EPRINSF Workshop presentation 402 Cancun

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Instability Monitoring and Control of Power
Systems
Workshop on Open Issues in Analysis and Control
of Power and Energy Processing SystemsIEEE CDC,
December 8, 2003, Maui, HI
Eyad H. Abed Electrical and Computer
Engineering and the Institute for Systems
Research University of Maryland, College Park
20742 abed_at_umd.edu
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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.

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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 theyre updated
  using a powerful and accurate failure detection
  system.

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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 low-level
  (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.

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Probe-Based Instability Monitoring Not Entirely
New, However

• Theres no deep theory at present.
• Only linearization-based statements are
  available.
• More comments later on needed work in this area.

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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.

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(a)
(b)
Power spectrum magnitude for Hopf bifurcation
when ?010 for two values of ? (a) ?10, (b)
?0.1 Kim Abed 2000.
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(a)
(b)
Power spectrum magnitude for stationary
bifurcation for two values of ? (a) ?10, (b)
?0.1 Kim Abed 2000.
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Application to a Power System Model
Consider a synchronous machine connected to an
infinite bus together with excitation control
Abed Varaiya, 1984. It was shown that this
system undergoes a subcritical Hopf bifurcation
as the control gain in the excitation system is
increased beyond a critical value. The
dynamics of the generator is given by
The dynamics of the generator is given by
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Output spectrum with noise probe signal, as
instability is approached. (Critical KA 193.7.)
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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.

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Chaotic probe signal
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Closed-loop precursor-based monitoring systems
Illustration for closeness to zero eigenvalue
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Combined model- and signal-based on-line
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.

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Model-based instability monitor with periodic
probe signal
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Example in a Simple Power System Model
Consider the simplified power system model
Venkatasubramanian et al. 1992 with generator,
voltage control, transmission line and matched
load
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Bifurcation boundaries in parameter space
(subcritical Hopf followed by saddle-node)
LExcitation gain and PLoad
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Advance of Hopf bifurcation as a function of
probe amplitude two-thirds power law
exhibited (Hassouneh et al., 2002)
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Severity of Instability/Bifurcation

• 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.

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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?

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Simple Message of this Talk

• We cant 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.

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A Related SubjectRevisiting Modal Participation
Factors

• Participation factors give measures of
  interaction between modes and states
• These can be useful in placement of controllers
  and sensors
• A better fundamental understanding of
  participation factors will contribute to health
  monitoring of heavily loaded power systems

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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.

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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.

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• 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.

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• 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.

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• 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.

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• 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 k-th
  state xk as the complex number
• pki lik rik

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New Approach and New Definitions

• Reference Abed, Lindsay and Hashlamoun
  (Automatica 2000).
• The linear system
• 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 re-defined participation factors using
  deterministic and probabilistic uncertainty
  models.

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New Approach and New Definitions, Contd.

• These new definitions involve computing the
  relative participation of a mode in a state
  averaged over the initial condition uncertainty.
• The averaging can be over a set (set-theoretic
  setting) or with respect to a probability density
  of the initial condition (probabilistic setting).
• When the uncertainty is symmetric with respect to
  the origin, the new definitions yield the
  original definition as a special case.

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New Approach and New Definitions, Contd.

• The new definitions and the old definitions focus
  on behavior around an equilibrium.
• It would be desirable to extend these concepts to
  the case of transient behavior initiated away
  from equilibrium, such as occurs when in a
  faulted system trajectory.