Download EPRI/NSF Workshop presentation 4/02 Cancun

Instability Monitoring and Control
of Power Systems
NSF Workshop on Applied Mathematics
for Deregulated Electric Power Systems
November 3-4, 2003, Arlington, VA
Eyad H. Abed
Electrical and Computer Engineering
and the Institute for Systems Research
University of Maryland, College Park 20742
[email protected]
Concepts of Instability Monitoring
• Detection of (incipient) instability by observing a
departure from synchronous operation. Use of global
phasor measurements of system trajectory and
detection of nature of the trajectory (e.g., concave or
convex).
• Determination of measures of proximity to the
border of the stability boundary in parameter space.
(Model based.)
• Determination of conditions for hitting the stability
boundary in parameter space. (Model based.)
• Probe signal or inherent noise-based detection of
impending instability.
Problems with Model-Based Instability
Monitoring
• As noted by Hauer (APEx 2000): “[recurring
problem of system oscillations and voltage
collapse] is due in part to … system behavior not
well captured by the models used in planning and
operation studies”
• In the face of component failures, system models
quickly become mismatched to the physical
network, and are only accurate if they’re updated
using a powerful and accurate failure detection
system.
Other Work Related to Probe and Ambient
Noise-Based Instability Monitoring
• In several papers, Hauer has discussed large system
experiments using probe signal injection and ambient
noise effects for stability and oscillation studies. This
includes HVDC modulation at mid-level (§125MW)
for probing of inividual oscillation modes, and lowlevel (§20MW) for broadband probing.
• Earthquake prediction research contains work on
studying the nature of pre-quake motions.
• Kevrekidis et al. (recent Phys. Rev. Lett. etc.) and
Sontag et al. (SCL) are looking for schemes for
automatically finding bifurcation points in uncertain
systems.
Probe-Based Instability Monitoring Not
Entirely New, … However:
• There’s no deep theory at present.
• Only linearization-based statements are
available.
• More comments later on needed work in this
area.
Noisy Precursors
• “Noisy precursors” were studied by Kurt Wiesenfeld
(1985, J. Stat. Phys.) in the context of noise
amplification near criticality (stability boundary).
Wiesenfeld found different noisy precursors for
different bifurcations, assuming a small white noise
disturbance.
• It is important to note that noisy precursors also give
a nonparametric indicator of impending instability.
• Noisy precursors are observed as rising peaks in the
power spectral density of a measured output signal of
a system with a persistent noise disturbance --- the
rising peak is seen as one or more eigenvalues
approach the imaginary axis.
Closed-loop precursor-based monitoring systems –
Illustration for closeness to zero eigenvalue
Non-noise-based precursors
• Resonant and nearly resonant (periodic)
perturbations: They can be shown to either
delay or advance bifurcations (instabilities)
– Supercritical bifurcations delayed
– Subcritical bifurcations advanced
• Chaotic signals containing a resonant
frequency have a similar effect.
• White noise can have such an effect, but it is
less pronounced.
Combined model- and signal-based online monitoring
• The effect of harmonic probe signals to advance a
subcritical (severe) bifurcations (instability) can be
very useful in early detection of an impending
instability.
• However, this would also introduce the system
instability into the power system before it would
otherwise occur (defeating the purpose).
• To circumvent this problem, the probe can be
applied to a model that is updated as system loading
and topology change, and detected impending
instabilities in the model can be used as an alarm to
trigger preventive control actions.
Participation Factors
(Modal) Participation factors are an important
element of Selective Modal Analysis (SMA)
(Verghese, Perez-Arriaga and Schweppe,
1982). See also books by Sauer and Pai,
Kundur, etc.
SMA is a very popular tool for system analysis,
order reduction and actuator placement in the
electric power systems area. Related concepts
occur in other engineering disciplines.
We have revisited the concept of participation
factors, and considered why it is useful in
sensor/actuator placement.
Basic Definition
Consider a linear time-invariant system
dx/dt = Ax(t),
where x2 Rn, and A is n£ n with n distinct
eigenvalues (l1,l2,…,ln).
It is often desirable to quantify and compare the
participation of a particular mode (i.e.,
eigenmode) in state variables. If the states are
physical variables, this lets us study the influence
of system modes on physical components.
Tempting to base the association of modes with
state variables on the magnitudes of the entries in
the right eigenvector associated with a mode.
Let (r1,r2,…,rn) be right eigenvectors of the
matrix A associated with the eigenvalues
(l1,l2,…,ln), respectively.
Using this criterion, one would say that
the mode associated with li is significantly
involved in the state xk if rik is large.
Two main disadvantages of this approach:
(i) It requires a complete spectral analysis
of the system, and is thus computationally
expensive;
(ii) The numerical values of the entries of
the eigenvectors depend on the choice of units
for the corresponding state variables.
Problem (ii) is the more serious flaw. It renders
the criterion unreliable in providing a measure of
the contribution of modes to state variables. This
is true even if the variables are similar physically
and are measured in the same units.
In SMA, the entries of both the right and left
eigenvectors are utilized to calculate participation
factors that measure the level of participation of
modes in states and the level of participation of
states in modes.
The participation factors defined in SMA are
dimensionless quantities that are independent of
the units in which state variables are measured.
Let (l1,l2,…,ln) be left (row) eigenvectors of the
matrix A associated with the eigenvalues
(l1,l2,…,ln), respectively.
The right and left eigenvectors are taken to
satisfy the normalization li rj = dij (Kronecker
delta).
Verghese, Perez-Arriaga and Schweppe define
the participation factor of the i-th mode in the kth state xk as the complex number
pki := lik rik
New Approach and New Definitions
Reference: Abed, Lindsay and Hashlamoun
(Automatica 2000).
The linear system
dx/dt = Ax(t)
usually represents the small perturbation dynamics of
a nonlinear system near an equilibrium.
The initial condition for such a perturbation is usually
viewed as being an uncertain vector of small norm.
We have taken two approaches to define participation
factors accounting for uncertainty in initial condition.
Severity of Instability/Biurcation
At least two types of severity issues:
• Severity of the nonlinear instability (nature
of the bifurcation). This depends strongly
on system nonlinearities.
• Spatial impact of the instability, which can
be checked by linear analysis using
participation factors and needed
generalizations.
Determining Severity of Instability in Advance
• Participation factors tell us which physical states
participate most in a mode (such as an unstable
mode).
• Can severity of a bifurcation be linked to criteria
in terms of participation factors (as well as
nonlinear calculations)?
• Is it true that if fewer states are tightly tied to a
mode then the chance of pervasive instability is
reduced?
• Can we use these concepts to build vibration
absorbers for power networks?
Simple Message of this Talk
• We can’t rely totally on models in power system
instability monitoring… the models become less
reliable as the system is stressed more and more.
• Signal-based tools need to be developed for
detecting instability problems before they start.
• A lot of deep theory needs to be developed to make
this happen, including ideas for time-space
propagation of instability.
• Finding synergies with other areas is needed --self-organized criticality, earthquake prediction,
lasers, etc.