is of dimension n+2 (even can be of n+1), in-
stead of 2n+2 , for n-dimensional power flow equations. More research work
is required in this task for power system applications.
We next discuss a practical package, CPFLOW (Continuation Power
Flow), a comprehensive tool for tracing power system steady-state behavior
due to parameter variations such as load variations, generation variations and
control variations [15]. CPFLOW simulates a realistic operating condition or
expected future operating conditions relative to parameter variations with
activation of control devices during the process of parameter variations. The
control devices include : (i) switchable shunts and static VAR compensators,
(ii) ULTC transformers, (iii) ULTC phase shifters, (iv)static tap changer and
phase shifters, (v) DC network.
CPFLOW can efficiently generate P-V, Q-V, and P-Q-V curves with the
capability that the controlling parameter λ can be one of the following
• general bus (P and/or Q) loads + real power generation at P-V buses
• area (P and/or Q) loads + real power generation at P-V buses
• system (P and/or Q) loads + real power generation at P-V buses
CPFLOW, computationally based on the continuation method, can trace
the power flow solution curve, with respect to any of the above three vary-
ing parameter, through the “nose” point (i.e. the saddle-node bifurcation
point) or the limit-induced bifurcation point, without any numerical diffi-
culty. CPFLOW can be used in a variety of applications such as (1) to analyze
voltage problems due to load and/or generation variations (e.g. voltage dip,
voltage collapse), (2) to evaluate maximum interchange capability and maxi-
mum transmission capability [31], (3) to simulate power system static behav-
ior due to load and/or generation variations with/without control devices,
and (4) to conduct coordination studies of control devices for steady-state
security assessment.
CPFLOW’s modelling capability is quite comprehensive. The current ver-
sion of CPFLOW can handle power systems up to 43,000 buses. CPFLOW
has been applied to a 40,000-bus power system with a complete set of op-
erational limits and controls. CPFLOW provides three options of parame-
terization schemes including arc-length parameterization. In order to achieve
computational efficiency, CPFLOW employs the tangent method in the first
phase of solution curve tracing and the secant method in the second phase.
However, if the number of corrector iterations becomes too large, then the
predictor switches back to the tangent since it is more accurate than the se-
cant. The Newton method is chosen in CPFLOW as the corrector. CPFLOW
computes the arc-length in the state space, which automatically forces the
predictor to take large steps on the “flat” part of the solution curve and small
steps on the “curly” part.](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-32-320.jpg)
![Bifurcation Analysis to Power Systems 19
6 Performance Indices for Assessing Voltage Collapse
We show in this section how local bifurcation theory can be applied to develop
performance indices for assessing voltage collapse. While continuation power
flow methods can generate P-V and Q-V curves in a reliable manner, they
may be too slow for certain applications such as contingency selection and
contingency analysis, design of preventive control for voltage collapse and on-
line voltage security assessments. To overcome these difficulties, a number of
performance indices intended to measure the severity of the voltage collapse
problem have been proposed in the literature. They can be divided into two
classes: state-space-based approach and the parameter-space-based approach.
The majority of performance indices developed for assessing voltage col-
lapse adopt the state-space-based approach. Among them, the minimum sin-
gular value in [65], the eigenvalue pursued in [37] and the condition number
in [54] of the system Jacobian intend to provide some measure of how far
the system is away from the point at which the system Jacobian becomes
singular. The performance index proposed in [61] and [64] is based on the
angular distance between the current stable equilibrium point and the clos-
est unstable equilibrium point in a Euclidean sense. the performance index
proposed in [25,26] measures the energy distance between the current stable
equilibrium point and the closest unstable equilibrium point using an en-
ergy function. These performance indices can be viewed as providing some
measure of the “distance” between the current operating point and the bi-
furcation point. Note that all these performance indices are defined in the
state space of power system models and they cannot directly answer ques-
tions such as: “Can the system withstand a 100 MVar increase on bus 20
without encountering voltage collapse?”
One basic requirement for useful performance indices is their ability to
reflect the degree of direct mechanism leading the underlying system toward
an undesired state. In the context of voltage collapse in power systems, a
useful performance index must have the ability to measure the amount of
load increase that the system can tolerate before collapse. The state-space-
based performance indices, however, generally do not exhibit any obvious
relation between their value and the amount of the underlying mechanism
that the system can tolerate before collapse.
In order to provide a direct relationship between its value and the amount
of load increases that the system can withstand before collapse, the perfor-
mance index must be developed in the parameter space (i.e., the load/generation
space). Development of performance indices in the parameter space is a rel-
atively new concept which may have been spurred by the local bifurcation
theory. In [16], a new performance index that provides a direct relation-
ship between its value and the amount of load demand that the system can
withstand before a saddle-node voltage collapse was developed. From an an-
alytical viewpoint, this performance index is based on the normal form of
saddle-node bifurcation points. It can be shown that, in the context of power](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-33-320.jpg)
![20 H.-D. Chiang
flow equations, the power flow solution curve passing through the nose point
is, at least locally, a quadratic curve. From a computational viewpoint, this
performance index makes use of the information contained in the power flow
solutions of the particular branch of interest. It only requires two power flow
solutions. The first power flow solution is the current operating point which
can be obtained from a state estimator. Only one additional power flow so-
lution and its derivative are needed to compute this performance index. One
of the features that distinguishes the proposed performance index is its de-
velopment in the load-generation space (i.e. the parameter space) instead of
the state space where the then existing performance indices were developed.
From an application viewpoint, the parameter-space-based performance
indices can be readily interpreted by power system operators to answer ques-
tions such as: “Can the system withstand a simultaneous increase of 70 MW
on bus 2 and 50 MVar on bus 6?”. Moreover, the computation involved in the
performance index is relatively inexpensive in comparison with those required
in the state-space-based ones. A look-ahead performance index intended for
on-line applications was developed in [17]. Given the following information;
(1) the current operating condition, say obtained from the state estimator
and the topological analyzer, (2) the near-term load demand at each bus,
say obtained from short-term load forecaster and predictive data, and (3)
the real power dispatch, say based on economic dispatch, the look-ahead per-
formance index provides a look-ahead load margin measure (in MW and/or
Mvar) which can be used to assess the system’s ability to withstand both
the forecasted load demands and real power variations. In addition, the in-
dex provides useful information as to how to derive effective load-shedding
schemes to avoid voltage instability.
We note that the parameter-space-based performance indices can not take
into account the physical limitations of typical control devices such as gen-
erator VAR limits and ULTC tap ratio limits; such that their computed
load/generation margins may bear some ‘distance’ from the exact margins.
Hence, the function of these performance indices is mainly for ranking the
severity of a list of credible contingencies or for ranking the effectiveness of
different control devices. Exact load/generation margins that accounts for all
control devices and their physical limitations can be accurately calculated by
using the continuation power flow approach. Recent work on the parameter-
space-based performance indices can be found, for example in [33,8,27,29].
7 Persistence of Local Bifurcations under Unmodelled
Dynamics
Many physical systems contain slow and fast dynamics. These slow and fast
dynamics are not easy to model in practice. Even if these dynamics can be
modelled properly, the resulting system model (the original model)is often ill-
conditioned. These difficulties have motivated development of several model](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-34-320.jpg)
![Bifurcation Analysis to Power Systems 21
reduction or simplification approaches to derive reduced models from the
original model. One popular model reduction approach (to derive a reduced
model) is to neglect both the fast and slow dynamics in an appropriate way.
On the other hand, traditional practice in system modelling has been to use
the simplest acceptable model that captures the essence of the phenomenon
under study. A common logic used in this practice is that the effect of a
system component or control device can be neglected when the time scale
of its response is very small or very large compared to the time period of
interest [69,72].
Electric power systems comprise a large number of components interacting
with each other in nonlinear manners. The dynamical response of these com-
ponents extends over a wide range of time scales. For example, the difference
between the time constants of excitation systems (fast control devices) and
that of governors (slow control devices) is a couple orders of magnitudes. The
dynamic behavior after a disturbance occurring on a power system involves
all the system components to varying degrees. The degree of involvement
from each component determines the appropriate system model necessary
for simulating the dynamic behaviors after the disturbance. For instance, an
extended power system dynamical mode l contain both fast variables, such
as the damping flux in the direct and quadrature axis of generators, and
slow variables, such as the field flux and the mechanical torque of genera-
tors. For simulating the dynamic behaviors of a power system after an event
disturbance, the effect of these fast and slow variables can be neglected in
the system modelling because the time scale of these variables is very small
or very large compared to the time period of the disturbance of interest. A
reduced system model is thus obtained from the original system model.
Several questions naturally arise regarding the validity of using the analy-
sis based on a reduced system model to determine the behavior of the original
system. These questions include the relation between the stability properties
of the reduced system and those of the original system, between the tra-
jectories of the reduced system and that of the original system, and so on.
We consider a nonlinear dynamical system with slow and fast un-modelled
dynamics of the form
ẋ = f(x, y, z, λ) + M1f0(x, y, z, λ, M1, M2, M̃1, M̃2)
ẏ = M̃1g(x, y, z, λ, M1, M2, M̃1, M̃2) slow (9)
M2ż = h(x, y, z, λ) + M̃2h0(x, y, z, λ, M1, M2, M̃1, M̃2) fast
where x ∈ Rn
, y ∈ Rm
, z ∈ Rp
, M1, M2, M̃1, M̃2 ∈ R+
, and f, f0, g, h, h0 are Cr
with r ≥ 2.
Associated with system (9), we define a reduced system which treats the
fast variables z as instantaneous variables and the slow variables y as con-](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-35-320.jpg)
![Bifurcation Analysis to Power Systems 23
system (with both fast and slow dynamics) (9) will encounter a saddle-node
bifurcation relative to a varying parameter if the associated reduced system
(10) (which is derived by neglecting both fast and slow dynamics) encounters
a saddle-node bifurcation relative to the varying parameter. A error bound
can be derived between the bifurcation point of the reduced system (10) and
that of the original system (9). Furthermore, it can be shown that the system
behaviors after the saddle-node bifurcation of the reduced system and that
of the original system are close to each other in state space [18]. The general
analytical results can be applied, among others, to justify the usage of simple
power system models for analyzing voltage collapse in electric power systems.
For instance, it provides a justification of the current practice that voltage
collapse can be analyzed based on a simple model of synchronous machines
(the so-called swing equation) rather than on a detailed model which includes
the dynamics of several control devices.
8 Concluding Remarks
We have presented in this chapter an overview on the application of local bi-
furcation analysis and theory to (i) develop models explaining power system
nonlinear behaviors and various power system instabilities such as voltage col-
lapse and low-frequency oscillations, to (ii) develop a powerful global analysis
tool based on continuation methods to trace power system quasi-steady-state
behaviors due to load and generation variations in realistic power system
models, and to (iii) develop performance indices for detecting and estimat-
ing local bifurcations of power systems. Furthermore, an overview on the
extension of saddle-node bifurcation, Hopf bifurcation and limit-induced bi-
furcation to include the analysis of the system dynamics after the bifurcation
has been presented.
Electric power systems comprise a large number of components whose
dynamical response extends over a wide range of time scales. Up to present,
most power system models used for bifurcation analysis involve only short-
term dynamical models (transient stability models). It raises the concern
about the validity of short-term dynamical models, which have disregarded
slow dynamics, for local bifurcation analysis. The effects of un-modelled dy-
namics due to fast and slow state variables on the local bifurcation analysis
of a power system model has been also discussed in this chapter.
During the last two decades, numerical bifurcation analysis of power sys-
tem models has been a subject of great interest to researchers and engineers
in the power system community. These numerical studies seem to favor the
claim that various types of bifurcations can occur in real power systems. Sev-
eral bifurcation-based models have been developed to provide an explanation
for various observed power system nonlinear behaviors and power system in-
stabilities. Furthermore, these numerical studies support the observation that
the complexity of power system dynamic behaviors is related more to the non-](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-37-320.jpg)
![Bifurcation Analysis with Application to
Power Electronics
Chi K. Tse and Octavian Dranga
Department of Electronic and Information Engineering
Hong Kong Polytechnic University
P. R. China
encktse@polyu.edu.hk
Abstract. The problem of sudden loss of stability (more precisely, sudden change
of operating behaviour) is frequently encountered in power electronics. A classic ex-
ample is the current-mode controlled dc/dc converter which suffers from unwanted
subharmonic operations when some parameters are not properly chosen. For this
problem, power electronics engineers have derived an effective solution approach,
known as ramp compensation, which has become the industry standard for current-
mode control of dc/dc converters. In this chapter, the problem is reexamined in
the light of bifurcation analysis. It is shown that such an analysis allows convenient
prediction of stability boundaries and facilitates the selection of parameter values
to guarantee stable operation. It also permits new phenomena to be discovered. An
example is given at the end of the chapter to illustrate how some bizarre operation
in a power-factor-correction (PFC) converter can be systematically explained.
1 Introduction
The term “stability” has a variety of meanings. In the strict mathematical sense,
one may state its meaning in terms of rigorously defined conditions. In engineering,
however, stability is often interpreted as a condition in which the system being
examined is operating in the expected regime. For instance, in power electronics, we
refer a stable operation to a specific periodic operation. When a power converter fails
to maintain its operation in this expected manner, it is considered unstable, even
though it may be operating in a perfectly predictable regime such as a subharmonic
or quasi-periodic regime. In conventional power electronics, all those subharmonic,
quasi-periodic and chaotic operations are regarded as being undesirable and should
be avoided. Thus, the traditional design objective must include the prevention of
any bifurcation within the intended operating range. In other words, any effective
design automatically has to avoid the occurrence of bifurcation for the range of
variation of the parameters [1].
Bifurcations and chaos have been observed and analysed for various kinds of
power electronics circuits [2]–[4]. For systems that have been shown to bifurcate
when a certain parameter is changed, the design problem is, in a sense, addressing
the “control of bifurcation”. Such a design problem can therefore be solved on the
basis of bifurcation analysis. One of our objectives in this chapter is to examine
the traditional stability problem from a bifurcation analysis perspective. We will
study, to some depth, dc/dc converters under current-mode control, which has been
G. Chen, D.J. Hill, and X. Yu (Eds.): Bifurcation Control, LNCIS 293, pp. 29–48, 2003.
Springer-Verlag Berlin Heidelberg](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-43-320.jpg)
![30 C.K. Tse and O. Dranga
1 "
1 "
1 " /
/
1
1
"
"
/.
0-
+
−
) i
"
#
Vin
R Q
S
'
&
/
/
/
11
1
+
−
'
*
+
−
1
1
1
1
1
1
)
*
compensation slope
+
−
+
v
−
C
L
'
i
Vref
S
D
clock
11
1
/
/
/
Iref
Fig.1. Boost converter under
current-mode control
a mature control technique applied in power electronics [5]–[8]. Specifically we will
show that the widely known ramp compensation technique is effectively a means
of controlling bifurcation, although it was never understood as such in the power
electronics community. One useful extension of this work is in the observation of the
subtle effect of controlling bifurcation (ramp compensation) on the overall system
dynamics. As we will show in this chapter, dynamical response can be undesirably
affected if excessive design margin is applied to avoid bifurcation. Hence, in deriving
effective control methods, consideration should be given to maintain adequately
fast response as well as sufficient clearance of bifurcation. In this chapter we will
demonstrate how an “adaptive avoidance of bifurcation” can be achieved by a simple
variable ramp compensation scheme. Furthermore, bifurcation analysis may lead to
discovery of new phenomena, as we will illustrate using the same current-mode
controlled converter but applying it to a power-factor-correction application.
The rest of the chapter is organized as follows. In the next section, we briefly
review the circuit operation of the boost converter under a typical current-mode
control and the instability condition of the inner current loop in terms of period-
doubling bifurcation. In Section 3, the conventional ramp compensation for stabi-
lization of the inner current loop is considered in the light of “avoiding bifurcation”.
Useful design curves will be provided for steady-state design. The effect of the use
of compensating ramp on the dynamics of the overall closed-loop system is then
considered formally, followed by experimental verifications. We also show how one
can make fruitful use of the basic concept of controlling bifurcation to derive a
simple effective control method that can ensure adequate margin from bifurcation
as well as sufficient transient speed. Finally, in Section 4, we examine the same
converter circuit when it is used for a power-factor-correction (PFC) application.
Based on results of our bifurcation analysis, we demonstrate an interesting practical
behaviour of this popular type of PFC converters, to which traditional theory offers
no simple explanation.](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-44-320.jpg)
![Bifurcation Analysis for Power Electronics 31
(a)
3
3
3
3
3
3
3
#
#
#
#
#
#
#
3
3
3
3
3
3
3
#
#
#
#
#
#
#
)
*
slope=
Vin
L
slope=
−(v−Vin)
L
t
Iref
i //
(b)
555555555
5
3
3
3
3
3
3
3
#
#
#
#
#
#
#
555555555
5
3
3
3
3
3
3
3
#
#
#
#
#
#
#
)
*
slope=−mc
slope=
Vin
L
slope=
−(v−Vin)
L
compensating ramp
&
&
&
&
t
Iref
i //
Fig.2. Illustration of
current-mode control
showing inductor cur-
rent (a) without and
(b) with ramp com-
pensation
2 Review of Operation and Bifurcation Analysis
2.1 Basic operation
Consider the boost converter shown in Fig. 1. The switch is turned on periodically,
and off according to the output of a comparator that compares the inductor current
with a reference level Iref . Specifically, while the switch is on, the inductor current
i climbs up, and as it reaches Iref , the switch is turned off, thereby causes the
inductor current to ramp down until the next periodic turn-on instant. Thus, the
average inductor current is programmed approximately by Iref . In the closed-loop
system, Iref is controlled via a feedback loop which attempts to keep the output
voltage fixed by adjusting Iref .
An important feature of the current-mode control is the presence of an inner
current loop. It is now widely known that this inner loop becomes unstable when
the duty ratio (designed steady-state value) exceeds 0.5 [7,8]. The usual practical
(conventional) remedy is to introduce a compensating ramp to the loop, as shown
in Fig. 1. The essential operation is illustrated by the waveforms shown in Fig. 2.](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-45-320.jpg)
![32 C.K. Tse and O. Dranga
2.2 Period-doubling bifurcation
The aforementioned inner-loop instability can in fact be examined from the view-
point of nonlinear dynamics. A handy starting point is the iterative function that
describes the inductor current dynamics.
We begin with the typical period-doubling bifurcation in the boost converter
without ramp compensation [2,4,9]. We let in and in+1 be the inductor current at
t = nT and (n+1)T respectively. Denote also the output voltage (voltage across the
output capacitor) by v. By inspecting the slopes of the inductor current in Fig. 2
(a), we get
Iref − in+1
(1 − D)T
=
v − Vin
L
and
Iref − in
DT
=
Vin
L
(1)
where D is the duty ratio which is defined as the fractional duration of a switch-
ing period when switch S is closed. Combining the above equations, we have the
following iterative function:
in+1 =
E
1 −
v
Vin
L
in +
Iref v
Vin
−
(v − Vin)T
L
(2)
If we are interested in the inner current loop dynamics near the steady state, we
may write
δin+1 =
E
−D
1 − D
L
δin + O(δi2
n) (3)
Clearly, the characteristic multiplier or eignenvalue, λ, is given by
λ =
−D
1 − D
(4)
which must fall between –1 and 1 for stable operation. In particular, the first period-
doubling occurs when λ = −1 which corresponds to D = 0.5. Consistent with what
is well known in power electronics, current-mode controlled converters must operate
with the duty ratio set below 0.5 in order to maintain a stable period-1 operation
[10].
In the application of current-mode control, the error signal derived from the
output voltage is often used to modify Iref directly (not the duty ratio as in the
case of voltage mode PWM control). It is thus helpful to look at the period-doubling
bifurcation in terms of the current reference Iref . Specifically we can express the
“criterion of a bifurcation-free operation”, D < 0.5, in terms of Iref by using the
steady-state equation relating R, D and Iref . For the boost converter, the equivalent
criterion of a bifurcation-free operation is
Iref <
Vin
R
5
DRT
2L
+
1
(1 − D)2
<
D=0.5
= Iref,c (5)
which can be derived from the power-balance equation
E
Iref −
∆i
2
L
Vin =
V 2
in
(1 − D)2R
(6)](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-46-320.jpg)
![40 C.K. Tse and O. Dranga
4 Application Example: Power-Factor-Correction
Boost Converter
In the foregoing, we have described the application of ramp compensation in con-
trolling bifurcation in dc/dc converters under current-mode control. In practice, the
current-mode controlled converter also finds application in shaping the input cur-
rent. In fact, the so-called boost rectifier or power-factor-correction (PFC) converter
is effectively a current-mode controlled boost converter [11]. The circuit schematic
is shown in Fig. 12. In this case, instead of setting Iref constant for a fixed load, we
let Iref vary according to the input voltage waveform. Thus, the input current is
being directly programmed to track the waveform of the input voltage. The result
is a nearly unity power factor.
4.1 Bifurcation analysis
The bifurcation analysis described earlier is directly applicable to the case of the
PFC boost converter. Effectively, since Iref follows the input voltage, its waveform is
a rectified sine wave whose frequency is much lower than the switching frequency.
Typically the frequency of this sine wave is 50 or 60 Hz. Thus, the situation is
analogous to the case of applying a time-varying ramp compensation to a current-
mode controlled boost converter. Suppose the input voltage is given by
vin(t) = V̂in |sin ωmt| (19)
where ωm is the line angular frequency. For algebraic brevity we express the input
voltage in terms of the phase angle θ, i.e., vin(θ) = V̂in |sin θ|.
As shown in Fig. 13, when the input voltage is in its first quarter cycle (i.e.,
0 ≤ θ < π/2), the value of Iref increases, which is equivalent to applying a negative
compensating ramp to Iref (i.e., Mc < 0). Moreover, when the input voltage is
in its second quarter cycle (i.e., π/2 < θ ≤ π), the value of Iref decreases, which
is equivalent to applying a positive compensating ramp to Iref (i.e., Mc > 0). At
θ = π/2, there is no ramp compensation. Therefore, based on the earlier analysis,
we can conclude that the system has asymmetric regions of stability for the two
quarter cycles.
Specifically, the second quarter cycle (i.e., π/2 ≤ θ < π) should be more remote
from period-doubling1
because of the presence of ramp compensation. To be precise,
we need to find the critical phase angle, θc, at which period-doubling occurs. Since
the duty ratio is equal to 1−vin/v and Mc is −(dIref /dt)L/V̂in| sin θ|, we have, from
(10),
|sin θc| =
v + 2L
dIref
dt
2V̂in
. (20)
Moreover, if the power factor approaches one, we have
Iref ≈ ˆ
Iin |sin θ| for 0 ≤ θ ≤ π (21)
1
The term period-doubling here refers to the switching period being doubled.](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-54-320.jpg)
![one of which was covered with a hairy brown integument, and was
rather sluggish in its movements, waddling awkwardly into its
burrow while it held aloft one of its hands in a most ridiculous
fashion. The other was a smaller crab, with a greenish body, and
having one of its pincer-claws, which was of a brilliant orange colour,
of a huge size compared with its fellow. Probably, after the lapse of a
few years, these flats will form part of the general forest land, when
the crabs may undergo further adaptive changes.
We saw little of King George during our stay, as being now advanced
in years he leads a retired life, passing his days in a small room in
the rear of the palace, and only coming out of doors after sunset for
a little airing. However, his grandson, "Wellington Gnu," who is
governor of Nukualofa, and heir presumptive to the throne, was
most civil and obliging. He is a remarkably fine-looking man, being
six feet two inches in height, and stout in proportion; his face beams
with amiability and intelligence; and he possesses all the manners
and bearing of a polished gentleman. Although the lineal heir to the
throne by direct descent, it is very doubtful whether he will succeed
the present king, as Maafu, his cousin, and the son of a deceased
brother of King George, is older in years, and is consequently by the
Tongan laws the legitimate heir to the throne.[3]
[3] Since the above was written I have heard of the death of
Maafu.
Wellington entertained us most hospitably, and drove us in his
buggies to various places of interest in the island. On one occasion
he took three of our officers to Moa, a native town situated near the
south-east extremity of the island. From there they went on to a
place eight miles to the southward, where there is a famous
megalithic structure of unknown origin, which has been described
and figured by Brenchley in his "Voyage of the Curaçoa." As our
experience differs somewhat from Brenchley's, I may be excused for
making a few remarks thereon. The monument—if such it can be
called—consists of three large slabs of coral rock, two of which are
planted vertically in the ground at a distance of about fifteen feet](https://image.slidesharecdn.com/1103188-250522062319-a511581c/85/Bifurcation-Control-Theory-And-Applications-1st-Edition-Guanrong-Chen-63-320.jpg)
Bifurcation Control Theory And Applications 1st Edition Guanrong Chen
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- 6. Lecture Notes in Control and Information Sciences 293 Editors: M. Thoma · M. Morari
- 8. Guanrong Chen ž David J. Hill ž XinghuoYu (Eds.) Bifurcation Control Theory and Applications With 125 Figures 1 3
- 9. Series Advisory Board A. Bensoussan · P. Fleming · M.J. Grimble · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · J.N. Tsitsiklis Editors Prof. Guanrong Chen Prof. David J. Hill City University of Hong Kong Department of Electronic Engineering 83 Tat Chee Avenue, Kowloon Hong Kong SAR P.R. China Prof. Xinghuo Yu Faculty of Engineering Royal Melbourne Institute of Technology GPO Box 2476V Melbourne VIC 3001 Australia ISSN 0170-8643 ISBN 3-540-40341-8 Springer-Verlag Berlin Heidelberg New York Library of Congress Cataloging-in-Publication Data Bifurcation control : theory and application / Guanrong Chen, David J. Hill, Xinghuo Yu (eds.). p. cm. -- (Lecture notes in control and information sciences ; 293) ISBN 3-540-40341-8 (alk. paper) 1. Control theory. 2. Bifurcation theory. I. Chen, G. (Guanrong) II. Hill, David J. (David John), 1949- III. Yu, Xing Huo. IV. Series. QA402.3.B52 2003 515’.64--dc21 2003054382 This work is subject to copyright. All rights are reserved, whether the whole or part of the mate- rial is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2003 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data conversion by the authors. Final processing by PTP-Berlin Protago-TeX-Production GmbH, Berlin Cover-Design: design & production GmbH, Heidelberg Printed on acid-free paper 62/3020Yu - 5 4 3 2 1 0
- 10. Preface Bifurcation control refers to the task of designing a controller that can modify the bifurcating properties of a given nonlinear system, so as to achieve some desirable dynamical behaviors. Typical bifurcation control objectives include delaying the onset of an inherent bifurcation, introducing a new bi- furcation at a preferable parameter value, changing the parameter value of an existing bifurcation point, modifying the shape or type of a bifurcation chain, stabilizing a bifurcated solution or branch, monitoring the multiplicity, amplitude, and/or frequency of some limit cycles emerging from bifurcation, optimizing the system performance near a bifurcation point, or a combination of some of these objectives. Bifurcation control not only is important in its own right, but also suggests an effective strategy for chaos control since bifurcation and chaos are usually “twins”; in particular, period-doubling bifurcation is a typical route to chaos in many nonlinear dynamical systems. Both chaos control and bifurcation control suggest a new technology that promises to have a major impact on many novel, perhaps not-so-traditional, time- and energy-critical engineering applications. In addition to the vast area of chaos control applications, bifurcation control plays a crucial role in special dynamical analysis and crisis control of many complex nonlinear sys- tems. The best known examples include high-performance circuits and devices (e.g., delta-sigma modulators and power converters), oscillation generation, vibration-based material mixing, chemical reactions, power systems collapse prediction and prevention, oscillators design and testing, biological systems modelling and analysis (e.g., the brain and the heart), and crisis management (e.g., jet-engine serge and stall), to name just a few. In fact, this new and challenging research and development area has become an attractive scien- tific inter-discipline involving control and systems engineers, theoretical and experimental physicists, applied mathematicians, and biomedical engineers alike. There are many practical reasons for controlling various bifurcations. In a system where a bifurcating response is harmful and dangerous, it should be significantly reduced or completely suppressed. This task includes, for exam- ple, avoiding voltage collapse and oscillations in power networks, eliminating deadly cardiac arrhythmias, guiding disordered circuit arrays (e.g., multi- coupled oscillators and cellular neural networks) to reach a certain level of desirable pattern formation, regulating dynamical responses of some mechan- ical and electronic devices (e.g., diodes, laser machines, and machine tools), removing undesirable vibrations, and so on. Bifurcation can also be useful and beneficial for some special applications, and it is interesting to see that there has been growing interest in utilizing the very nature of bifurcation, particularly in some engineering applications
- 11. VI Preface involving oscillations analysis and utilization. A prominent feature of bifurca- tion is its close relation with various vibrations (periodic oscillations or limit cycles), which sometimes are not only desirable but may actually be neces- sary. Mechanical vibrations and some material and liquid mixing processes are good examples in which bifurcations (and chaos) are very desirable. In biological systems, bifurcation control seems to be an essential mechanism employed by the human heart in carrying out some of its tasks particularly on atrial fibrillation. Some medical evidence lends support to the idea that control of certain bifurcating cardiac arrhythmias may contribute to the new design of a safer and more effective intelligent pacemaker. A further idea, suggested as useful in power systems, is to use the onset of a small oscillation as an indicator for proximity to collapse. In control systems engineering, the deliberate use of nonlinear oscillations has been applied effectively for system identification. Motivated by many potential real-world applications, current research on bifurcation control has proliferated in recent years, along with the promising progress of chaos control. With respect to theoretical considerations, bifurca- tion control poses a substantial challenge to both system analysts and control engineers. This is due to the extreme complexity and sensitivity of bifurcating dynamics, which intrinsically is associated with the reduction in long-term predictability and short-term controllability of chaotic systems in general. Notwithstanding many technical obstacles, both theoretical and practi- cal developments in this area have experienced remarkable progress in the last decade. It is now known that bifurcations can be controlled via var- ious methods. Some representative approaches employ linear or nonlinear state-feedback controls, perhaps with time-delayed feedback, apply a washout filter-aided dynamic feedback controller, use harmonic balance approxima- tions, and utilize quadratic invariants in normal forms. Surprisingly, however, there exist no control-theory-oriented books written by control engineers for control engineers available in the market that are devoted to the subject of Bifurcation Control. In particular, there has been no exposure of these very active research topics in the Lecture Notes Series in Control and Information Science. This edited book, therefore, aims at filling in the gap and present- ing current achievements in this challenging field at the forefront of research, with emphasis on the engineering perspectives, methodologies, and poten- tial applications of bifurcation controls. It is intended as a combination of overview, tutorial and technical reports, reflecting state-of-the-art research of significant problems in this field. The anticipated readership includes uni- versity professors, graduate students, laboratory researchers and industrial practitioners, as well as applied mathematicians and physicists in the areas of electrical, mechanical, physical, chemical, and biomedical engineering and sciences. We received enthusiastic assistance from several individuals in the prepa- ration of this book. In particular, we are very grateful to Noel Patson, who
- 12. Preface VII helped a great deal in taking care of many painful editorial tasks. We would also like to thank Prof. T. Thoma and Dr. T. Ditzinger, Editors of Springer- Verlag, for their continued support and kind cooperation. Finally, we wish to express our sincere thanks to all the authors whose significant scientific contributions have directly led to the publication of this timely treatise. Guanrong Chen, David J. Hill, Xinghuo Yu Hong Kong and Melbourne, January, 2003
- 13. Contents Bifurcation Control Preface .!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.! V Bifurcations in Control, Power, Electronics and Mechanical Systems Application of Bifurcation Analysis to Power Systems .! . !.! . !.! . !.! . ! 1 Hsiao-Dong Chiang Bifurcation Analysis with Application to Power Electronics .!.!.! 29 Chi K. Tse, Octavian Dranga Distance to Bifurcation in Multidimensional Parameter Space: Margin Sensitivity and Closest Bifurcations. !.! . !.! . !.! . !.! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! 49 Ian Dobson Static Bifurcation in Mechanical Control Systems . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! 67 Harry G. Kwatny, Bor-Chin Chang, Shiu-Ping Wang Bifurcation and Chaos in Simple Nonlinear Feedback Control Systems . . . . . . . . . . . . . . . . . . . . . . . . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! 83 Wallace K. S. Tang Bifurcation Dynamics in Control Systems.! . !.! . !.! . !.! . !.! . !.! . !.! . !.! . !.! . !.! . !.! 99 Pei Yu Controlling Bifurcations and Bifurcation Control Analysis and Control of Limit Cycle Bifurcations .!.!.!.!.!.!.!.!.!.!.!.!.!127 Michele Basso, Roberto Genesio Global Control of Complex Power Systems .! . !.! . !.! . !.! . !.! . !.! . !.! . !.! . !.! . !.!155 David J. Hill, Yi Guo, Mats Larsson, Youyi Wang Preserving Transients on Unstable Chaotic Attractors .!.!.!.!.!.!.!.! 189 Tomasz Kapitaniak, Krzysztof Czolczynski Bifurcation Control in Feedback Systems .!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!205 Diego M. Alonso, Daniel W. Berns, Eduardo E. Paolini, Jorge L. Moiola
- 14. X Contents Emerging Directions in Bifurcation Control . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . ! . !229 Hua O. Wang, Dong S. Chen Applications Bifurcation Analysis for Control Systems Applications .!.!.!.!.!.!.!.!249 Mario di Bernardo Feedback Control of a Nonlinear Dual–Oscillator Heartbeat Model .!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.! 265 Michael E. Brandt, Guanyu Wang, Hue-Teh Shih Local Robustness of Bifurcation Stabilization with Application to Jet Engine Control .!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!.!275 Xiang Chen, Ali Tahmasebi, Guoxiang Gu Bifurcations and Chaos in Turbo Decoding Algorithms .! . !.! . !.! . !.!301 Zarko Tasev, Petar Popovski, Gian Mario Maggio, Ljupco Kocarev
- 15. Application of Bifurcation Analysis to Power Systems Hsiao-Dong Chiang School of Electrical and Computer Engineering Cornell University, Ithaca, NY 14853, USA chiang@ee.cornell.edu Abstract. Electric power systems are physically some of the largest and most complex nonlinear systems in the world. Bifurcations are rather mundane phenom- ena in power systems. The pioneer work on the local bifurcation analysis of power systems can be dated back to the 1970’s and earlier. Within the last 20 years or so nonlinear dynamical theory has become a subject of great interest to researchers and engineers in the power system community. Powerful computational tools for bifurcation analysis have been applied during this period to study important non- linear problems arising in power systems, and in some cases, to relate this study to observed nonlinear phenomena in power systems. In this chapter, we will present an overview on the application of local bifurcation analysis and theory to (i) develop models explaining power system nonlinear behaviors and various power system in- stabilities such as voltage collapse and low-frequency oscillations, to (ii) develop a powerful global analysis tool based on continuation methods to trace power sys- tem quasi-steady-state behaviors due to load and generation variations in realistic power system models, and to (iii) develop performance indices for detecting and estimating local bifurcations of power systems. An overview on the extension of saddle-node bifurcation, Hopf bifurcation and limit-induced bifurcation to include the analysis of the system dynamics after the bifurcation is presented. In addi- tion, the effects of un-modelled dynamics due to fast and slow variables on local bifurcations is presented. 1 Introduction Electric power systems are comprised of a large number of components in- teracting with each other, exhibiting nonlinear dynamic behaviors with a wide range of time scales. Physically, an electric power system is an inter- connected system composed of generating stations (which convert fuel en- ergy into electricity), primary and secondary distribution substations (that distribute power to loads (consumers), and transmission lines, i.e., transmis- sion network, that tie the generating stations and distribution substations together. The fundamental function of power systems is meeting customer load demands in a reliable and economical manner. To this end, various types of control devices, local and centralized, and protection systems are placed throughout the system. The local control devices attached to gener- ating plants, such as excitation control system and turbine control system, G. Chen, D.J. Hill, and X. Yu (Eds.): Bifurcation Control, LNCIS 293, pp. 1–28, 2003. Springer-Verlag Berlin Heidelberg
- 16. 2 H.-D. Chiang are automatic and relatively high speed. On the other hand, the local con- trol devices, such as ULTC transformers, ULTC phase-shifters, synchronous var compensator (SVC), shunt capacitor (SC) installed in the transmission network are relatively low speed. Electric power systems are physically some of the largest and most com- plex nonlinear systems in the world. Their nonlinear behaviors are difficult to analyze and predict due to several factors such as (i) the extraordinary size of the systems, (ii) the nonlinearity in the components and control de- vices in the systems, (iii) the dynamical interactions within the systems, (iv) the uncertainty in the load behaviors, (iv) the complexity and different time- scale of power system components (equipments and control devices). These complicating factors have forced power system engineers to analyze power systems through extensive computer simulations. Large-scale computer sim- ulation programs are widely used in power utilities for studying power sys- tem steady-state behaviors and dynamic responses relative to disturbances. By nature, a power system continually experiences disturbances. These may be classified under two main categories: event disturbances and load dis- turbances. Event disturbances, i.e., contingencies, include loss of generating units or transmission components (lines, transformers, substations) due to short-circuits caused by lightning, high winds, failures such as incorrect relay operations or insulation breakdown, sudden large load changes, or a combina- tion of such events. Event disturbances usually lead to a change in the network configuration of the power system du e to actions from protective relays and circuit breakers. Load disturbances, small or large, on the other hand, in- clude the variations in load demands (e.g. the daily load cycle), termed load variations, the rescheduling of real power generations, the scheduled power transfers across the transmission network between two regions or two areas in the interconnected system, or a combination of the above three types of load disturbances. The network configuration usually remains unchanged af- ter load disturbances. Power systems are planned and operated to withstand the occurrence of certain credible disturbances. A major activity in utility system planning and operations is to examine the impact of a set of credible disturbances on power system dynamical behaviors such as stability and to develop counter-measures. A power system subject to load disturbances can be modelled as a set of parameter-dependent nonlinear differential and algebraic equations with parameter variation. Power systems are normally operated near a stable equi- librium point. When the system load parameters are away from their bifur- cation values and their variations are occurring continuously but slowly, it is very likely that • the stable equilibrium point of the underlying power system changes po- sition but remains a stable equilibrium point, and • the old stable equilibrium point lies inside the stability region of the new stable equilibrium point.
- 17. Bifurcation Analysis to Power Systems 3 Consequently, the power system dynamics starting from the old stable equi- librium point will converge to the new stable equilibrium point and will make the system state track its new stable equilibrium point, whose position is changed continuously but slowly, and yet the system remains stable under this load disturbance. The typical ways in which a study power system may lose stability, under the influence of load variations are through the following: • the stable equilibrium point and another equilibrium point coalesce and disappear in a saddle-node bifurcation as parameter varies, or • the stable equilibrium point and another equilibrium point coalesce and exchange stability in a limit-induced bifurcation (a type of transcritical bifurcation) as parameter varies, • the stable equilibrium point and an unstable limit cycle coalesce and disappear and an unstable equilibrium point emerges in a subcritical Hopf bifurcation as parameter varies, • the stable equilibrium point bifurcates into an unstable equilibrium point surrounded by a stable limit cycle in a supercritical Hopf bifurcation as parameter varies. It is now well recognized that bifurcations are rather mundane phenomena that can occur in many physical and man-made systems where nonlinearity is present. The pioneer work on the local bifurcation analysis of power systems can be dated back to the 1970’s and earlier [62,63,41]. Within the last 20 years or so nonlinear dynamical theory has become a subject of great inter- est to researchers and engineers in the power system community. Powerful computational tools for bifurcation analysis have been applied during this period to study important nonlinear problems arising in power systems, and in some cases, to relate this study to observed nonlinear phenomena in power systems [45,46,59,53]. In addition, some counter-measures to avoid bifurca- tions have been developed to design control schemes for prevention of power system instabilities [23,24,52,73,34,35]. From the engineering viewpoint, one important task in performing bifur- cation analysis to nonlinear systems, such as electric power systems, is the analysis of both the mechanism leading to disappearance of stable equilib- rium points due to a bifurcation and the system dynamical behaviors after the bifurcation. After a bifurcation occurs, the system state will evolve ac- cording to the system dynamics. The dynamics after bifurcation determine whether the system remains stable or become unstable; and what is the type of system instability. Local bifurcation theory does not describe the dynam- ical behaviors after a bifurcation. We will review some work on the analysis of the system dynamics after typical local bifurcations in this chapter. Electric power systems comprise a large number of components interact- ing with each other in nonlinear manners. The dynamical response of these components extends over a wide range of time scales. The different time- scale components of power systems all have their corresponding influences on power system dynamical responses. It has become convenient to divide
- 18. 4 H.-D. Chiang the time span of dynamic response simulations into short-term (transient), mid-term and long-term, covering the post-disturbance times of up to a few seconds, 5 minutes and 20 minutes or so, respectively. Up to present, most power system models used for bifurcation analysis involve only short-term dynamical models (transient stability models). It raises the concern about the validity of short-term dynamical models, which have disregarded slow dynamics, for local bifurcation analysis. The effects of un-modelled dynamics on the local bifurcation analysis of a power system model is also discussed in this chapter. P-V and Q-V curves have been widely used by power system analysis engineers to study voltage stability [60]. These curves represent one impor- tant aspect of the saddle-node bifurcation occurring in power systems due to variations of loads and generations. While global analysis tools based on continuation methods developed in the last decade can generate P-V and Q-V curves in a reliable manner, these tools may be too slow for certain power system on-line applications. To overcome this difficulty, a number of performance indices intended to measure the severity of the voltage stability problem have been proposed in the literature. We will examine several exist- ing performance indices and discuss a performance index which has rendered practical applications. This performance index is based on the normal form theory of saddle-no de bifurcation point. In this chapter, we will present an overview on the application of local bi- furcation analysis and theory to (i) develop models explaining power system nonlinear behaviors and various power system instabilities such as voltage col- lapse and low-frequency oscillations, to (ii) develop a powerful global analysis tool based on continuation methods to trace power system quasi-steady-state behaviors due to load and generation variations in realistic power system models, and to (iii) develop powerful computational tools for detecting and estimating local bifurcations of power systems. An overview on the extension of saddle-node bifurcation, Hopf bifurcation and limit-induced bifurcation to include the analysis of the system dynamics after the bifurcation will be pre- sented. In addition, the effects of un-modelled dynamics due to fast and slow variables on local bifurcations will be analyzed. 2 Local Bifurcations and Power System Behaviors Recently, local bifurcation theory has being applied to interpret observed non- linear dynamical behaviors in power systems. In some cases, local bifurcation theory has been extended to include the analysis of dynamics after a local bifurcation. A comprehensive bifurcation and chaos analysis of a 3-bus power system was carried out in [11]. Numerical bifurcation analysis of a simplified model of a 9-bus power system and a 39-bus power system were conducted in [12]. Other numerical bifurcation analysis of simple power systems can be found in, for example, [1,4]. It has been found that the bifurcation phenomena
- 19. Bifurcation Analysis to Power Systems 5 observed in the 3-bus and 9-bus power system are similar. These bifurcation phenomena have been observed in the 39-bus power system as well, includ- ing Hopf, period-doubling, and cyclic fold bifurcations. Furthermore, some bifurcation phenomena not appearing in the 3-bus and 9-bus systems have surfaced in the 39-bus system. These numerical studies favor the claim that various types of bifurcations can occur in real power systems. Local bifurcation theory has been applied to provide an explanation for various observed power system nonlinear behaviors and power system insta- bilities such as voltage collapse and low-frequency electro-mechanical oscil- lations that occur in electric power networks. Abed and Varaiya [2] were probably the first to suggest a possible role for Hopf bifurcations in explain- ing the low-frequency electro-mechanical oscillation phenomena. Later Chen and Varaiya [10] numerically demonstrate that degenerate Hopf bifurcation can occur in a simple power system model. In [20], Dobson and Chiang inves- tigated a generic mechanism leading to disappearance of stable equilibrium points due to a saddle-node bifurcation and the subsequent system dynam- ics after the bifurcation for one-parameter dynamical systems. A saddle-node voltage instability model to analyze the process of voltage instability was then proposed to explain voltage stability/instability due to slow load variations in three stages. Iravani and Semlyen investigate the transition of growing torsional oscillations into limit cycles occurring in torsional dynamics based on Hopf bifurcation [28]. Their studies indicate that the range of instability (growing oscillations) based on Hopf bifurcation is noticeably narrower than the one predicted by an eigen-analysis method. Another dynamic phenomenon of concern in power systems that can be explained via Hopf bifurcation is subsynchronous resonance (SSR). This is a condition where the electric network exchanges energy with a turbine genera- tor at one or more of the natural frequencies of the combined system below the synchronous frequency of the system. The IEEE SSR working group suggests that if the locus of a particular eigenvalue approaches or crosses the imaginary axis, then a critical condition is identified that will require the application of one or more SSR counter measures. The critical condition is closely re- lated to the Hopf condition. To provide high-quality electricity, utilities must endeavor to minimize the effects of large fluctuating loads associated with large motors and furnaces on the transmission network. One of the effects, characterized by visible fluctuations in other customers’ electric lighting, is called voltage flicker. This phenomenon is often categorized as cyclic and non-cyclic. From a dynamic system viewpoint, cyclic flickers may relate to limit cycles or quasi-periodic motion in power systems and non-cyclic flickers to quasi-periodic or chaotic trajectories. Another advance of applying bifurcation analysis to power systems can be manifested in the development of several bifurcation-based models to explain several instances of recent power system voltage instability and/or collapse. This kind of blackout has occurred in several countries such as Belgium,
- 20. 6 H.-D. Chiang Canada, France, Japan, Sweden and the United States [49,60]. Voltage in- stability and/or collapse, a frequent concern on modern power systems, are generally caused by either of two types of system disturbances: load distur- bances or contingencies, i.e., event disturbances. Among several examples of voltage collapse, the 1987 occurrence in Japan [43] was due to large load vari- ations, while the collapse in Sweden in 1982 [71] was caused by a contingency. The dynamic process of voltage instability or collapse usually starts with a power system weakened by a contingency due to a transmission line or gen- erator outage, or by an unusually high peak load (a high load variation), or by a combination of such events. The system may be further weakened due to an inappropriate transmission under-load tap-changer (ULTC) setting, or insufficient reactive power supports, or load restorations that have been temporary reduced because of low voltage. Three bifurcation-based voltage collapse models will be discussed in some details in this chapter. 3 Local Bifurcations in Power Systems A power system model relative to a disturbance comprises a set of first-order differential equations and a set of algebraic equations ẋ = f(x, y, u, λ) (1) 0 = g(x, y, λ) where λ ∈ R1 is a parameter, x is a dynamic state variable and y a static “state” variable, such as the load variables of voltage magnitude and angle. The vector field depends on the value of parameter and will change its dimen- sion accordingly. It describes the internal dynamics of devices such as gener- ators, their associated control systems, certain loads, and other dynamically modelled components. The set of algebraic equations describe the electrical transmission system (the interconnections between the dynamic devices) and internal static behaviors of passive devices (such as static loads, shunt ca- pacitors, fixed transformers and phase shifters). The differential equations (1) can describe as broad a range of behaviors as the dynamics of the speed and angle of generator rotors, flux behaviors in generators, the response of generator control systems such as excitation systems, voltage regulators, tur- bines, governors and boilers, the dynamics of equipments such as synchronous VAR compensators (SVCs), DC lines and their control systems, and the dy- namics of dynamically modelled loads such as induction motors. The state variables x typically include generator rotor angles, generator velocity de- viations (speeds), mechanical powers, field voltages, power system stabilizer signals, various control system internal variables, and voltages and angles at load buses (if dynamical load models are employed at these buses). The forcing functions u acting on the differential equations are terminal voltage magnitudes, generator electrical powers, and signals from boilers and au- tomatic generation control systems. Some control system internal variables
- 21. Bifurcation Analysis to Power Systems 7 have upper bounds on their values due to their physical saturation effects. Let z be the vector of these constrained state variables; then the saturation effects can be expressed as 0 ≤ z(t) ≤ ẑ (2) We term the above model as a set of parameter-dependent differential and algebraic equations with hard constraints. A detailed description of equations (1) - (2) for each component can be found, for example, in [36,42,55]. For a 500-generator power system, the number of differential equations can easily reach as many as 10,000. From a nonlinear dynamical system viewpoint, (1-2) is an one-parameter dynamical system while, in power system applications, it can represent a power system that operates with one of the following conditions: 1. the real (or reactive) power demand at one load bus varies while the others remain fixed, 2. both the real and reactive power demand at a load bus vary and the variations can be parameterized. Again the others remain fixed, 3. the real and/or reactive power demand at some collection of load buses varies and the variations can be parameterized while the others are fixed, 4. the real power transfer at one transmission corridor (e.g. interface transfer and import/export) varies while the others remain fixed, 5. the real power transfer at some collection of transmission corridors (e.g. interface transfer and import/export) varies while the others remain fixed. The only typical ways in which a power system may lose stability (un- der the influence of one parameter variation) are through the saddle-node bifurcation, or limit-induced bifurcations or the Hopf bifurcation. In [21], it has been shown that for generic one-parameter dynamical systems, before a saddle-node bifurcation the equilibrium point x1(λ) is type-one. By type-one, we mean that the corresponding Jacobian matrix has exactly one eigenvalue with a positive real part and the rest of the eigenvalues have negative real parts. Furthermore, x1(λ) lies on the stability boundary of xs(λ). The Ja- cobian matrix, when evaluated at xs(λ), has all of its eigenvalues with only negative real parts and among them, one of the eigenvalues is close to zero. At the bifurcation occurring at say, the bifurcation value λ = λ∗ , equilibrium points xs(λ) and x1(λ) coalesce to form an equilibrium point x∗ (= xs(λ∗ ) = x1(λ∗ )). The Jacobian matrix evaluated at x∗ has one zero eigenvalue and the real parts of all t he other eigenvalues are negative. If the parameter λ increases beyond the bifurcation value λ∗ , then x∗ disappears and there are no other equilibrium points nearby. Another local bifurcation peculiar to power systems is the so-called limit- induced bifurcation. Physically, the generation reactive power capability is limited. The reactive power capability of a generator can reach a limit due to the excitation current limit or the stator thermal limit. Power systems
- 22. 8 H.-D. Chiang are vulnerable to voltage collapse when generation reactive power limits are reached. Given a load/generation variation pattern, the effect of reaching a generator reactive power limit is to immediately change the system equation. From a static analysis viewpoint the generator whose reactive power limit is reached may be simply modelled by replacing the equation describing a P-V bus by the equation describing a P-Q bus. In [21], numerical examples and general arguments were developed to show that a sufficiently heavily loaded but stable power systems can become immediately unstable via a transcritical bifurcation when a reactive power limit is encountered. We term this type of bifurcation as limit-induced bifurcation. We note that when a transcritical bifurcation occurs at say, the bifurcation value λ = λ∗ , the stable equilibrium point xs(λ) and a type-one unstable equilibrium point xu(λ) coalesce to form an equilibrium point x∗ (= xs(λ∗ )). The Jacobian matrix evaluated at x∗ has a single, simple eigenvalue and the real parts of all the other eigenvalues are negative. After the bifurcation, the two equilibrium points change stability to become a type-one equilibrium point and a stable equilibrium point. Hopf bifurcation can occur on generic one-parameter dynamical systems. Before a subcritical Hopf bifurcation, the unstable limit cycle xl 1(λ, t) lies on the stability boundary of xs(λ). The Jacobian matrix, when evaluated at xs(λ), has all of its eigenvalues with only negative real parts and among them, a pair of complex eigenvalues are close to zero. At the bifurcation occurring at say, the bifurcation value λ = λ∗ , the equilibrium point xs(λ) and the unstable limit cycle xl 1(λ, t) coalesce to form an equilibrium point x∗ (= xs(λ∗ ) = xl 1(λ∗ )). The Jacobian matrix evaluated at x∗ has two zero eigenvalues and the real parts of all the other eigenvalues are negative. If the parameter λ increases beyond the bifurcation value λ∗ , then x∗ becomes a type-two equilibrium point and there are no other equilibrium points or limit cycles nearby. As for the supercritical Hopf bifurcation, we make the following remarks. Before the bifurcation. The Jacobian matrix, when evaluated at xs(λ), has all of its eigenvalues with only negative real parts and among them, a pair of complex eigenvalues are close to zero. A t the bifurcation occurring at say, the bifurcation value λ = λ∗ , the equilibrium point xs(λ) is an equilibrium point x∗ (= xs(λ∗ ). The Jacobian matrix evaluated at x∗ has two zero eigenvalues and the real parts of all the other eigenvalues are negative. If the parameter λ increases beyond the bifurcation value λ∗ , then x∗ becomes a unstable equilibrium point x∗ (λ) surrounded by a stable limit cycle. 4 Dynamics after Local Bifurcations Local bifurcation theory has been developed to describe mechanisms leading to disappearance of stable equilibrium points due to a local bifurcation. Re- cently, local bifurcation theory has being extended to include the analysis of system dynamics after a local bifurcation. The knowledge of subsequent sys-
- 23. Bifurcation Analysis to Power Systems 9 tem dynamics is essential to determine whether the system remains stable or becomes unstable, and the type of system instability if the system is deemed unstable. This section briefly summarizes the model for analyzing the system dynamics after a saddle-node bifurcation. Models for analyzing the system dynamics after the limit-induced bifurcation and Hopf bifurcations are also described. 4.1 Saddle-node bifurcation In [20], Dobson and Chiang investigated a generic mechanism leading to disappearance of stable equilibrium points due to a saddle-node bifurcation and the subsequent system dynamics for one-parameter dynamical systems. When a saddle-node bifurcation occurs at say, the bifurcation value λ = λ∗ , equilibrium points xs(λ) and x1(λ) coalesce to form an equilibrium point x∗ (= xs(λ∗ ) = x1(λ∗ )). The Jacobian matrix evaluated at x∗ has one zero eigenvalue and the real parts of all the other eigenvalues are negative. The eigenvector p that corresponds to the zero eigenvalue points in the direction along which the two vectors xs(λ) and x1(λ) approached each other. There is a curve made up of system trajectories which is tangent to eigenvector p at x∗ . This curve is called the center manifold of x∗ and is the union of a system trajectory Wc − converging to x∗ , the equilibrium point x∗ and a system trajectory Wc + diverging from x∗ . Next, we consider the case that λ remains fixed at bifurcation value λ∗ . When the system trajectory is near Wc + at the moment that the bifurcation occurs and if λ remains fixed at its bifurcation value λ∗ , then the system trajectory after the bifurcation moves near Wc +. The system dynamics due to the bifurcation are then determined by the position of Wc + in state space. If Wc + is positioned so that some of the voltage magnitudes decrease along Wc +, then we associate the movement along Wc + with voltage collapse. This is the center manifold voltage collapse model due to saddle-node bifurcation. This model has two advantages from a computational point of view. 1. Since p is tangent to Wc + at x∗ , the initial direction of Wc + near x∗ is determined by p which can be computed f rom the Jacobian matrix at x∗ . 2. Since Wc + is a system trajectory, the dynamics of voltage collapse can be predicted by integrating system equations (1) starting on Wc + near x∗ . 4.2 Limit-induced bifurcation In [21], Dobson and Lu studied a mechanism leading to disappearance of sta- ble equilibrium points due to a limit-induced bifurcation and the subsequent system dynamics. Before the bifurcation, the system is operated around the stable equilibrium point xs(λ). At the limit-induced bifurcation, the stable
- 24. 10 H.-D. Chiang equilibrium point xs(λ) and a type-one unstable equilibrium point xu(λ) co- alesce to form an equilibrium point. The Jacobian matrix evaluated at the equilibrium point has a single, simple eigenvalue and the real parts of all the other eigenvalues are negative. The operating point becomes immediately un- stable when the limit is reached and the system state will move away from the type-one unstable equilibrium point which has a one-dimensional unsta- ble manifold, say Wu . Geometrically speaking, the unstable manifold Wu is tangent at the type-one equilibrium point to the system eigenvector associ- ated with the positive eigenvalue. After the bifurcation, the system state will move along the unstable manifold Wu . It may converge to the near-by stable equilibrium point or diverge along the unstable manifold Wu . 4.3 Hopf bifurcation A mechanism leading to disappearance of stable equilibrium points due to Hopf bifurcation and the subsequent system dynamics is presented below. When a Subcritical Hopf bifurcation occurs at say, the bifurcation value λ = λ∗ , the stable equilibrium point xs(λ) and a unstable limit cycle xl 1(λ, t) coalesce to form an equilibrium point x∗ (= xs(λ∗ )). The Jacobian matrix evaluated at x∗ has two zero eigenvalues and the real parts of all the other eigenvalues are negative. The subspace spanned by the two eigenvectors p1 and p2 that correspond to the two zero eigenvalues points in the direction along which xs(λ) and xl 1(λ, t) approached each other. The subspace is the center manifold of x∗ , say Wc . Next, we consider the case that λ remains fixed at bifurcation value λ∗ . Recall that before the bifurcation occurs, the system state is tracking its stable equilibrium point. Therefore, at the moment the bifurcation occurs, the system state is in a neighborhood of x∗ . Hence, if the system trajectory is near Wc at the moment that the bifurcation occurs and if λ remains fixed at its bifurcation value λ∗ , then the system trajectory after the bifurcation moves along Wc . The system dynamics due to the bifurcation are then determined by the position of Wc in state space. If Wc is positioned so that some of the voltage magnitudes decrease along Wc , then we associate the movement along Wc with voltage collapse. This is the center manifold voltage collapse model due to subcritical Hopf bifurcation. When a Supercritical Hopf bifurcation occurs at say, the bifurcation value λ = λ∗ , the equilibrium point xs(λ) is an equilibrium point x∗ (= xs(λ∗ )). The Jacobian matrix evaluated at x∗ has two zero eigenvalues and the real parts of all the other eigenvalues are negative. If the parameter λ increases be- yond the bifurcation value λ∗ , then x∗ becomes a unstable equilibrium point x∗ (λ) surrounded by a stable limit cycle, say xl 2(λ, t). Next, we consider the case that λ changes slowly after the bifurcation value λ∗ and that at the moment the bifurcation occurs, the system state is in a neighborhood of x∗ . Note that i t is very likely that the system state will lie inside the stability re- gion of the stable limit cycle xl 2(λ, t), making the system trajectory attracted to xl 2(λ, t). Thus, the system dynamics after a supercritical Hopf bifurcation
- 25. Bifurcation Analysis to Power Systems 11 moves along the stable limit cycle xl 2(λ, t) until another bifurcation occurs on xl 2(λ, t) at another bifurcation value. So, the system dynamics due to the bi- furcation are then determined by the position of the stable limit cycle xl 2(λ, t) in state space. If xl 2(λ, t) is positioned so that some of the voltage magnitudes oscillate along the stable limit cycle xl 2(λ, t) and become unacceptable, then we associate the movement with voltage instability. This is the limit cycle voltage instability model due to the supercritical Hopf bifurcation. In summary, after a local bifurcation occurs, the system state will evolve according to the system dynamics as described above. The dynamics after bi- furcation determine whether the system remains stable or becomes unstable, and the type of system instability. The analysis of a typical local bifurca- tion of a stable equilibrium!point!in a power system with slowly varying parameters has two parts: 1. Before the bifurcation when the quasi-static model applies 2. After the bifurcation when the corresponding dynamic model applies We point out that the quasi-static model is not applicable after the bifurca- tion and cannot be used to explain the dynamical behaviors after the bifur- cation. 4.4 Models for voltage collapse Voltage collapse is characterized by a slow variation in the system oper- ating point in such a way that voltage magnitudes at load buses gradually decrease until a sharp, accelerated change occurs. In this section, we present two bifurcation-based models to explain voltage collapse in power systems due to slow load variations. It will be then shown that one of the two models encompasses several existing models for explaining voltage collapse. Recall that “slow load variations” means the dynamics of load variations are rela- tively slower than the dynamics occurring in the state vector. 4.5 SAD voltage collapse model Stage 1: the system is in quasi-steady state and is tracking a stable equilibrium point. Stage 2: the system reaches its “steady-state” stability limit when t he stable equilibrium point undergoes a saddle-node bifurcation or a limit-induced bifurcation. Stage 3: depending on the type of bifurcation encountered in Stage 2, the system dynamics after bifurcation are captured either by the cen- ter manifold trajectory of the saddle-node bifurcation point or by the unstable manifold of the limit-induced bifurcation point. Stage 1 is related to the feasibility of “power flow” solutions (i.e., the existence of a system operating point in a feasible region). Stage 2 deter- mines the steady-state stability limit based on the saddle-node bifurcation
- 26. 12 H.-D. Chiang point or the limit-induced bifurcation point. Stage 3 describes the system dynamical behavior after bifurcation to assess whether the system, after bi- furcation, remains stable or becomes unstable; and determines the types of system instability (voltage collapse and/or angle instability). Hence, the volt- age collapse model describes both the static aspect (stages 1 and 2) and the dynamic aspect (stage 3) of the problem. 4.6 Hopf voltage collapse model Stage 1: the system is in quasi-steady state and is tracking a stable equilibrium point. Stage 2: the system reaches its steady-state stability limit when the stable equilibrium point undergoes a subcritical Hopf bifurcation. Stage 3: the system dynamics after bifurcation are captured by a two- dimensional center manifold. The Hopf voltage collapse model also describes both the static aspect (stages 1 and 2) and the dynamic aspect (stage 3) of the problem. This model dic- tates that mathematically speaking, the steady-state stability limit may be determined by the subcritical Hopf bifurcation, instead of the Saddle-node bifurcation. One implication is that the load margins will be less that one might expect if the nose point was taken as the point of voltage collapse. Since detecting Hopf bifurcation requires the knowledge of the eigenvalues of the system Jacobian, the traditional repetitive power flow approach cannot detect Hopf bifurcation. The saddle-node bifurcation has been a widely accepted model for voltage collapse analysis. Most computational tools developed so far have been con- centrated on the identification of saddle-node bifurcation point, also termed point of collapse [6]. The SAD voltage collapse model includes the saddle- node bifurcation point as a point of collapse. In fact, it can be shown that the SAD voltage collapse model encompasses many existing models used to explain voltage collapse such as the multiple power flow model, the power flow feasibility model, the static bifurcation model, the singular Jacobian model and the system sensitivity model. Indeed, Stage 1 is related to the feasibility of the power flow solution [37,40]. It has been shown that stage 2 itself is a generalization of many existing models used to explain voltage collapse [13]. From an algebraic point of view, the point (x(λ0), λ0) abbreviated by (x0, λ0) is a saddle node of (1) if the following conditions hold: 1. f(x0, λ0) = 0. 2. fx(x0, λ0) has a simple eigenvalue 0. 3. fλ(x0, λ0) ;∈ Range space of (fx(x0, λ0)). 4. there is a parameterization (x(t), λ(t)) with x(t0 ) = x0, λ(t0) = λ0 and d2 λ(t0) dt2 ;= 0.
- 27. Bifurcation Analysis to Power Systems 13 We now use the above algebraic characterizations to examine several models proposed by several researchers for predicting voltage collapse. Note that the voltage collapse models based on the determinant of system Jacobian in [67,56] can be viewed as providing necessary conditions for the first two conditions for saddle nodes. We show in the following that the model based on the sensitivity of system Jacobian in [19] also provides a necessary condition for the first two conditions of saddle nodes. More specifically, we show that the sensitivity of state vector with respect to its parameter at a saddle node is infinity. Suppose that (x∗ , λ∗ ) is a solution of (1) and fx(x∗ , λ∗ ) is non- singular, the implicit function theorem guarantees the existence of a unique solution curve (x(λ), λ) passing through (x∗ , λ∗ ), i.e . x∗ = x(λ∗ ). And we have the following identity: dx(λ) dλ = − fx(x(λ), λ)− 1 fλ(x(λ), λ) (3) Although the matrix fx(x(λ), λ)− 1 does not exist at the saddle node (x(λ0 ), λ0 ), the property that fλ(x0, λ0) ;∈ Range space of (fx(x0, λ0)) ensures that there is a unique solution curve passing through (x(λ0 ), λ0 ) and dx(λ) dλ becomes infinite there and a small change in λ yields a large change in +x+. This result also explains the “knee” phenomenon in the P-V curve and Q-V curve. The voltage collapse model based on multiple power flow solutions [61,64, 57] uses the presumption that the existence of a pair of very close power flow solutions indicates that the system is about to undergo a voltage collapse. We note that before the bifurcation, there are two equilibrium points (power flow solutions) close to each other. One is stable and the other is type-one. As these two points approach each other an annihilation occurs at the saddle node bifurcation, while at the same time the system Jacobian becomes singular. The above two voltage collapse models can be extended to general slow time-varying one-parameter dynamical systems. The assumption that the parameter λ freezes at a bifurcation point may be inadequate to reflect real power system behavior. A more realistic model is to allow a slowly time- varying parameter both before and after the bifurcation. In particular, the assumption that the system parameter “freezes” at the bifurcation point of interest can be removed. The system dynamics after the bifurcation are then captured by the center manifold of the bifurcation point x∗ with respect to (2). ẋ = f(x, λ) (4) λ̇ = Mg(x, λ) where M is a small number and g is a locally Lipschitz function. In order to extend the voltage collapse model to system (4), it will be necessary to examine the adequacy of the center manifold of x∗ with respect to system (1) in capturing the dynamics after a bifurcation. In this regard,
- 28. 14 H.-D. Chiang we examine the relationship between the trajectories of (1) and (4) which start at the same point near the center manifold of x∗ . It can be shown that system trajectories of (4) follow the system trajectories of (1) during each choice of time interval provided the rate of parameter variation is sufficiently small as shown in the following: Proposition 1 Let x1(t), x2(t) be the system trajectories of (1) and of (4) starting from the initial condition (x0 1, t0) and (x0 2, t0) respectively. Let U be a compact set of the state space containing the center manifold as far as it is of interest, and let K be the Lipschitz constant of (4) on the set U. If x0 1 = x0 2 ∈ U, then |x1(t) - x2(t)| ≤ CM K [eK(t−t0) - 1], where M is a constant. The above result shows that we may approximate the system dynamics after a local bifurcation point by applying (1) instead of (4). Hence, if the parameter changes slowly enough, then the solutions of (4) which lie near the center manifold of (1) at the time of bifurcation will subsequently track the center manifold trajectory of (1) in the state space. This analysis validates the voltage collapse model for power system with slowly variation loads. 5 Computational Tools Power systems are subject to parameter variations. It is important to study the impacts of parameter variations on power system behaviors by tracing the quasi-steady-state of realistic power system models subject to parameter variations. A powerful global analysis tool based on continuation methods can meet this requirement. Continuation methods, sometimes called curve tracing or path following, are useful tools to generate solution curves for general nonlinear algebraic equations with a varying parameter. The theory of continuation methods has been studied extensively and has its roots in algebraic topology and differential topology. Continuation methods have four basic elements: parameterization, predictor, corrector and step-size control. The application of continuation methods to power system analysis has been very actively investigated in recent decades, see for example [38,14,48,3, 7,15,32]. The most attractive feature of a continuation method is that it al- lows users to globally analyze a given power system relative to parameter variations in a reliable and efficient manner. In [12], a survey of existing and pioneering continuation methods applied to power system analysis, which may contain thousands of nonlinear algebraic equations with some limits on some of the state variables, is presented. This survey also includes a com- parison among different implementations of continuation methods for power system applications according to predictor type, corrector type, step-size con- trol strategy, parameterization schemes and modelling capability. A widely used approach in the power industry to investigate potential voltage stability problems, with respect to a given parameter increase pattern, is the use of repetitive power flow calculations. The main advantages of a
- 29. Bifurcation Analysis to Power Systems 15 power system analysis tool based on continuation methods over the repetitive power flow calculations are the following: • Computation 1. it is more reliable than the repetitive power flow calculations in ob- taining the solution curve and the nose point via the parameterization scheme; 2. it is faster than the repetitive power flow calculations via an effec- tive predictor-corrector, adaptive step-size selection algorithm and efficient I/O operations. • Function 1. it is more versatile than the repeated power flow approach via param- eterizations such that general bus real and/or reactive loads, area real and/or reactive loads, or system-wide real and/or reactive loads, and real generation at P-V buses, e.g., determined by economic dispatch or participation factor, can vary. Consider a comprehensive (static) power system mode expressed in the following form: 0 = f(x, λ) (5) where λ ∈ R1 is a (controlling) parameter subject to variation. Using terminology from the field of nonlinear dynamical systems, system (5) is a one-parameter nonlinear system. We next discuss an indirect method to simulate the approximate behavior of the power system (5) due to load and/or generation variation. Before reaching the “nose” point, the power system with a slowly varying parameter traces its operating point which is a solution of the following equation whose corresponding Jacobian has all eigenvalues with negative real parts: f(x, λ) = 0, x ∈ Rn , f ∈ Rn , λ ∈ R (6) These n equations of n+1 variables define in the n+1-dimensional space a one-dimensional curve x(λ) passing through the operating point of the power system (x0 , λ0 ). The indirect method is to start from (x0 , λ0 ), and produce a series of solution points (xi , λi ) in a prescribed direction until the “nose” point is reached. However, it is known that the set of power flow equations near its “nose” point is ill-conditioned, making the Newton method diverge in the neighborhood of “nose” points. There are several possible means to resolve the numerical difficulty arising from the ill-conditioning. One effective way is as follows: First, treat the parameter λ as another state variable xn+1 = λ. Second, introduce the arc-length s on the solution curve as a new parameter in the continuation process. This parameterization process gives x = x(s), λ = λ(s) = xn+1.
- 30. 16 H.-D. Chiang The step-size along the arc-length s yields the following constraint: n V i=1 {(xi − xi(s))2 } + (λ − λ(s))2 − (∆s)2 = 0 Third, solve the following n + 1 equations for the n + 1 unknowns x and λ f(x, λ) = 0 (7) n V i=1 {(xi − xi(s))2 } + (λ − λ(s))2 − (∆s)2 = 0 (8) It can be shown that the above set of augmented power flow equations is well- conditioned, even at the “nose” point. These augmented power flow equations can be solved to obtain the solution curve passing through the “nose” point without encountering the numerical difficulty of ill-conditioning. The task of computing maximum loading points (it saddle-node points or limit-induced bifurcation points) relative to a given load/generation variation pattern has important applications in power system operations and planning. The maximum loading points have a strong relationship with the operating points where voltage collapse may occur. A widely used approach in power industry to determine the maximum loading point, with respect to a given load/generation increase pattern, is the repetitive power flow calculations to generate the so-called P-V or Q-V curve relative to the variation pattern. An operating point on the curve is said to be the maximum loading point of the system if the point is the first point on the curve where power flow calculation does not converge. Note that, due to the its shape in the bifurcation diagram, the saddle-node bifurcation point is termed nose point in power engineering community. Depending on the physical meaning of the underlying parameter and the power network conditions, nose points have been physically related to maximum loading points, or to maximum transfer capability points, or to voltage collapse points. Several issues arise regarding this approach [44]. These issues however can be resolved by applying the local bifurcation theory. First, the point where the power flow diverges (which is a numerical failure caused by a numerical method) does not necessarily represent the maximum loading points (which is a physical limitation). Second, the point where power flow calculations fail to converge may vary, depending on which numerical method was used in the power flow calculation. In other words, based on the criterion of power flow divergence, different numerical methods may come up with different calculated maximum loading points of the system while the maximum loading point physically is unique. We note that the set of power flow equations is ill- conditioned near nose points making Jacobian-based numerical methods such as the Newton method diverge in the neighborhood of nose points. It is well recognized that the Jacobian at a nose point has one zero eigenvalue, causing the set of power flow equations ill-conditioned near nose points. Recently,
- 31. Bifurcation Analysis to Power Systems 17 considerable progress has been made in calculating nose points in a reliable and exact way by using continuation methods and the characteristic equations of nose points. Continuation methods are reliable to overcome the singularity of the Jacobian near nose points and can provide partial initial conditions for solving the characteristic equations. The standard formulation for the characteristic equations of nose points for a set of n-dimensional power flow equations is a set of (2n+1)-dimensional nonlinear equations. Solutions to the characteristic equations give the nose point (n-dimensional), the bifurcation value and the left or right eigenvector (n-dimensional corresponding to the zero eigenvalue. To solve the characteris- tic equations, continuation methods can only provide good initial conditions for an estimated nose point and an estimated bifurcation value. What is missing is a good initial guess for the eigenvector which is an additional fac- tor affecting convergence to the solution. Another method which solves an extended (2n+1)-dimensional system of equations characterizing the saddle- node bifurcation point was proposed in [6] and more recently in [7]. The methods attempt to compute the saddle-node bifurcation point directly. The success of the above two methods depends greatly on a good initial guess of the desired saddle-node bifurcation point. Otherwise, the methods may diverge or converge to another saddle-node bifurcation point. This is because these methods are static in nature, they do not make use of any information on the particular branch of solutions and they do not confine their iterative process to the desired branch of solutions. A simpler set of characteristic equations for nose points of power flow equations can be developed by exploring a decoupled parameter-dependent property of power flow equations. In [44], a test function was developed to characterize nose points of power flow equations. The test function possesses a monotonic property in the neighborhood of nose points that it is positive on one side of the bifurcation value while it is negative one the other side. Hence, it offers an effective way to bracket the parameter value during a search procedure of bifurcation values to guarantee a solution exists inside the bracket. This test function in conjunction with the set of power flow equations constitute a set of (n+1)-dimensional characteristic equations for saddle-node bifurcation points of general nonlinear equations with decoupled parameter [39]. Distinguishing features of the new set of characteristic equations are that they are of dimension n+1, instead of 2n+1, for n-dimensional power flow equations and that the required initial conditions (bifurcations point and bifurcation value) can be completely provided by the continuation method. The task of computing Hopf bifurcations points has physical importance in power system analysis and control. This task, though more involved in com- putation, receives less attention than the task of computing nose points. For n-dimensional power flow equations, the standard formulation for the char- acteristic equations of Hopf bifurcation points is a set of (2n+2)-dimensional nonlinear equations. A simpler set of characteristic equations for Hopf bi-
- 32. 18 H.-D. Chiang furcation points of power flow equations can be developed by exploring a decoupled parameter-dependent property of power flow equations. The new set of characteristic equations is of dimension n+2 (even can be of n+1), in- stead of 2n+2 , for n-dimensional power flow equations. More research work is required in this task for power system applications. We next discuss a practical package, CPFLOW (Continuation Power Flow), a comprehensive tool for tracing power system steady-state behavior due to parameter variations such as load variations, generation variations and control variations [15]. CPFLOW simulates a realistic operating condition or expected future operating conditions relative to parameter variations with activation of control devices during the process of parameter variations. The control devices include : (i) switchable shunts and static VAR compensators, (ii) ULTC transformers, (iii) ULTC phase shifters, (iv)static tap changer and phase shifters, (v) DC network. CPFLOW can efficiently generate P-V, Q-V, and P-Q-V curves with the capability that the controlling parameter λ can be one of the following • general bus (P and/or Q) loads + real power generation at P-V buses • area (P and/or Q) loads + real power generation at P-V buses • system (P and/or Q) loads + real power generation at P-V buses CPFLOW, computationally based on the continuation method, can trace the power flow solution curve, with respect to any of the above three vary- ing parameter, through the “nose” point (i.e. the saddle-node bifurcation point) or the limit-induced bifurcation point, without any numerical diffi- culty. CPFLOW can be used in a variety of applications such as (1) to analyze voltage problems due to load and/or generation variations (e.g. voltage dip, voltage collapse), (2) to evaluate maximum interchange capability and maxi- mum transmission capability [31], (3) to simulate power system static behav- ior due to load and/or generation variations with/without control devices, and (4) to conduct coordination studies of control devices for steady-state security assessment. CPFLOW’s modelling capability is quite comprehensive. The current ver- sion of CPFLOW can handle power systems up to 43,000 buses. CPFLOW has been applied to a 40,000-bus power system with a complete set of op- erational limits and controls. CPFLOW provides three options of parame- terization schemes including arc-length parameterization. In order to achieve computational efficiency, CPFLOW employs the tangent method in the first phase of solution curve tracing and the secant method in the second phase. However, if the number of corrector iterations becomes too large, then the predictor switches back to the tangent since it is more accurate than the se- cant. The Newton method is chosen in CPFLOW as the corrector. CPFLOW computes the arc-length in the state space, which automatically forces the predictor to take large steps on the “flat” part of the solution curve and small steps on the “curly” part.
- 33. Bifurcation Analysis to Power Systems 19 6 Performance Indices for Assessing Voltage Collapse We show in this section how local bifurcation theory can be applied to develop performance indices for assessing voltage collapse. While continuation power flow methods can generate P-V and Q-V curves in a reliable manner, they may be too slow for certain applications such as contingency selection and contingency analysis, design of preventive control for voltage collapse and on- line voltage security assessments. To overcome these difficulties, a number of performance indices intended to measure the severity of the voltage collapse problem have been proposed in the literature. They can be divided into two classes: state-space-based approach and the parameter-space-based approach. The majority of performance indices developed for assessing voltage col- lapse adopt the state-space-based approach. Among them, the minimum sin- gular value in [65], the eigenvalue pursued in [37] and the condition number in [54] of the system Jacobian intend to provide some measure of how far the system is away from the point at which the system Jacobian becomes singular. The performance index proposed in [61] and [64] is based on the angular distance between the current stable equilibrium point and the clos- est unstable equilibrium point in a Euclidean sense. the performance index proposed in [25,26] measures the energy distance between the current stable equilibrium point and the closest unstable equilibrium point using an en- ergy function. These performance indices can be viewed as providing some measure of the “distance” between the current operating point and the bi- furcation point. Note that all these performance indices are defined in the state space of power system models and they cannot directly answer ques- tions such as: “Can the system withstand a 100 MVar increase on bus 20 without encountering voltage collapse?” One basic requirement for useful performance indices is their ability to reflect the degree of direct mechanism leading the underlying system toward an undesired state. In the context of voltage collapse in power systems, a useful performance index must have the ability to measure the amount of load increase that the system can tolerate before collapse. The state-space- based performance indices, however, generally do not exhibit any obvious relation between their value and the amount of the underlying mechanism that the system can tolerate before collapse. In order to provide a direct relationship between its value and the amount of load increases that the system can withstand before collapse, the perfor- mance index must be developed in the parameter space (i.e., the load/generation space). Development of performance indices in the parameter space is a rel- atively new concept which may have been spurred by the local bifurcation theory. In [16], a new performance index that provides a direct relation- ship between its value and the amount of load demand that the system can withstand before a saddle-node voltage collapse was developed. From an an- alytical viewpoint, this performance index is based on the normal form of saddle-node bifurcation points. It can be shown that, in the context of power
- 34. 20 H.-D. Chiang flow equations, the power flow solution curve passing through the nose point is, at least locally, a quadratic curve. From a computational viewpoint, this performance index makes use of the information contained in the power flow solutions of the particular branch of interest. It only requires two power flow solutions. The first power flow solution is the current operating point which can be obtained from a state estimator. Only one additional power flow so- lution and its derivative are needed to compute this performance index. One of the features that distinguishes the proposed performance index is its de- velopment in the load-generation space (i.e. the parameter space) instead of the state space where the then existing performance indices were developed. From an application viewpoint, the parameter-space-based performance indices can be readily interpreted by power system operators to answer ques- tions such as: “Can the system withstand a simultaneous increase of 70 MW on bus 2 and 50 MVar on bus 6?”. Moreover, the computation involved in the performance index is relatively inexpensive in comparison with those required in the state-space-based ones. A look-ahead performance index intended for on-line applications was developed in [17]. Given the following information; (1) the current operating condition, say obtained from the state estimator and the topological analyzer, (2) the near-term load demand at each bus, say obtained from short-term load forecaster and predictive data, and (3) the real power dispatch, say based on economic dispatch, the look-ahead per- formance index provides a look-ahead load margin measure (in MW and/or Mvar) which can be used to assess the system’s ability to withstand both the forecasted load demands and real power variations. In addition, the in- dex provides useful information as to how to derive effective load-shedding schemes to avoid voltage instability. We note that the parameter-space-based performance indices can not take into account the physical limitations of typical control devices such as gen- erator VAR limits and ULTC tap ratio limits; such that their computed load/generation margins may bear some ‘distance’ from the exact margins. Hence, the function of these performance indices is mainly for ranking the severity of a list of credible contingencies or for ranking the effectiveness of different control devices. Exact load/generation margins that accounts for all control devices and their physical limitations can be accurately calculated by using the continuation power flow approach. Recent work on the parameter- space-based performance indices can be found, for example in [33,8,27,29]. 7 Persistence of Local Bifurcations under Unmodelled Dynamics Many physical systems contain slow and fast dynamics. These slow and fast dynamics are not easy to model in practice. Even if these dynamics can be modelled properly, the resulting system model (the original model)is often ill- conditioned. These difficulties have motivated development of several model
- 35. Bifurcation Analysis to Power Systems 21 reduction or simplification approaches to derive reduced models from the original model. One popular model reduction approach (to derive a reduced model) is to neglect both the fast and slow dynamics in an appropriate way. On the other hand, traditional practice in system modelling has been to use the simplest acceptable model that captures the essence of the phenomenon under study. A common logic used in this practice is that the effect of a system component or control device can be neglected when the time scale of its response is very small or very large compared to the time period of interest [69,72]. Electric power systems comprise a large number of components interacting with each other in nonlinear manners. The dynamical response of these com- ponents extends over a wide range of time scales. For example, the difference between the time constants of excitation systems (fast control devices) and that of governors (slow control devices) is a couple orders of magnitudes. The dynamic behavior after a disturbance occurring on a power system involves all the system components to varying degrees. The degree of involvement from each component determines the appropriate system model necessary for simulating the dynamic behaviors after the disturbance. For instance, an extended power system dynamical mode l contain both fast variables, such as the damping flux in the direct and quadrature axis of generators, and slow variables, such as the field flux and the mechanical torque of genera- tors. For simulating the dynamic behaviors of a power system after an event disturbance, the effect of these fast and slow variables can be neglected in the system modelling because the time scale of these variables is very small or very large compared to the time period of the disturbance of interest. A reduced system model is thus obtained from the original system model. Several questions naturally arise regarding the validity of using the analy- sis based on a reduced system model to determine the behavior of the original system. These questions include the relation between the stability properties of the reduced system and those of the original system, between the tra- jectories of the reduced system and that of the original system, and so on. We consider a nonlinear dynamical system with slow and fast un-modelled dynamics of the form ẋ = f(x, y, z, λ) + M1f0(x, y, z, λ, M1, M2, M̃1, M̃2) ẏ = M̃1g(x, y, z, λ, M1, M2, M̃1, M̃2) slow (9) M2ż = h(x, y, z, λ) + M̃2h0(x, y, z, λ, M1, M2, M̃1, M̃2) fast where x ∈ Rn , y ∈ Rm , z ∈ Rp , M1, M2, M̃1, M̃2 ∈ R+ , and f, f0, g, h, h0 are Cr with r ≥ 2. Associated with system (9), we define a reduced system which treats the fast variables z as instantaneous variables and the slow variables y as con-
- 36. 22 H.-D. Chiang stants. This is done by setting M1, M2, M̃1, M̃2 = 0 ẋ = f(x, y, z, λ) ẏ = 0 (10) 0 = h(x, y, z, λ) We pose and study the following problems: (p1) If the reduced system (10) has a saddle-node bifurcation point at (x∗ , y∗ , z∗ , λ∗ ) = (x∗ , y∗ , z∗ (x∗ , y∗ , λ∗ ), λ∗ ) relative to the varying parame- ter λ, then does this imply that the original system (9) with the varying parameter λ also has a saddle-node bifurcation point in a neighborhood of (x∗ , y∗ , z∗ , λ∗ )? If the answer is yes, then (p2) what is the relationship between these two saddle-node bifurcation points? Furthermore, (p3) what is the relationship between the system behaviors after the saddle- node bifurcation of the reduced system (10) and that of the original system (9)? We propose to solve the above three problems via the following three steps. In the first step, we consider a nonlinear dynamical system with slow un-modelled dynamics of t he form : ẋ = f(x, y∗ , λ) + M̃f0(x, y, λ, M, M̃) ẏ = Mg(x, y, λ, M, M̃) (11) where x ∈ Rn , and y ∈ Rm is a slowly varying vector, M, M̃ are small numbers and λ ∈ R is a parameter which is subject to variation. A reduced system associated with (11) can be derived by treating y ∈ Rm as a constant vector: ẋ = f(x, y, λ) ẏ = 0 (12) In this step several analytical results to address the above three issues can be developed. We consider in the second step a nonlinear dynamical system with fast un-modelled dynamics of the form: ẋ = f(x, y, λ, M) Mẏ = g(x, y, λ, M) (13) where x ∈ Rn , y ∈ Rm , λ, M > 0 ∈ R, and f, g are Cr with r ≥ 2. A reduced system by “neglecting” the fast dynamics y can be defined by setting M = 0 in (13) ẋ = f(x, y, λ, 0) 0 = g(x, y, λ, 0) (14) In the third step we connect the analytical results derived in the first two steps to show that, under fairly general conditions, the general nonlinear
- 37. Bifurcation Analysis to Power Systems 23 system (with both fast and slow dynamics) (9) will encounter a saddle-node bifurcation relative to a varying parameter if the associated reduced system (10) (which is derived by neglecting both fast and slow dynamics) encounters a saddle-node bifurcation relative to the varying parameter. A error bound can be derived between the bifurcation point of the reduced system (10) and that of the original system (9). Furthermore, it can be shown that the system behaviors after the saddle-node bifurcation of the reduced system and that of the original system are close to each other in state space [18]. The general analytical results can be applied, among others, to justify the usage of simple power system models for analyzing voltage collapse in electric power systems. For instance, it provides a justification of the current practice that voltage collapse can be analyzed based on a simple model of synchronous machines (the so-called swing equation) rather than on a detailed model which includes the dynamics of several control devices. 8 Concluding Remarks We have presented in this chapter an overview on the application of local bi- furcation analysis and theory to (i) develop models explaining power system nonlinear behaviors and various power system instabilities such as voltage col- lapse and low-frequency oscillations, to (ii) develop a powerful global analysis tool based on continuation methods to trace power system quasi-steady-state behaviors due to load and generation variations in realistic power system models, and to (iii) develop performance indices for detecting and estimat- ing local bifurcations of power systems. Furthermore, an overview on the extension of saddle-node bifurcation, Hopf bifurcation and limit-induced bi- furcation to include the analysis of the system dynamics after the bifurcation has been presented. Electric power systems comprise a large number of components whose dynamical response extends over a wide range of time scales. Up to present, most power system models used for bifurcation analysis involve only short- term dynamical models (transient stability models). It raises the concern about the validity of short-term dynamical models, which have disregarded slow dynamics, for local bifurcation analysis. The effects of un-modelled dy- namics due to fast and slow state variables on the local bifurcation analysis of a power system model has been also discussed in this chapter. During the last two decades, numerical bifurcation analysis of power sys- tem models has been a subject of great interest to researchers and engineers in the power system community. These numerical studies seem to favor the claim that various types of bifurcations can occur in real power systems. Sev- eral bifurcation-based models have been developed to provide an explanation for various observed power system nonlinear behaviors and power system in- stabilities. Furthermore, these numerical studies support the observation that the complexity of power system dynamic behaviors is related more to the non-
- 38. 24 H.-D. Chiang linearity of individual power system models than to the dimensionality of the system. However, these numerical studies only establish a presumption. The next logical step is to investigate the nature, extent and significance of these (local) bifurcations in realistic power system models; if not in real power systems. To this regard, several issues must be addressed. The first issue, related to its nature, is whether the model used reflects a realistic power system. The second issue is under what conditions can realistic power system models encounter bifurcations (such as saddle-node bifurcation, Hopf bifurcation). The third issue, related to its extent, is whether the regions in the parameter space as well as in the state space where bifurcation can occur lie near normal operating points of power systems. The forth issue, related to its significance, is whether the magnitudes of dynamical behaviors after bifurcations are observable in power system behaviors. Other issues remained to be addressed include the following • Under what conditions can realistic power system models encounter global bifurcations? • Under what conditions can realistic power system models encounter limit- induced bifurcations? • How can bifurcation affect power system nonlinear behaviors? • How to evaluate the merits of each explanation of power system instabil- ities when there are several competing explanations? • What kind of actions can be taken to prevent bifurcations? The above issues related to the nature, extent and significance of bifur- cations in realistic power system models can only be addressed using both powerful computational tools and analytical tools. This presents a great chal- lenge for researchers to develop a highly effective computational environment for analyzing bifurcations in large-scale power systems, which are described by a large set of nonlinear equations with parameter-dependent differential and algebraic equations with hard limits. References 1. Abed, E. H., Wang, H. O., Alexander, J. C., Hamdan, A. M. A., Lee, H. C. (1993) Dynamic bifurcations in a power system model exhibiting voltage col- lapse. Int. J. of Bifur. Chaos, 3:1169–1176 2. Abed, E. H., Varaiya, P. P. (1984) Nonlinear oscillations in power system. Int. J. of Electr. Power Energy Syst., 6:37–43 3. Ajjarapu, V. A., Lee, B. (1992) The continuation power flow: A tool for steady state voltage stability analysis. IEEE Trans. Power Syst., 7:416–423 4. Ajjarapu, V. A., Lee, B. (1992) Bifurcation theory and its application to non- linear dynamical phenomena in an electric power system. IEEE Trans. Power Syst., 7:424–431 5. Budd, C. J., Wilson, J. P. (2002) Bogdanov-Takens bifurcation points and Sil’nikov homoclinicity in a simple power-system model of voltage collapse. IEEE Trans. Circ. Syst.-I, 49:575–590
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- 43. Bifurcation Analysis with Application to Power Electronics Chi K. Tse and Octavian Dranga Department of Electronic and Information Engineering Hong Kong Polytechnic University P. R. China encktse@polyu.edu.hk Abstract. The problem of sudden loss of stability (more precisely, sudden change of operating behaviour) is frequently encountered in power electronics. A classic ex- ample is the current-mode controlled dc/dc converter which suffers from unwanted subharmonic operations when some parameters are not properly chosen. For this problem, power electronics engineers have derived an effective solution approach, known as ramp compensation, which has become the industry standard for current- mode control of dc/dc converters. In this chapter, the problem is reexamined in the light of bifurcation analysis. It is shown that such an analysis allows convenient prediction of stability boundaries and facilitates the selection of parameter values to guarantee stable operation. It also permits new phenomena to be discovered. An example is given at the end of the chapter to illustrate how some bizarre operation in a power-factor-correction (PFC) converter can be systematically explained. 1 Introduction The term “stability” has a variety of meanings. In the strict mathematical sense, one may state its meaning in terms of rigorously defined conditions. In engineering, however, stability is often interpreted as a condition in which the system being examined is operating in the expected regime. For instance, in power electronics, we refer a stable operation to a specific periodic operation. When a power converter fails to maintain its operation in this expected manner, it is considered unstable, even though it may be operating in a perfectly predictable regime such as a subharmonic or quasi-periodic regime. In conventional power electronics, all those subharmonic, quasi-periodic and chaotic operations are regarded as being undesirable and should be avoided. Thus, the traditional design objective must include the prevention of any bifurcation within the intended operating range. In other words, any effective design automatically has to avoid the occurrence of bifurcation for the range of variation of the parameters [1]. Bifurcations and chaos have been observed and analysed for various kinds of power electronics circuits [2]–[4]. For systems that have been shown to bifurcate when a certain parameter is changed, the design problem is, in a sense, addressing the “control of bifurcation”. Such a design problem can therefore be solved on the basis of bifurcation analysis. One of our objectives in this chapter is to examine the traditional stability problem from a bifurcation analysis perspective. We will study, to some depth, dc/dc converters under current-mode control, which has been G. Chen, D.J. Hill, and X. Yu (Eds.): Bifurcation Control, LNCIS 293, pp. 29–48, 2003. Springer-Verlag Berlin Heidelberg
- 44. 30 C.K. Tse and O. Dranga 1 " 1 " 1 " / / 1 1 " " /. 0- + − ) i " # Vin R Q S ' & / / / 11 1 + − ' * + − 1 1 1 1 1 1 ) * compensation slope + − + v − C L ' i Vref S D clock 11 1 / / / Iref Fig.1. Boost converter under current-mode control a mature control technique applied in power electronics [5]–[8]. Specifically we will show that the widely known ramp compensation technique is effectively a means of controlling bifurcation, although it was never understood as such in the power electronics community. One useful extension of this work is in the observation of the subtle effect of controlling bifurcation (ramp compensation) on the overall system dynamics. As we will show in this chapter, dynamical response can be undesirably affected if excessive design margin is applied to avoid bifurcation. Hence, in deriving effective control methods, consideration should be given to maintain adequately fast response as well as sufficient clearance of bifurcation. In this chapter we will demonstrate how an “adaptive avoidance of bifurcation” can be achieved by a simple variable ramp compensation scheme. Furthermore, bifurcation analysis may lead to discovery of new phenomena, as we will illustrate using the same current-mode controlled converter but applying it to a power-factor-correction application. The rest of the chapter is organized as follows. In the next section, we briefly review the circuit operation of the boost converter under a typical current-mode control and the instability condition of the inner current loop in terms of period- doubling bifurcation. In Section 3, the conventional ramp compensation for stabi- lization of the inner current loop is considered in the light of “avoiding bifurcation”. Useful design curves will be provided for steady-state design. The effect of the use of compensating ramp on the dynamics of the overall closed-loop system is then considered formally, followed by experimental verifications. We also show how one can make fruitful use of the basic concept of controlling bifurcation to derive a simple effective control method that can ensure adequate margin from bifurcation as well as sufficient transient speed. Finally, in Section 4, we examine the same converter circuit when it is used for a power-factor-correction (PFC) application. Based on results of our bifurcation analysis, we demonstrate an interesting practical behaviour of this popular type of PFC converters, to which traditional theory offers no simple explanation.
- 45. Bifurcation Analysis for Power Electronics 31 (a) 3 3 3 3 3 3 3 # # # # # # # 3 3 3 3 3 3 3 # # # # # # # ) * slope= Vin L slope= −(v−Vin) L t Iref i // (b) 555555555 5 3 3 3 3 3 3 3 # # # # # # # 555555555 5 3 3 3 3 3 3 3 # # # # # # # ) * slope=−mc slope= Vin L slope= −(v−Vin) L compensating ramp & & & & t Iref i // Fig.2. Illustration of current-mode control showing inductor cur- rent (a) without and (b) with ramp com- pensation 2 Review of Operation and Bifurcation Analysis 2.1 Basic operation Consider the boost converter shown in Fig. 1. The switch is turned on periodically, and off according to the output of a comparator that compares the inductor current with a reference level Iref . Specifically, while the switch is on, the inductor current i climbs up, and as it reaches Iref , the switch is turned off, thereby causes the inductor current to ramp down until the next periodic turn-on instant. Thus, the average inductor current is programmed approximately by Iref . In the closed-loop system, Iref is controlled via a feedback loop which attempts to keep the output voltage fixed by adjusting Iref . An important feature of the current-mode control is the presence of an inner current loop. It is now widely known that this inner loop becomes unstable when the duty ratio (designed steady-state value) exceeds 0.5 [7,8]. The usual practical (conventional) remedy is to introduce a compensating ramp to the loop, as shown in Fig. 1. The essential operation is illustrated by the waveforms shown in Fig. 2.
- 46. 32 C.K. Tse and O. Dranga 2.2 Period-doubling bifurcation The aforementioned inner-loop instability can in fact be examined from the view- point of nonlinear dynamics. A handy starting point is the iterative function that describes the inductor current dynamics. We begin with the typical period-doubling bifurcation in the boost converter without ramp compensation [2,4,9]. We let in and in+1 be the inductor current at t = nT and (n+1)T respectively. Denote also the output voltage (voltage across the output capacitor) by v. By inspecting the slopes of the inductor current in Fig. 2 (a), we get Iref − in+1 (1 − D)T = v − Vin L and Iref − in DT = Vin L (1) where D is the duty ratio which is defined as the fractional duration of a switch- ing period when switch S is closed. Combining the above equations, we have the following iterative function: in+1 = E 1 − v Vin L in + Iref v Vin − (v − Vin)T L (2) If we are interested in the inner current loop dynamics near the steady state, we may write δin+1 = E −D 1 − D L δin + O(δi2 n) (3) Clearly, the characteristic multiplier or eignenvalue, λ, is given by λ = −D 1 − D (4) which must fall between –1 and 1 for stable operation. In particular, the first period- doubling occurs when λ = −1 which corresponds to D = 0.5. Consistent with what is well known in power electronics, current-mode controlled converters must operate with the duty ratio set below 0.5 in order to maintain a stable period-1 operation [10]. In the application of current-mode control, the error signal derived from the output voltage is often used to modify Iref directly (not the duty ratio as in the case of voltage mode PWM control). It is thus helpful to look at the period-doubling bifurcation in terms of the current reference Iref . Specifically we can express the “criterion of a bifurcation-free operation”, D < 0.5, in terms of Iref by using the steady-state equation relating R, D and Iref . For the boost converter, the equivalent criterion of a bifurcation-free operation is Iref < Vin R 5 DRT 2L + 1 (1 − D)2 < D=0.5 = Iref,c (5) which can be derived from the power-balance equation E Iref − ∆i 2 L Vin = V 2 in (1 − D)2R (6)
- 47. Bifurcation Analysis for Power Electronics 33 where ∆i = DTVin/L and all symbols have their usual meanings. The critical value (upper bound) of Iref for the uncompensated case is thus given by Iref,c = Vin R E RT 4L + 4 L (7) Hence, period-doubling occurs when Iref exceeds the above-stated limit. To prevent period-doubling, we must therefore control Iref . Indeed, the use of a compensating ramp, as we will see, is to raise the upper bound of Iref , thereby widening the operating range. Fig.3. Bifurcation diagrams obtained numerically for the boost converter under current-mode control, showing the “delaying” of the onset of bifurcation by ramp compensation. (a) No ramp compensation; (b) with compensating ramp m = 0 c 1V /L; (c) m = 0 3V /L; (d) m = 0 8V /L. For all cases, C = 20 in c in in µF, L . . c = 1 . 5 mH, R = 40 Ω V = 5 V and T = 100 in µs. . , 3 Control of Bifurcation by Ramp Compensation 3.1 Design to avoid bifurcation With compensation, the reference current is first subtracted from an artificial ramp before it is used to compare with the inductor current, as shown in Fig. 2. By
- 48. 34 C.K. Tse and O. Dranga inspecting the inductor current waveform, we obtain the modified iterative function for the inner loop dynamics as δin+1 = E Mc 1 + Mc − D (1 − D)(1 + Mc) L δin + O(δi2 n) (8) where Mc = mcL/Vin is the normalized compensating slope, and mc is defined in Fig. 2. Now, using (8), we get the eigenvalue or characteristic multiplier, λ, for the compensated inner loop dynamics as λ = Mc 1 + Mc − D (1 − D)(1 + Mc) (9) Hence, by putting λ = −1, the critical duty ratio, at which the first period-doubling occurs, is obtained, i.e., Dc = Mc + 0.5 Mc + 1 (10) Using (5) and the above expression for Dc, we get the critical value of Iref for the compensated system as Iref,c = Vin R 5 RT 2L Mc + 0.5 Mc + 1 + 4(Mc + 1)2 < > Iref (11) Note that Iref,c increases monotonically as the compensating slope increases. Hence, it is obvious that compensation effectively provides more margin for the system to operate without running into the bifurcation region. Figure 4 shows some plots of the critical value of Iref against R, for a few values of Mc. The choice of the magnitude of the compensating ramp constitutes a design problem which aims at avoiding bifurcation. In a likewise manner, we may consider the input voltage variation and produce a similar set of design curves that provide information on the choice of the compensating slope for ensuring no bifurcation for a range of input voltage. This is shown in Fig. 5 Also, for a general reference, the boundary curves in terms of normalized parameters are shown in Fig. 6. 3.2 Effects on dynamical response The transient response of a power converter can be compromised if bifurcation is kept too remote in order to give a large safe margin, especially when the operating range required is very wide, since guaranteeing “no bifurcation” for a wide range of parameter values would inevitably make the safe margin excessively large at one extreme end of the range. It is therefore of interest to study the effect of the presence of compensating ramp on the closed-loop dynamics of the overall system. We will take a simple av- eraging approach to derive the eigenvalues of the stable closed-loop system, mainly to reveal the transient speed for different values of the compensating slope. Specif- ically we can write down the normalized state equations for the boost converter as dx dτ = −x γ + (1 − d)y γ (12) dy dτ = −(1 − d)x ζ + E ζ (13)
- 49. Bifurcation Analysis for Power Electronics 35 Fig.4. Specific boundary curves Iref,c versus R for current-mode controlled boost converter without compensation and with normalized compensating slope Mc = 0.2, 0.4, 0.6, 0.8 and 1 Fig.5. Specific boundary curves Iref,c versus Vin for current-mode controlled boost converter without compensation and with compensation slope Mc = 0.2, 0.4, 0.6, 0.8 and 1 where the normalized variables and parameters are defined by x = v/Vref , y = i/(Vref /R), E = Vin/Vref , τ = t/T, γ = CR/T, and ζ = L/RT. Here, we choose the steady-state output voltage as Vref . The closed-loop control can be modelled
- 50. 36 C.K. Tse and O. Dranga Fig.6. Specific boundary curves plotted with normalized parameters Fig.7. Plots of Re(λc) versus Mc for ζ = 0.128, γ = 343 (corresponding to C = 440 µF, L = 250 µH, R = 39 Ω and T = 1/20000 s) and E = 3.5/8 approximately by (see Fig. 2) i + ∆i 2 ≈ Iref # &% $ 5 Po Vin + Ioffset − k(v − Vref ) < −mcdT (14) where Po is the output power (i.e., Po = V 2 ref /R), k is the voltage feedback gain, and Ioffset accounts for the steady-state shift due to the difference between the average and the peak value of the inductor current. This control equation can readily be
- 51. Bifurcation Analysis for Power Electronics 37 translated, in terms of the normalized parameters, into d = 1 − 2ζ D 1 E − y − κ(x − 1) K − 2McE x − E(1 − 2Mc) (15) where κ = kR. Hence, putting (15) into the state equations, we get the closed- loop state equations which can then be used to study the closed-loop dynamics. Specifically, we can obtain the Jacobian matrix JF as JF = 2ζ D 2Y − 1 E K + 2McE E(1 + Mc)γ D2ζ E + 2McE K E(1 + 2Mc)γ −2κ E(1 + 2Mc) −2 E(1 + 2Mc) x=X,y=Y (16) Note that in the steady state, x = X = 1 and y = Y = 1/E. Suppose the eigenvalues of the closed-loop system, λc, are complex. The real part of λc can be easily found as Re(λc) = − E2 (1 + 2Mc) + 2E(γ + Mc) − 2κζ 2E2(1 + 2Mc)γ (17) In practice, E < 1 and γ 6 1. Also, for stable operation, κ has to be kept small enough so that Re(λc) < 0. Under such condition, we can readily show that d dMc |Re(λc)| < 0. In other words, the transient becomes slower as Mc increases. Some plots of Re(λc) versus Mc are shown in Fig. 7. Consider the current-mode controlled boost converter. We first observe that for a higher input voltage, the system is more remote to bifurcation. In fact, we can use (11) to determine if a current-mode controlled boost converter, which is designed for a certain Iref (corresponding to a given power level), may bifurcate and be chaotic for a given input voltage. Remarks — It should be stressed that the overall dynamics is modelled by (12) and (13), while the inner current loop dynamics is described by (8). Inconsistent conclusion may be drawn from studying the two dynamical equations. Specifically, from (8), we observe that increasing Mc will make the inner loop dynamics “faster”. However, the foregoing analysis of the overall system dynamics reveals that for some range of parameter values, the system actually becomes slower as Mc increases. Ob- viously, (8) is inadequate for the purpose of examining the overall system dynamics. 3.3 Experimental measurements An experimental prototype of a boost converter under current-mode control has been constructed, as schematically shown in Fig. 1. The circuit parameters are: L = 250 µH, C = 440 µF, R = 39 Ω, v = 8 V (steady-state), and T = 50 µs (i.e., 20 kHz). The feedback circuit has an appropriate integral control to adjust the steady-state level of Iref in the event of a change in Vin. Such an arrangement is common in current-mode control of dc/dc converters. Now, from (1 − D)v = Vin, we know that the uncompensated system will walk out of the stable region if Vin is reduced to below about 4 V, since the output voltage is kept at 8 V by the integral control. We may apply compensation to restore stability.
- 52. 38 C.K. Tse and O. Dranga (a) (b) Fig.8. Measured output voltage v (a) and inductor current i (b) of boost converter under current-mode control without compensation showing chaos as input voltage drops to 3.5 V (a) (b) Fig.9. Measured output voltage v (a) and inductor current i (b) of boost converter under current-mode control with fixed ramp compensation Mc = 0.2 showing sta- bilized operation (a) (b) Fig.10. Measured output voltage v (upper) and inductor current i (lower) of boost converter under current-mode control with fixed ramp compensation Mc = 0.8 showing stabilized operation but slow transient
- 53. Bifurcation Analysis for Power Electronics 39 (a) (b) Fig.11. Measured output voltage v (upper) and inductor current i (lower) of boost converter under current-mode control with variable ramp compensation showing improved transient for 5 V input The results show that, without compensation (Fig. 8), the system becomes chaotic when the input voltage falls to 3.5 V. Moreover, with compensation, the system remains stable. We further verify, from Figs. 9 and 10, that excessive com- pensation lengthens the response time. Further elaboration will be given in the next subsection. 3.4 Variable ramp compensation In order to keep bifurcation away while maintaining fast response, the control should incorporate a special function that dynamically adjusts the compensating ramp. The aim is to give just enough compensation under all input voltage conditions. Thus, the controller may contain, in addition to a conventional proportional-integral gain, a variable ramp generator providing necessary, but not excessive, compensa- tion. For this simplified scenario (i.e., fixed load and output voltage), the compen- sating ramp needs only be controlled according to Mc(Vin) ≥ v 2Vin − 1 or mc(Vin) ≥ v − 2Vin 2L (18) which is derivable from (10). In our experiment, a variable-ramp control circuit has been constructed in discrete form. Figure 11 shows the measured waveforms for the boost converter under such control. The series of waveforms shown in Figs. 9 through 11 serve to illustrate effect of applying ramp compensation to the system dynamics. Specifically, from (18), the value of Mc needed for a 3.5 V input is about 0.14, and no compensation is at all needed for a 5 V input. Thus, with Mc = 0.8 (Fig. 10), the system is over- compensated and hence suffers a slower transient compared to the case with Mc = 0.2 (Fig. 9). Furthermore, even for Mc = 0.2, the system is still over-compensated when the input is 5 V. Thus, we can see a much faster response with the variable ramp compensation (Fig. 11) since it applies just enough compensation for the 3.5 V input and none for the 5 V input.
- 54. 40 C.K. Tse and O. Dranga 4 Application Example: Power-Factor-Correction Boost Converter In the foregoing, we have described the application of ramp compensation in con- trolling bifurcation in dc/dc converters under current-mode control. In practice, the current-mode controlled converter also finds application in shaping the input cur- rent. In fact, the so-called boost rectifier or power-factor-correction (PFC) converter is effectively a current-mode controlled boost converter [11]. The circuit schematic is shown in Fig. 12. In this case, instead of setting Iref constant for a fixed load, we let Iref vary according to the input voltage waveform. Thus, the input current is being directly programmed to track the waveform of the input voltage. The result is a nearly unity power factor. 4.1 Bifurcation analysis The bifurcation analysis described earlier is directly applicable to the case of the PFC boost converter. Effectively, since Iref follows the input voltage, its waveform is a rectified sine wave whose frequency is much lower than the switching frequency. Typically the frequency of this sine wave is 50 or 60 Hz. Thus, the situation is analogous to the case of applying a time-varying ramp compensation to a current- mode controlled boost converter. Suppose the input voltage is given by vin(t) = V̂in |sin ωmt| (19) where ωm is the line angular frequency. For algebraic brevity we express the input voltage in terms of the phase angle θ, i.e., vin(θ) = V̂in |sin θ|. As shown in Fig. 13, when the input voltage is in its first quarter cycle (i.e., 0 ≤ θ < π/2), the value of Iref increases, which is equivalent to applying a negative compensating ramp to Iref (i.e., Mc < 0). Moreover, when the input voltage is in its second quarter cycle (i.e., π/2 < θ ≤ π), the value of Iref decreases, which is equivalent to applying a positive compensating ramp to Iref (i.e., Mc > 0). At θ = π/2, there is no ramp compensation. Therefore, based on the earlier analysis, we can conclude that the system has asymmetric regions of stability for the two quarter cycles. Specifically, the second quarter cycle (i.e., π/2 ≤ θ < π) should be more remote from period-doubling1 because of the presence of ramp compensation. To be precise, we need to find the critical phase angle, θc, at which period-doubling occurs. Since the duty ratio is equal to 1−vin/v and Mc is −(dIref /dt)L/V̂in| sin θ|, we have, from (10), |sin θc| = v + 2L dIref dt 2V̂in . (20) Moreover, if the power factor approaches one, we have Iref ≈ ˆ Iin |sin θ| for 0 ≤ θ ≤ π (21) 1 The term period-doubling here refers to the switching period being doubled.
- 55. Exploring the Variety of Random Documents with Different Content
- 56. were enabled, after a stiffish climb up the face of a rather steep hillside, to attain a ridge 1,700 feet high, which separates the watersheds of the east and west sides of the island. After a brief stay here, we descended the other side by a steep and almost obliterated track for about three hours more, when we reached Livoni, the site of an old Kaicolo stronghold. Here, amid the ruins of the town, we found a farmhouse of recent construction, the property of a Mr. McCorkill, who had obtained a three years' lease of the land, and was about to try his hand at stock-farming. He had two hundred head of cattle, for which he obtained a ready sale at Levuka, but the difficulties of transit were considerable, and he did not seem hopeful as to the success of his enterprise. He was, moreover, apprehensive that his house, which was built close to the bank of a mountain torrent, was on a rather insecure site, and that the next heavy flood in the rainy season would sweep away all his household belongings. He pointed out close to his own house the characteristic raised foundations of an old native temple (Bure Kalou), and told us that his workmen, in clearing the ground for his garden and paddock, frequently turned up human skulls and other bones. He also very kindly promised to send me some Kaicolo crania on the first opportunity; a promise which he amply redeemed some weeks later by presenting me with two excellent specimens. After a short stay in his house, and refreshed by a drink of delicious milk, we continued our walk down this valley, following the course of the river, which, as we advanced, rapidly increased in size, and pursued so sinuous a course that we were obliged to cross and recross it five or six times before we arrived at Buréta—a native village on the west side of Ovalau—which we gained just before nightfall. A further walk of about a mile brought us to the residence of a friend of Mr. Parr's (Captain Morne), a retired merchant captain, and now the owner of a large estate, by whom we were most hospitably entertained and lodged for the night. This gentleman was doing a large trade in pine-apples, of which he has about twenty acres under cultivation. He sends the produce
- 57. periodically to Sydney by steamer, packed in wooden boxes, where they fetch about 20s. per dozen. On the following morning we spent some hours in strolling about the estate, and in a creek near the outlet of the Livoni River we saw the curious mud-fish Periophthalmus jumping about on the moist beach in the ludicrous manner which has been so well described by Mr. Moseley in his "Notes of a Naturalist," its pectoral fins being used for terrestrial progression. We made the return journey by the south side of the island, Captain Morne very considerately sending us on in one of his boats as far as the south-west extreme of Ovalau, from whence a three hours' walk along the seashore brought us back to Levuka. TOTOONGA VALLEY, OVALAU, FIJI. On the morning of the 11th of October we got under way from Levuka, and spent the day in steaming over to Suva, a commodious harbour, situated on the south-east side of Viti Levu, where it was
- 58. our intention to coal ship from a stationary hulk which supplies the steamers plying between Sydney and the Fijian ports. It is said that Suva, from the accommodation which its harbour affords, and from its position on an easily accessible part of the largest island of the group, is destined to become the seat of government and the future capital of Fiji; but at the time of our visit the settlement was very insignificant, and looked a mere speck in the great extent of wooded land which seemed from our anchorage to spread before us in a vast semicircle. Leaving the ship on the morning of the following day, I started for a walk on shore, taking my gun, insect bottle, and collecting boxes. I at first directed my steps inland along the main road, and for about three miles proceeded over an upland plain of undulating land, thickly covered with tall reeds, and showing here and there patches of brush in the wet hollows. In the last-mentioned localities a good many birds, chiefly parrots, were to be heard screaming shrilly, but owing to the denseness of the foliage, few were visible. In the afternoon I returned to the settlement, and from thence proceeded along the beach towards the low point which shelters the harbour from the north-east winds. Here, as the tide fell and laid bare broad flats of mud and coral, several flocks of sandpipers, whose general plumage resembled that of the snipe, came in from seaward, settled, and commenced to feed. A brace of duck and a large grey tern were the only other birds seen. We learned that the country in the immediate vicinity of Suva was exceedingly unproductive. The soil was very thin, and the sub-soil was a stiff pasty clay of a grey colour—in places resembling soapstone—and so impervious to drainage as to render all attempts at agriculture hitherto abortive. We left Suva on October 13th, and sailed for Tongatabu, searching on the way for certain reefs and banks of doubtful existence, which it was desirable on proper evidence to expunge from the charts.
- 59. During the traverses which we made in sounding for these, I had a good opportunity of plying the tow-net. Among the forms thus obtained were a minute conferva, a brilliantly phosphorescent pyrosoma, measuring three inches in length, and a small shell-less pteropod, the Eurybia gaudichaudi. A specimen of the latter, which I examined in a glass trough, measured one-twelfth of an inch across the body. After giving it about half-an-hour's rest, it protruded its epipodia and tentacles, and commenced to swim about vigorously. The caudal portion of the body was furnished with cilia, and the digestive organs presented the appearance of a dark-red opaque mass, surrounded by a transparent envelope of a gelatinous consistency, whose surface exhibited a reticulated structure. Tongatabu, Friendly Islands, 8th to 18th of November.—The credit of discovering the Tonga Islands rests with Tasman, who saw them on the 20th of January, 1643, and subsequently anchored his ship on the north-west side of the large island, Tongatabu. Cook saw the islands during his second voyage in October 1773, and on his third voyage in 1777 he made a stay of three months at the group, for more than a month of which time he was anchored at Tongatabu, the principal and most southward island of the group. The islands were subsequently visited by D'Entrecasteaux, Maurelle (1781), Lieutenant Bligh of the Bounty, Captain Edwards of the Pandora (1791), and other explorers of the eighteenth century. In the month of November 1806, an English privateer, the Port-au- Prince, arrived at Lifonga, one of the Hapai Islands, where the ship was seized by the natives, and most of the crew massacred. Among the few whose lives were spared was a young man named Mariner, who acquired the friendship of the chief, Finow, and lived peacefully with the natives for the space of four years, accumulating during that time a vast amount of information concerning their manners and habits. Mariner's narrative was subsequently published in a book written by Dr. John Martin, which is still regarded as the standard work on the Tonga Islands.
- 60. The Wesleyan missionaries established themselves here in the year 1822, and were well received; and some years subsequently a French Roman Catholic mission was also successfully established. At the time of our visit the entire population of the Tonga Islands, including Tongatabu, Hapai, and Vavau, amounted to 25,000, while that of Tongatabu alone was 12,000. Of the latter number, 8,000 belonged to the Wesleyan, and 4,000 to the Catholic, Church. We anchored in the harbour of Tongatabu, off the town of Nukualofa, on the 8th of November, at about midday. The anchorage looked very bare indeed, there being only one vessel beside ours, a merchant barque belonging to Godeffroy and Co., of Hamburg, the well-known South Sea Island traders. The most striking objects on shore, as viewed from our position in the anchorage, were the Wesleyan Church—an old dilapidated wooden building crowning the summit of a round-topped hill, about sixty feet high, and said to be the highest point on the island—and the king's palace, a very neat-looking villa-edifice abounding in plate-glass windows, and surrounded by a low wall, in which remained two breaches, intended for the reception of massive iron gates, which, through a series of untoward circumstances, are not likely to be ever placed in position. It appears that some time ago the king gave a carte blanche order for two pairs of gates to be sent out from England, and when, after a long series of delays, owing to mistakes in the shipping arrangements, they at length reached Tongatabu, he was rather unpleasantly surprised to find that the excessive charges for freightage had run up the entire cost to the sum of £800. They were then found to be so large and massive as to be quite unsuited for the purpose for which they were intended, so they were thrown down on the ground in a disjointed condition, where they now lie, rusting and half-buried in weeds. Somewhat in the rear of the royal palace is seen a rather imposing private dwelling-house, the residence of Mr. Baker, formerly a Wesleyan minister, and now the political prime minister of the kingdom.
- 61. In the afternoon some of us walked out to see the old fortified town of Bea, which is distant from Nukualofa about four miles in a southerly direction, and is reached by a very good cart-road. This town—or, more properly speaking, village, for it is now but thinly populated—was formerly the stronghold of a party of Tongans, who objected to the introduction of Christianity, and were consequently obliged to defend themselves against the followers of the Wesleyan missionaries. The village is encircled by a rampart and moat, which have for many years past been allowed to go to decay, so that the moat is now partly obliterated with weeds and rubbish, and the strong palisades, which in former times added considerably to the defensive strength of the ramparts, have almost entirely disappeared. As we entered the village by a cutting which pierced the ramparts on the north side, we saw the spot where Captain Croker, of H.M.S. Favourite, was shot down in 1848, when heading an armed party of bluejackets, with whom he was assisting the missionary party in an attack upon the irreconcilables. It seems to have been altogether a most disastrous and ill-advised undertaking, and of its effects some traces still remain in an assumption of physical superiority over their white fellow-creatures, which may be seen among some of the Tongans. Nowhere have I seen the cocoa-nut trees growing in such luxuriance as at Tongatabu. Here they grow over the whole interior of the island, as well as near the seashore; a circumstance which may be attributed to the mean level of the island being only a few feet above high-water mark, and to the coral sub-soil extending over the entire island. The latter is everywhere penetrated to a greater or less degree by the sea-water, as evidenced by the brackish water which is reached on sinking a well to a depth of two or three yards. We made shooting excursions for several miles to the eastward and westward of Nukualofa, and on one of the latter we met with an intelligent native, who excited in us hopes of obtaining some good duck-shooting, and undertook to bring us to the right place. Under
- 62. his guidance we reached a series of extensive salt-water lagoons, which seemed likely places enough. However, on this occasion he proved to be a false prophet; and as he was anxious to make amends for our disappointment, he induced us to follow him into the bush in quest of pigeons. Of these, on reaching a thick part of the forest, we heard a good many; but owing to the dense foliage of the shrubs, which obscured our view aloft, we got very few glimpses of the birds, which, as a rule, keep to the summits of the tallest trees. Nevertheless, by dint of "cooing," to evoke responses from the birds and thus ascertain their whereabouts, we at length succeeded in shooting a good specimen of the great "fruit pigeon." Our guide, "Davita," was most elaborately tattooed from the waist to the knees. He was a well-to-do man, and the chief of a district; and was also, as he informed us, a member of the "royal guard," whose duty it is to act as sentries in front of the door of the king's palace. "Davita" accompanied us back to the town, and after receiving his honorarium and bidding us good-bye, he went off to procure his military uniform, and subsequently, as we walked by the palace on our way to the boat, we saw our friend in full toggery doing sentry. He was a very fine man, but did not look half so well in a soldier's uniform as in his native garb, which consisted simply of a waistcloth, above and below which appeared the margins of his beautiful blue tattooing. There are evidences of recent elevation of the land both to the eastward and westward of Nukualofa. I noticed above high-water mark extensive flats of almost barren land, composed of level patches of coral, the interstices of which were gradually getting filled up with coral detritus, and the decayed remains of stunted plants. The mangrove bushes here seemed with difficulty to eke out an existence, their roots being no longer bathed in sea water; but on the other hand a few Ivi trees (Aleurites sp.?) had gained a footing. An amazing quantity of crabs of the genus Gelasimus inhabit these desolate flats, where they will have an opportunity of gradually adapting themselves to a terrestrial existence. I noticed two species,
- 63. one of which was covered with a hairy brown integument, and was rather sluggish in its movements, waddling awkwardly into its burrow while it held aloft one of its hands in a most ridiculous fashion. The other was a smaller crab, with a greenish body, and having one of its pincer-claws, which was of a brilliant orange colour, of a huge size compared with its fellow. Probably, after the lapse of a few years, these flats will form part of the general forest land, when the crabs may undergo further adaptive changes. We saw little of King George during our stay, as being now advanced in years he leads a retired life, passing his days in a small room in the rear of the palace, and only coming out of doors after sunset for a little airing. However, his grandson, "Wellington Gnu," who is governor of Nukualofa, and heir presumptive to the throne, was most civil and obliging. He is a remarkably fine-looking man, being six feet two inches in height, and stout in proportion; his face beams with amiability and intelligence; and he possesses all the manners and bearing of a polished gentleman. Although the lineal heir to the throne by direct descent, it is very doubtful whether he will succeed the present king, as Maafu, his cousin, and the son of a deceased brother of King George, is older in years, and is consequently by the Tongan laws the legitimate heir to the throne.[3] [3] Since the above was written I have heard of the death of Maafu. Wellington entertained us most hospitably, and drove us in his buggies to various places of interest in the island. On one occasion he took three of our officers to Moa, a native town situated near the south-east extremity of the island. From there they went on to a place eight miles to the southward, where there is a famous megalithic structure of unknown origin, which has been described and figured by Brenchley in his "Voyage of the Curaçoa." As our experience differs somewhat from Brenchley's, I may be excused for making a few remarks thereon. The monument—if such it can be called—consists of three large slabs of coral rock, two of which are planted vertically in the ground at a distance of about fifteen feet
- 64. apart, while the third forms a horizontal span, resting on its edges in slots made in the summits of the vertical slabs. The height of the structure, of which the picture gives a good idea, is about fifteen feet. We were, I regret to say, unable to obtain any information— legendary or otherwise—concerning the origin of this remarkable structure. He also took us on a very pleasant excursion to a village called Hifo, which lies about eleven miles to the south-west of Nukualofa. The party consisted of Wellington Gnu (pronounced "Mou"), David Tonga, the principal of the native school, Captain Maclear, and myself. Our means of locomotion consisted of two buggies, in which we started on the outward journey by a circuitous route, so as to take in the village of Bea and four or five others on our way. On arriving at Hifo, we halted in the centre of the village, on an open patch of sward under the shade of several large vi trees (Spondias dulcis), on whose branches were hanging large numbers of fox bats (Pteropus keraudrenii), of which we obtained specimens. We were now formally introduced to the chief of Hifo, who at once announced that a feast would speedily be prepared in honour of our visit, and pending the necessary culinary arrangements, invited us to walk through his dominions. In an adjacent bay we were pointed out the place where Cook had formerly anchored his vessel, a matter of great interest to the Tongans, who are keenly alive to the fact that the period of Cook's visit formed the great turning-point in their history. As we returned to the village we found that the natives had collected in great numbers under the shade of the trees before mentioned; so we squatted down on the grass, taking up our places with the chief's party, so as to occupy the base-line of a large horseshoe-shaped gathering of natives. The ceremony began with the preparation of the kava, in which respect the Tongans now differ from the Fijians in reducing the root to a pulpy condition by pounding it between stones instead of the rather disgusting process of mastication. While the national beverage was being prepared, a large procession of
- 65. women, gaily dressed, and bearing garlands, shells, and similar offerings, filed solemnly into the centre of the group, and deposited their presents at the feet of Captain Maclear and myself, who were the distinguished guests on this occasion. Sometimes a frolicsome girl would place a garland round one of our necks, and then trip away, laughing merrily. When the kava was ready, a fine-looking elderly man, the second in authority in the village, acted as master of the ceremonies, and gave the orders for carrying out the various details of the function. As the cupbearer advanced with each successive bowl of liquor, this venerable functionary called out in order of precedence the names of the different persons who were to be served, beginning with the visitors, and continuing to indicate each one by name, until every one of the whole vast assemblage— men and women—had partaken. As soon as the kava drinking was over, a procession of young men advanced into the midst of the assemblage, bearing on their shoulders palm-leaf baskets which contained pigs roasted whole, large bunches of bananas, and cocoa- nuts, which they deposited seriatim at our feet. The district chief then made a short speech, informing us, through Wellington's interpretation, that these precious gifts were also at our disposal. Captain Maclear replied, to the effect that we gratefully accepted the present, and requested that it might be distributed for consumption among the villagers. Accordingly the feast was spread, and eating, drinking, and merry-making became general. Occasionally one of the girls would rise from her place, and after lighting a cigarette, of which the cylinder was composed of pandanus leaf instead of paper, would give a few puffs from her own swarthy lips, and then present it courteously to one of us. The act was looked on as a delicate way of paying a compliment, and was on each occasion loudly applauded, the damsel, as she returned among her friends, seeming as if overcome with confusion at her own temerity. When the time fixed for our departure arrived, a most affectionate shaking of hands took place, and we bade good-bye to the happy little village of Hifo, delighted with the kindness, hospitality, and good nature of these far-famed Friendly Islanders.
- 66. ANCIENT STONE MONUMENT AT TONGATABU (p. 173). On the last day previous to our departure from Tongatabu, we made an excursion to the south side of the island, under the guidance of Mr. Symonds, the British Consul, and Mr. Hanslip, the consular interpreter, in order to examine some caves which were said to be of an unusually wonderful nature. They had, of course, never been thoroughly explored, and were consequently said to be of prodigious extent, forming long tunnels through the island. One story was to the effect that an adventurous woman had penetrated one branch of the cave, entering on the south side of the island, and threading its dark recesses for many days, until she finally emerged into the light of day somewhere near Nukualofa, on the north side of the island. A pleasant drive of about ten miles brought us to the shore of a small bay exposed to the prevailing wind, and receiving on its beach the full fury of the swell of the main ocean. The foreshore was strewn with coral débris, and above high-water mark were quantities of pumice-stone, probably washed up from the sides of the
- 67. neighbouring volcanic island of Uea. On either side, the bay was hemmed in by bold projecting crags of coral rock, whose faces indicated, by parallel tide erosions, that they had been elevated by sudden upheaval into their present position. About one hundred yards from the beach, and forty feet above the sea-level, was the entrance to the caves, a narrow aperture in the upraised coral rock, leading by a rapid incline into a spacious vaulted chamber, from whose gloomy recesses dark and forbidding passages led in various directions. In the floor of the chamber were deep pools of water, probably communicating with the sea, and said to be tenanted by a species of blind eel, about two feet long, which we were told the natives sometimes caught with hook and line, and fed upon. I was provided with fishing-tackle for capturing a specimen of this singular creature; but as several of our party were induced to relieve themselves of the intolerable heat of the cave by bathing in these pools, the fish were probably scared away, and I was unable to obtain a single specimen. The rock pierced by the caverns was everywhere of coral formation, and as water freely penetrated through from the soilcap above, the roof and floor were abundantly decorated with stalactites and stalagmites in all their usual fantastic splendour. I noticed that many parts of the floor of the cave were speckled with white spots resembling bird-droppings, on which drops of water were frequently falling from the roof above, and I formed the opinion that the white colour of these spots was due to the drops of water which pattered on them having traversed a portion of the ground above, from which they did not receive a charge of lime salts, and consequently washing clean the portion of the coral floor on which they fell, instead of depositing thereon a calcareous stalagmite. This surmise was strengthened by observing the absence of stalactites depending from the roof in these situations. Numbers of small swifts, apparently the same species which is common on the island (Collocalia spodiopygia), flitted about the vaulted parts of the cave, looking in the torchlight like bats, which at
- 68. first sight I felt sure they must be, until our native guide succeeded in catching one specimen, which resolved our doubts. We traversed the more open parts of the cave to a distance of about one hundred yards from the entrance; but finding further progress all but impracticable, from the narrowness of the passages, and the quantity of water of uncertain depth to be encountered, we soon gave up the attempt, and were glad to return to the cool and clear atmosphere of the upper air. During the voyage from Tonga to Fiji, we spent a good deal of time in hunting up the reputed positions of certain doubtful "banks," viz., the "Culebras" and "La Rance" banks, with a view to clearing up the question as to their having any real existence except in the too vivid imaginations of the discoverers. On the 24th of November, when in latitude 24° 25′ S., longitude 184° 0′ W., we steamed over the position assigned by the chart to the "La Rance" bank, and here our sounding line ran out to three hundred fathoms without touching bottom, thus sufficiently establishing the non-existence of any such "bank." Our position at this time may be roughly stated as some two hundred miles to the southward of Tongatabu. During the greater portion of the day, the sea-surface exhibited large patches of discoloured water, due to the presence of a fluffy substance of a dull brown colour, which in consistency and general arrangement resembled the vegetable scum commonly seen floating on the stagnant water of ditches. This matter floated on the surface in irregularly-shaped streaky patches, and also in finely-divided particles impregnated the sea-water to a depth of several feet. Samples were obtained by "dipping" with a bucket as well as with the tow-net, and when submitted to microscopic examination it proved to be composed of multitudes of minute confervoid algæ. On slightly agitating the water in a glass jar, the fluffy masses broke up into small particles, which, under a magnifying power of sixty diameters, were seen to be composed of spindle-shaped bundles of filaments. Under a power of five hundred diameters, these filaments were further resolved into straight or slightly-curved rods, articulated but not branching, and divided by transverse septa into cylindrical
- 69. cells, which contained irregularly-shaped masses of granular matter. These rods, which seemed to represent the adult plant, measured 1⁄2000 inch in width. On careful examination of many specimens, some filaments were observed, portions of which seemed to have undergone a sort of varicose enlargement, having a width two or three times that of the normal filaments. These propagating filaments (if I am right in so calling them) were invested by a delicate tubular membrane, and were filled with a granular semi- transparent matter, in which were imbedded a number of discoid bodies which were being discharged one by one from the ruptured extremity of the tube. These bodies measured 1⁄1000 of an inch in diameter: when viewed edgewise they presented a lozenge-shaped appearance, and they were devoid of cilia or striæ. A jar full of the sea-water was put by until the following day, when it was found that the confervoid matter had all risen to the surface, forming a thick scum of a dull green colour, while the underlying water was of a pale purple colour, resembling the tint produced by a weak solution of permanganate of potash. From the 24th to the 29th of November, during which time the ship traversed a distance of three hundred miles, we were surrounded by these organisms; during the first three days the large patches were frequently in sight, and for the rest of the time the sea presented a dusty appearance, from the presence of finely-divided particles. On the evening of the 25th an unusually dense patch was sighted, and mistaken for a reef, being reported as such by the look-out man aloft. On the 28th November I encountered among the proceeds of the tow-net another minute alga, of quite a different appearance from that just described. It was composed of vermiform rods 1⁄1000 inch in width, and breaking up into cylindrical segments with biconcave ends. We returned to Levuka on the 4th of December, and stayed in harbour for ten days. At this time we had dismal wet weather, and
- 70. consequently little was done in the way of exploration. I received a visit from a Mr. Boyd of Waidou, a colonist, who has resided for the last sixteen years in Fiji, and who has spent a great deal of his time in collecting natural history specimens. He very kindly presented me with some crania, three of natives of Mallicollo, New Hebrides, and two from Merilava in Bank's Group. We anchored at Suva for part of a day, in order to fill up with coal, and then proceeded on our voyage to Sydney. I made frequent use of the tow-net during this cruise, obtaining thereby a great quantity and variety of surface organisms. Among these were representatives of Thalassicolla, Pyrocystis, Phyllosoma, Sagitta, Eurybia, Atlanta, etc. I obtained one specimen of a curious Annelid. It was two inches in width, had two prominent ruby- coloured eyes, and was marked along its snakelike body by a double row of conspicuous black dots. One day, as were lying almost becalmed, a few hundred miles from the Australian coast, we passed into the midst of a great flock of brown petrels, who were sitting on the water grouped in the form of a chain, and apparently feeding. I had the tow-net out, and after dragging it for about half a mile, brought it in, and found it to contain a mass of yellow-coloured cylindrical and oval bodies belonging to the group Thalassicollidæ. The cylindrical bodies were about one inch in long diameter, by 1⁄8 of an inch in width, and those of an oval shape were about 3⁄16 inch in long diameter. They proved to be mere gelatinous sacks, without any appearance of digestive or locomotory organs. The thin membranous wall was dotted over thickly with dark cells of a spherical or oval shape, each of which contained from three to nine light-coloured nuclei. On examining one of the oval bodies under a magnifying power of forty diameters, the clear transparent nature of the interior of the organism allowed the cells on the distal side to be seen out of focus with misty outlines, while the cells on the proximal wall, which was in focus, came out sharp and clear, and vice versâ.
- 72. W CHAPTER IX. THE EAST COAST OF AUSTRALIA. E remained at Sydney, refitting ship and enjoying the unaccustomed pleasures of civilized society, from the 23rd of January, 1881, until the 16th of April, 1881, but as little of general interest occurred during this period, and as Sydney with its surroundings is a place about which so much has been written by better pens than mine, I think I shall be exercising a judicious discretion by passing over this period in silence, and resuming the narrative from the time when we started on our next surveying cruise. On leaving Sydney we received a welcome addition to our numbers in the person of Mr. W. A. Haswell, a professional zoologist, residing at Sydney, who expressed a wish to accompany us as far as Torres Straits, in order that he might have opportunities of studying the crustacean fauna of the east coast of Australia. He was consequently enrolled as an honorary member of our mess, and Captain Maclear kindly accommodated him with a sleeping place in his cabin. I am indebted to Mr. Haswell for much valuable information concerning the marine zoology of Australia. Steaming northwards, along the east coast of Australia, the first place at which we anchored was Port Curtis, in Queensland, where we took up a berth in the outer roads close to the Gatcombe Head lighthouse. The place bore a rather desolate appearance. There was no building in sight except the lighthouse. The beach was lined with
- 73. a dense fringe of mangrove bushes, behind which rose a straggling forest of gums and grass trees (Xanthorrhœa), and for a long time we saw no living thing excepting several large fish-eagles (Haliæetus leucogaster), and an odd gull that hovered about our stern, picking up the garbage that drifted away from the ship. On the following morning two of us landed and set to work to explore the mudflats, which, stretching out for a long distance from the beach, were laid bare by the ebb tide. As we ranged along in search of marine curiosities, we encountered a solitary individual attired in the light and airy costume of a pajama sleeping suit, and carrying a Westly-Richards rifle on his shoulder. We soon made his acquaintance, and found that he was in quest of wild goats, the descendants of some domestic animals originally let loose by the keeper of the lighthouse. He was an Englishman named Eastlake, and held the position of "government immigration agent" on board a ninety-ton schooner, the Isabella, which at the time was anchored just outside the lighthouse point, awaiting a favourable wind to enable her to put to sea. She was engaged in the "labour traffic" and was just then about to return to the Solomon Islands with some "time-expired" native labourers. The Queensland government compels every vessel engaged in the "labour traffic" to carry an immigration agent, who is accredited to and salaried by the government. His duty is to see that the natives who are shipped from the islands for transit to Queensland come of their own free will, and under a proper contract, and that during the voyage they are treated well and are furnished with proper accommodation, and are dieted according to a scale laid down by the government. In the afternoon I accompanied Mr. Eastlake on board. The Isabella, a vessel of ninety tons, was allowed to carry eighty-five natives besides her crew of some half-a-dozen hands. She had now on board about a dozen natives of New Hebrides, who had completed their time as contract labourers in Queensland, and were about to be returned to their island home. The skipper of the vessel was an old Welshman, who, in the true spirit of hospitality, did the honours of
- 74. the ship, and pressed me to partake of such luxuries as the stores in his cuddy afforded. Among the articles which the New Hebrides men had purchased in Queensland with the proceeds of their labours were a number of old muskets, which they seemed to set great store by. These weapons are probably destined to be brought into action against some future "labour vessel," or "slaver," as they are commonly called by the Australians, which may violate the provision of the "Kidnapping Act" by forcible abduction of natives. We worked the dredge from the ship as she swung round her anchor in seven fathoms of water, and also dragged it from a boat in shallower water inshore. Conspicuous by their abundance amongst the contents of the dredge, and by their curious habit of making a loud snapping noise with the large pincer-claw, were the shrimps of the genus Alpheus. When placed in water in a glass jar, the sound produced exactly resembles the snap which is heard when a tumbler is cracked from unequal expansion by hot water. We also obtained a good many whitish fleshy Gorgoniæ, and among Polyzoa the genera Crisia and Eschara afforded a good many specimens. A moderate- sized brownish Astrophyton was generally found entangled in the swabs, but in most cases some of its brittle limbs had parted company with the disc, so that we got scarcely a single perfect specimen. A good many crabs were found on the foreshore; among others were species of the genera Ozius, Gelasimus, and Thalassina; the latter a lobster-like crustacean which burrows deeply in the mud about the mangrove bushes, and throws up around the aperture of its burrow a conical pile of mud. On April 23rd we got under way, and steamed for five miles further up the bay, anchoring immediately off the settlement of "Gladstone." Nothing could exceed the hospitality shown to us by the inhabitants of this quiet little Utopia. Our stay of five days was occupied by an almost continuous round of festivities, during which we were driven about the country, had a cricket-match, shooting expeditions, two balls in the Town Hall, and sundry other amusements. The
- 75. settlement contains a population of only 300, and seems to have been of late years rather receding than advancing in numbers, as many of the settlers had moved on to other more promising centres of industry. There was the old story of a projected railway which was to open up the country, develop its hidden resources, connect it with the neighbouring town of Rockhampton—distant about eighty miles —and give a fresh impetus to trade; but the hopes of its construction were visionary. We made several shooting excursions in quest of bird specimens, and found the pied grallina (G. picata), the butcher bird (a species of Grauculus), the garrulous honeyeater (Myzantha garrula), the laughing jackass (Dacelo gigas), and many doves and flycatchers abundant in the immediate vicinity of the settlement. Walking one day through the forest about two miles inland, we came upon a grove of tall eucalyptus trees, on the upper branches of which were myriads of paroquets, making an almost deafening noise as they flew hither and thither, feeding on the fragrant blossoms. Among them were three species of Trichoglossus, viz., T. novæhollandiæ, T. rubritorquis, and T. chrysocolla. We also shot specimens of the friar bird (Tropidorhynchus corniculatus), and several honeyeaters, flycatchers, and shrikes; so that as a place for bird collecting it was exceedingly rich, both in numbers and species. We got under way on the 30th of April, in the morning, and on the following day anchored off the largest and most northern of the Percy Islands. I landed with Haswell in the afternoon, and after exploring the beach in search of marine specimens, we directed our steps towards the interior of the island. We followed a narrow winding foot track, which led us to a rudely-built hut, in which dwelt an old Australian colonist named Captain Allen, to whom the island virtually belongs. He had a small kitchen garden in the bed of a valley, through which ran a tiny stream; and his live stock consisted of a herd of goats and a number of poultry. We understood that he intended eventually to undertake regular farming operations, but that he at present merely occupied the land in order to retain the
- 76. "pre-emptive" right until the Queensland government should be in a position to sell or let it. It appeared that as yet it was not certain whether the colonial government had a clear title to the group of islands, or whether—being on the Great Barrier Reef, and detached from the mainland by a considerable distance—it was still under the control and jurisdiction of the imperial government. We noticed very few birds: among these were a Ptilotis, a flycatcher, a crow, and a heron; but we were told that in the less frequented parts of the island there were brush turkeys, native pheasants, and black cockatoos. Among the rocks bordering the shore, a large white-tailed rat— probably of the genus Hydromys—was said to be abundant. The only other mammal recorded was a large fox-bat, a skeleton of which was found hanging on a mangrove bush. We left our anchorage at the Percy Islands on the morning of the 2nd of May, and on the forenoon of the 3rd steamed into the sheltered waters of Port Molle, i.e., into the strait which separates Long Island from the main shore of Queensland; and we finally came to an anchor in a shallow bay on the west side of Long Island, where we lay at a distance of about half-a-mile from the shore. The island presented the appearance of undulating hills, covered for the most part with a thick growth of tropical forms of vegetation, but exhibiting a few patches of land devoid of trees, and bearing a rich crop of long tangled grasses. On landing, we found that there was no soil, properly so-called, but that the forest trees, scrub, and grass sprung from a surface layer of shingle, which on close inspection contrasted strangely with the rich and verdant flora which it nourished. Small flocks of great white cockatoos flew around and above the summits of the tallest trees, and by the incessant screaming which they maintained, gave one the idea that the avifauna was more abundant than we eventually found it to be. On the beach we collected shells of the genera Nerita, Terebra, Siliquaria, and Ostræa, and among the dry hot stones above high-
- 77. water mark we found in great numbers an Isopod Crustacean, and as the females were bearing ova, Haswell took the opportunity to make some researches into the mode of development of the embryo. I spent another day accompanying Navigating-Lieutenant Petley, who was then cruising from point to point in one of our whale-boats, determining on the positions for main triangulation. In the course of the day we visited the lighthouse on Dean Island, and on arriving there found a large concourse of blacks on the hill above, looking on our intrusion with great consternation. The lighthouse people told us that the natives, from their different camps on the island, had observed our approach while we were yet a long distance off, and hastily concluding that we were a party of black police coming to disperse (i.e., shoot) them, had fled with precipitation from all parts of the island, to seek the protection of the white inhabitants of the lighthouse. It appeared that some few years previously the natives of Port Molle had treacherously attacked and murdered the shipwrecked crew of a schooner, and in requital for this the Queensland Government had made an example of them by letting loose a party of "black police," who, with their rifles, had made fearful havoc among the comparatively unarmed natives. The "black police," or "black troopers," as they are more commonly called, are a gang of half-reclaimed aborigines, enrolled and armed as policemen, who are distributed over various parts of the colony, and are under the immediate direction of the white police inspectors. Their skill as bush "trackers" is too well known to need description, and the peculiar ferocity with which they behave towards their own countrymen is due to the fact that they are drawn from a part of the continent remote from the scene of their future labours, and from tribes hostile to those against which they are intended to act. Through their instrumentality the aborigines of Queensland are being gradually exterminated. In the official reports of their proceedings, when sent to operate against a troublesome party of natives, the verb "to disperse" is playfully substituted for the harsher term "to shoot."
- 78. But to return to our friends at Dean Island. Our peaceful aspect, and a satisfactory explanation on the part of the white people in charge of the lighthouse, soon set matters right, and the wretched blacks were now so delighted at finding their fears to be groundless, that they crowded about us—male and female—to the number of forty or fifty, brought us some boomerangs for barter, and finally shared our lunch of preserved meat and coffee, of which we partook on the rocks near where the boat was moored. I was surprised at noticing a large proportion of children, a circumstance which does not support one of the views put forward to account for the rapid decrease in numbers of the race. Most of the men had a certain amount of clothing, scanty and ragged though it was, but the children were all stark naked, and some of the women were so scantily attired that the requirements of decency were not at all provided for. They seemed to be fairly well nourished, and from their cheerful disposition I should imagine that they were not undergoing any privations which to them would be irksome. On re-embarking, we sailed along the western shore of the island, and again landed in a small bay about a mile to the northward of the lighthouse. We then proceeded to ascend a hill, on which Petley wished to erect a mark for surveying purposes. The natives, although quick enough about following us along the seashore, showed no inclination to follow us up the hillside, and before we had gone a few hundred yards they had all dropped off. Possibly the fear of snakes was the deterring influence. Port Molle proved to be an excellent place for obtaining examples of the marine fauna of this part of the coast. A great extent of reefs was exposed at low spring tides, exhibiting Corals of the groups Astræa, Meandrina, Porites, Tubipora, Orbicella, and Caryophyllia, besides a profusion of soft Alcyonarian Polyps. Holothurians were abundant, as were also some large Tubicolous Annelids, with very long gelatinous thread-like tentacles. We also got a few Polynæ, and several other annelids of the family Amphinomidæ. A Squilla, with
- 79. variegated greenish markings on the test, made itself remarkable by the vigour with which it resented one's attempts, for the most part unintentional, to invade the privacy of its retreat. An active black Goniograpsus was a common object on the reefs, and the widely distributed Grapsus variegatus was also met with. Haswell obtained from the interior of the large Pinna shells examples of a curious small lobster-like crustacean, which is of parasitic—or perhaps rather commensal—habit, like Pinnotheres. Not uncommon in the rock pools was a bivalve shell of the genus Lima, which on being disturbed swims about in a most lively manner by flapping its elongated valves, exhibiting at the same time a scarlet mantle fringed with a row of long prehensile tentacles. Shells of the genera Arca, Tridacna, and Hippopus were common, and three or four species of Cypræa were seen. We dredged several times with one of the steam-cutters in depths varying from twelve to twenty fathoms, obtaining several species of Comatulas, two or three Astrophytons, Starfishes, Ophiurids, Echini of the genera Salmacis and Goniocidaris, small Holothurians, many species of Annelids, two or three Sponges, a great variety of handsome Gorgoniæ, Hydroids of the group Sertularia and Plumularia, Polyzoa of the genera Eschara, Retepora, Myriozoum, Cellepora, Biflustra, Salicornaria, Crisia, Scrupocellaria, Amathia, etc., and Crustaceans of the genera Myra, Hiastemis, Lambrus, Alpheus, Huenia, and many others. Among the Annelids was one with long glassy opalescent bristles surrounding the oral aperture, and projecting forwards to a distance of one and a half inches from the prostomium. Another Annelid (species unknown) was peculiar in having two long barb-like tentacles projecting backwards from the under part of the head. On examining the proboscis of the latter, while it was resting in sea-water in a glass trough, Haswell noticed a number of singular bodies being extruded from the mouth, which he eventually ascertained, to his great astonishment, were the partially developed young of the worm.
- 80. One of the large Astrophytons which came up with the dredge was seen to exhibit nodular swellings on several parts of the arms, but principally at the points of bifurcation. Each of these swellings was provided with one or more small apertures, and had the general appearance of being a morbid growth. On incising the dense cyst- wall a cavity was exposed, containing a tiny red gastropodous mollusc (of the genus Stilifer), enveloped in a mass of cheesy matter, which contained moreover one or two spherical white pellets of (probably) fæcal matter. Haswell obtained about a dozen specimens of the shell from a single astrophyton. Port Denison is only forty miles to the northward of Port Molle, so that we accomplished the passage in about six hours, and before dusk took up a berth in the shallow bay about a mile and a half from the shore, and three-quarters from the end of a long wooden pier, which was built some years ago in the vain hope of developing the shipping trade of the port. The township of "Bowen" is built on a larger scale than "Gladstone"—of which we had such pleasant reminiscences—but did not appear to be in a more flourishing condition, a "gold-rush" further to the northward having drawn off part of the population, and some of the trade which had previously gone through the port. On the outskirts of the town were some large encampments of the blacks, who lived in a primitive condition, and afforded an interesting study for an ethnologist. Like most of the Australian aborigines, their huts were little better than shelter screens to protect them from the wind and sun. In some instances the twigs on the lee side of a bush, rudely interlaced with a few leafy boughs torn from the neighbouring trees, afforded all the shelter that was required. Both men and women, especially the latter, seemed to be in a filthy, degraded state. They had just received their yearly gifts of blankets from the Queensland Government—I believe the only return which they receive for the appropriation of their land. It appears, however, that they do not much appreciate the donation, for soon after the general issue many of the blankets are bartered with the whites for tobacco and grog. Some of the young men are really fine-looking fellows, and seemed to feel all the pride of life and
- 81. liberty as they strutted about encumbered with a variety of their native weapons, among which I saw the nulla, waddy, shield, huge wooden sword, spear without throwing-stick, and different patterns of boomerangs. They are very expert in the use of the latter. It was the first time that I had seen the boomerang thrown, and I can safely say that its performances, when manipulated by a skilful hand, fully realized my expectations. I noticed that whatever gyrations it was intended to execute, it was always delivered from the hand of the thrower with its concave side foremost—a circumstance I was not previously aware of. Some of the children were amusing themselves in practising the art, using instead of the regular boomerang short pieces of rounded stick bent to about the usual angle of the finished weapon; and I was surprised at noticing that even these rude substitutes could be made to dart forward, wheel in the air, and return to near the feet of the thrower. I had always imagined up to this time that the flat surface was an essential feature in the boomerang. The foreshore at low-water afforded us examples of a great many flat Echinoderms of the genus Peronella, Starfishes of the genus Asteracanthus, and Crustaceans of the genera Macrophthalmus, Matuta, Mycteris, etc. We made several hauls of the dredge in four to five fathoms of water, obtaining a quantity of large Starfishes and Gorgonias, and Crustaceans of the family Porcellanidæ. We left Port Denison on the 24th of May, and continued our coasting voyage northward, anchoring successive nights off Cape Bowling Green, Hinchinbrock Island, Fitzroy Island, Cooktown, and Lizard Island. We landed at the island last mentioned for a few hours. On the shore of the bay in which we anchored was a "Beche-de-mer" establishment, belonging to a Cooktown firm, and worked by a party of two white men, three Chinese, and six Kanakas. The buildings consisted of two or three rudely-built dwelling huts, and a couple of sheds for curing and storing the trepangs. We learned from the "Boss" that his men had been working the district for the previous twelve months, and having now cleared off the trepangs from all the
- 82. neighbouring reefs, he expected soon to move on to some other location further north. The Beche-de-mer industry seems simple enough to conduct. The sluggish animals are picked off the reefs at low tide, and at the close of each day the produce as soon as landed is transferred to a huge iron tank, propped up on stones, in which it is boiled. The trepangs are then slit open, cleaned, and spread out on gratings in a smoke- house until dry, when they are ready for shipping to the Chinese market. The best trepangs are the short stiff black ones with prominent tubercles. Since the above notes were written, a horrible catastrophe occurred at Lizard Island. The bulk of the party had gone on a cruise among the islands to the northward, leaving the station in charge of a white woman—wife of one of the proprietors—and two Chinamen. A party of Queensland blacks came over from the mainland, massacred these three wretched people, and destroyed all the property on the station. On the evening of the 29th of May we anchored off Flinders Island, in latitude 14° 8′ S., and before darkness came on we spent a few hours in exploring. The shore on which we landed was covered with large blocks of quartzite stained with oxide of iron, and disseminated among them were many large irregularly-shaped masses of hæmatite. Immediately above the beach, and among the familiar screw-pines, we saw a few fan palms, the first met with on our northern voyage. Groping among the rocks of the foreshore, I encountered a multitude of crabs of the genera Porcellana and Grapsus, and caught after much trouble a large and uncommonly fierce specimen of the Parampelia saxicola. On anchoring, the dredge had been lowered from the ship, and when hauled up after the ship had swung somewhat with the tide, a curious species of Spatangus, a Leucosia, and a somewhat mutilated Phlyxia, were obtained.
- 83. Early on the following morning I accompanied Captain Maclear and Mr. Haswell on a boat trip to Clack Island (five miles from our anchorage). We were anxious to see and examine some drawings by the Australian aborigines, which were discovered in the year 1821 by Mr. Cunningham, of the Beagle, (see "King's Australia," vol. ii., p. 25), and since probably unvisited. After about an hour's sailing we reached the island—a bold mass of dark rock resembling in shape a gunner's quoin; but we now found it no easy matter to find a landing-place. On the south-east extremity was a precipitous rocky bluff about eighty feet in height, against whose base the sea broke heavily, while the rest of the island—low and fringed with mangroves —was fenced in by a broad zone of shallow water, strewn with boulders and coral knolls, over which the sea rose and fell in a manner dangerous to the integrity of the boat. After many trials and much risk to the boat, we at length succeeded in jumping ashore near the south-east or weather extremity of the island. Here we found abundant traces of its having been frequently visited by natives, but it did not appear as if they had been there during at least half-a-dozen years prior to the time of our visit. We saw the drawings, as described by Cunningham, covering the sides and roofs of galleries and grottoes, which seemed to have been excavated by atmospheric influences in a black fissile shale. This shale, which gave a banded appearance to the cliff, was disposed in strata of about five feet in thickness, and was interbedded with strata of pebbly conglomerate—the common rock of the islet. In these excavations, almost every available surface of smooth shale was covered with drawings, even including the roofs of low crevices where the artist must have worked lying prone on his back, and with his nose almost touching his work. Most of the drawings were executed in red ochre, and had their outlines accentuated by rows of white dots, which seemed to be composed of a sort of pipe-clay. Some, however, were executed in pale yellow on a brick-red ground, and in many instances the objects depicted were banded with rows of white dots crossing each other irregularly, and perhaps intended in a rudimentary way to convey the idea of light and shade. The objects delineated (of which I made such sketches as I was able) were
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