EPRI/NSF Workshop presentation 4/02 Cancun

1
Participation Factors, Actuator Placement, and
Some Nonlinear Control Issues in Power Systems
Presentation at EPRI/NSF Workshop on Global
Dynamic Optimization of the Power Grid April
10-12, 2002, Playcar, Mexico
Eyad H. Abed Elec. and Comp. Eng. and Inst.
Systems Res. University of Maryland
2
Summary

• In this talk, the notion of participation factors
  is revisited, the relation to issues of
  controller placement in power networks is
  considered, and some related nonlinear control
  issues are discussed. Participation factors are
  an important element of Selective Modal Analysis
  (SMA), a methodology introduced in 1982 by
  Perez-Arriaga, Verghese and Schweppe 1-2. 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. Participation
  factors, a key element of SMA, provide a
  mechanism for assessing the level of interaction
  between system modes and system state variables.
• Following 1-2, participation factors are
  considered in two basic senses. In the first
  sense, a participation factor measures the
  relative contribution of a mode to a state. In
  the second, a participation factor measures the
  relative contribution of a state to a mode. To
  motivate their original definition of
  participation factors, the authors of 1-2
  assumed a particular initial condition and
  calculated the relative contribution of modes to
  a state variable or of state variables to a mode
  (depending on the sense of participation factor
  being considered). It isn't clear at the outset
  that these two senses should lead to identical
  formulas for participation factors. However, the
  precise definitions in 1-2 for these two
  senses of participation factors did indeed result
  in identical mathematical expressions. The same
  conclusion is found to apply in the present work,
  under assumptions valid for a large class of
  problems.
• The new definitions of participation factors
  given here (see 3 for a detailed discussion)
  involve averaging operations with respect to the
  initial state, which means that the initial
  condition is taken to be an uncertain vector. The
  averaging operation can be a set-theoretic
  average or a probabilistic average, which is
  tantamount to assuming a probability distribution
  for the initial state vector and then taking an
  expectation of a measure of the relative
  contribution of modes to a state variable or of
  state variables to a mode. The more versatile of
  these two averaging approaches is found to be the
  probabilistic approach.

3

• When the probability law of the initial state is
  symmetric with respect to the coordinate axes,
  the new definition reduces to the original
  definition of 1-2. However, the new and
  original definitions generally do not coincide
  otherwise.
• The next issue discussed in the presentation is
  the use of participation factors in controller
  and sensor placement. The issue of controller
  placement is of great significance in electric
  power networks, especially with respect to
  placement of expensive FACTS devices. It is
  important to have methodologies that lead to
  optimal, or nearly optimal, placement of
  controllers in the sense that a maximum impact
  results on the system modes whose damping level
  needs to be improved. One idea that appears in
  this context is that placement of controllers in
  the physical vicinity of state variables with
  high participation in a particular mode leads to
  maximum damping improvement when using a local
  controller which is feeding back local
  information. This fact is proved in the talk.
  Also, the applicability of an optimal control
  technique based on low authority controllers and
  convex optimization is discussed. The issue of
  combined controller and sensor placement in power
  networks and other systems is considered in 4.
  That reference also gives a generalized notion of
  participation factors, taking into account the
  inputs, outputs and state variables along with
  the system modes.
• The last topic in the presentation deals with
  connections with robust nonlinear control for
  systems close to their stability boundary and
  with detection of such situations in a
  nonparametric (i.e., non-model based) way. The
  detection of near instability conditions using
  signal-based methods is very important since
  systems close to their stability boundary are
  highly sensitive to modeling errors, and since
  crossing the stability boundary can lead to
  system disintegration. Using a concept of noisy
  precursors, it may be possible to detect
  impending instability prior to its onset 5.
  Stabilization using well-placed sensors and
  controllers can then be triggered to move the
  system away from the stability boundary.

4

• References
• 1 I.J. Perez-Arriaga, G.C. Verghese and F.C.
  Schweppe, Selective modal analysis with
  applications to electric power systems, Part I
  Heuristic introduction, IEEE Trans. Power
  Apparatus and Systems, Vol. 101, 1982, pp.
  3117-3125.
• 2 G.C. Verghese, I.J. Perez-Arriaga and F.C.
  Schweppe, Selective modal analysis with
  applications to electric power systems, Part II
  The dynamic stability problem, IEEE Trans. Power
  Apparatus and Systems, Vol. 101, 1982, pp.
  3126-3134.
• 3 E.H. Abed, D. Lindsay and W.A. Hashlamoun,
  On participation factors for linear systems,
  Automatica, Vol. 36, No. 10, October 2000, pp.
  1489-1496.
• 4 H. Yaghoobi and E.H. Abed, Optimal actuator
  and sensor placement for modal and stability
  monitoring, Proc. American Control Conference,
  June 2-4, 1999, San Diego, pp. 3702-3707.
• 5 T. Kim and E.H. Abed, Closed-loop monitoring
  systems for detecting impending instability,
  IEEE Trans. Circuits and Systems, - I
  Fundamental Theory and Applications, Vol. 47, No.
  10, October 2000, pp. 1479-1493.

5
Participation Factors

• The main subject considered here is (modal)
  participation factors an important element of
  Selective Modal Analysis (SMA) (Verghese,
  Perez-Arriaga and Schweppe, 1982).
• 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 will revisit the concept of participation
  factors, and consider why it is useful in
  actuator placement.

6
Basic Background and Original 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 the
  participation of a particular mode (i.e.,
  eigenmode) in a state variable. If the states are
  physical variables, this lets us study the
  influence of system modes on physical components.

7

• 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.

8

• 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.

9

• 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.

10

• 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
• Original logic/motivation for definition
• x(t) eAtx0
• where x0 is the initial condition. This yields
• x(t) å (li x0) exp(lit) ri

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• Now suppose the initial condition x0 is ek, the
  unit vector along the k-th coordinate axis. Then
  the evolution of the k-th state becomes
• xk(t) åi (lik rik) exp(lit)
• This formula indicates that pki can be viewed as
  the relative participation of the i-th mode in
  the k-th state at t0.
• A similar motivation was given for viewing pki as
  the relative participation of the k-th state in
  the i-th mode at t0.

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New Approach and New Definitions

• 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 take two approaches to define participation
  factors accounting for uncertainty in initial
  condition

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Two approaches to handling uncertainty in initial
condition

• Set-valued uncertainty and average relative
  participation at time 0
• Probabilistic uncertainty and mean participation
  at time 0
• The second approach is more generally applicable.

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Set-valued uncertainty in initial condition

• Take x0 to lie in a connected set S containing
  the origin x0 2 S
• The case of greatest interest is when SRn.
• Sets S that are symmetric in the sense of the
  next definition are of particular significance.
• Definition 1. The set S is symmetric if it is
  symmetric with respect to each of the hyperplanes
  xk0, k1,,n. That is, for any k21,,n and
  z(z1,,zk,,zn)2 Rn, z2 S implies that
• (z1,,-zk,,zn)2 S.

15

• Note that the average contribution at time t0 of
• the i-th mode to state xk vanishes and so is not
  useful as a notion of participation factor
• avg (li x0) rik 0
• Definition 2. The participation factor for the
  mode associated with li in state xk with respect
  to a symmetric uncertainty set S is
• pki avgx0 2 S (li x0) rik / x0k
• whenever this quantity exists. Here, avgx0 2 S is
  an operator that computes the average of a
  function over the set S (in the sense of Cauchy
  principal value).

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• This quantity measures the average relative
  contribution at time t0 of the i-th mode to
  state xk. In the definition, the i-th mode is
  interpreted as the eli t term in the expansion
  for xk.
• Also, the denominator on the right side of the
  definition is simply the sum of the contributions
  from all modes to xk(t) at t0.

17

• Next, we evaluate Definition 2 for pki under the
  assumption of a symmetric uncertainty set S.
• Let Vol(S) sx0 2 S dx0
• denote the volume of the set S. If S has infinite
  volume, then the construction below is performed
  for a finite symmetric subset, and then a limit
  is taken as discussed in Definition 2. From Def.
  2,
• pki avg (lik x0k) rik/x0k avgx0åj¹ k (lij
  x0j) rik/x0k
• lik rik sx0 2 S åj¹ k (lij x0j)
  rik/x0k dx0 / Vol(S)
• lik rik åj¹ k lij rik sx0 2 S x0j/x0k dx0
  / Vol(S)
• lik rik

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• The last step follows from the observation that,
  because S is symmetric according to Definition 1,
• sx0 2 S x0j/x0k dx0 0
• for any j¹ k, where the integral is interpreted
  in the sense of Cauchy principal value.
• Thus Definition 2 for pki reduces to the
  original 1982 Verghese/Perez-Arriaga/Schweppe
  definition in the case of a symmetric uncertainty
  set S.
• Problem For nonsymmetric S, Def. 2 isnt useful.

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Probabilistic uncertainty in initial condition

• Assumption 1. The components x0j, j1,,n,
• of the initial condition vector x0 are
  independent random variables with probability
  density functions fX0j(x0j).
• Definition 3 (Participation of modes in states).
  Suppose Assumption 1 holds. Define pki, the
  participation at time t0 of the mode li in
• state xk, as the expectation
• pki E (li x0) rik / x0k
• whenever this expectation exists.

20

• In applications of this definition to specific
  problems, it is useful to rewrite the defining
  formula as follows
• pki E (li x0) rik / x0k
• E åj1n (lij x0j) rik / x0k
• E (lik x0k) rik / x0k E åj¹ k (lij
  x0j) rik / x0k
• lik rik åj¹ k lij rik E x0j / x0k .
  ()
• Note It is straightforward to verify that this
  formula for pki is dimensionless and doesnt
  depend on the units chosen for state variables.

21

• The following well-known fact from probability
  theory will be useful in the examples below.
• Lemma 1. (Law of Iterated Expectation) Let X and
  Y be random variables and let g(X,Y) be a
  function of X and Y. Then
• Eg(X,Y)EY EX g(X,Y)Y ,
• where EX and EY emphasize that the inner
  expectation is conditioned on Y and taken with
  respect to X, and the outer expectation is
  unconditional and taken with respect to Y.

22

• For the purposes of this paper, the following
  observation, based on Lemma 1, is particularly
  valuable.
• Remark. If X and Y are independent random
  variables and the probability density of at least
  one of X or Y is symmetric with respect to the
  origin, then
• E XY 0, and
• E X / Y 0.

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• Example 1. Suppose that the marginal densities
• fX0j(x0j) are symmetric with respect to x0j0,
  i.e., that they are even functions of
• x0j, for j1,,n. With this assumption, Eq. ()
  gives
• pki lik rik åj¹ k lij rik E x0j / x0k
• lik rik
• Thus, again the new definition reduces to the
  original 1982 Verghese/Perez-Arriaga/Schweppe
  definition in the case of symmetric uncertainty,
  here embodied in the symmetry and independence of
  the marginal densities.

24

• Example 2. Suppose that the state variables are
  restricted to be nonnegative, and, more
  specifically, that the density functions
  fX0j(x0j) are Rayleigh
• fX0j(x)x/bj exp -x2/2bj
• if xx0j 0, and fX0j(x)0 otherwise.
• The mean of X0j is EX0j p bj / 20.5,
  j1,,n.
• After some calculations, pki for this example is
  found to be
• pki lik rik åj¹ k (p/2) (bj/bk)0.5 lij rik
• Thus, this time the new definition does not
  reduce to the original 1982 Verghese/Perez-Arriaga
  /Schweppe definition --- note that the
  uncertainty was nonsymmetric.

25
Participation of states in modes

• Denote by V the matrix of right eigenvectors of
  A
• Vv1 v2 vn
• Because of the normalization of right and left
  eigenvectors, we have that V-1 has the lj as its
  rows.
• Perform the change of variables
• z V-1 x.
• Then z follows the dynamics
• dz/dt (t) V-1AVz(t)
• L z
• where Ldiag l1,, ln

26

• The new states zi, i1,,n evolve according to
• zi(t) eli t z0i,
• To define the participation of the original
  states xk, k1,,n in the mode zi, z0i is written
  in terms of x0 and then an appropriate
  expectation is taken
• zi(t) eli t li x0
• eli t åj1n (lij x0j).
• This equation, which shows the contribution of
  each component of the initial state x0j, j1,,n,
  to the i-th mode, motivates the following
  definition for the participation factor governing
  participation of states in modes.

27

• Definition 4 (Participation of states in modes).
  Suppose Assumption 1 holds. Define pki, the
  participation at time t0 of the state xk in the
  mode li, as the expectation
• pki E lik x0k / z0i
• whenever this expectation exists.
• After some work, this gives
• pki lik rik åj¹ k lij rjk E z0j / z0i
• Again, in the case of symmetric uncertainty, we
  can show that this formula gives the original
  notion of participation.

28
Controller Placement Implications?

• Consider the linear control system
• dx/dt Ax(t) Bu(t)
• where u(t) is a vector of controls, uk(t),
  k1,,nu.
• Which of the controls has the highest
  effectiveness in moving a given system mode?
• Conventional wisdom
• Choose controller(s) near the physical state
  variable(s) that participates most in the mode of
  interest.

29
An Analytical View of Controller Placement
vis-à-vis Participations

• Following Hassibi, How and Boyd (1999), we focus
  on the the setting of low-authority controller
  (LAC) design the actuators have limited
  authority, and hence cannot significantly shift
  the eigenvalues of the system.
• In this setting, the closed-loop eigenvalues can
  be approximated analytically using perturbation
  theory for linear operators.

30

• A Naïve Calculation
• Suppose BI, nun, and each control uk-gk xk.
  Then
• dx/dt Ax(t) - G x(t),
• where G diag g1,,gn .
• What is the influence of each gain gk on each li?
• We need the following result from Kato (1982)
  Let AA(g) have distinct e-vals depending
  smoothly on the gk. Then
• li(g) liåk1n (li Ak ri / li ri) gk
  O(g2)
• where
• Ak A(0) / gk, k1,,n.

31

• For this setting, this quickly leads to
• li(g) liåk1n (pki / li ri) gk O(g2)
• li åk1n pkigk / li ri O(g2)
• So the k giving a larger pki ostensibly leads to
  a greater effectiveness per unit gain in
  providing eigenvalue mobility.

32

• Notes
• Low authority controller setting was needed in
  this calculation.
• This result is related to an interpretation of
  participation factors as eigenvalue sensitivities
  to diagonal entries of state dynamics matrix (Van
  Ness et al.).
• More interesting control structures can be
  considered input/output participation factors
  (PhD thesis of Yaghoobi, Univ. Maryland, 2000)
  paper of Boyd et al. noted above seeks sparse
  controller structures via optimization.

33

• (4) Other participation factor concepts have been
  introduced by other groups
• Fouad, Vittal et al. introduced a concept of
  higher order participation factors
• Verghese et al. considered participation factors
  for limit cycles of nonlinear systems
• Control energy concept etc.

34
The Input/Output Selection Problem

• The I/O selection problem concerns making
  decisions on the
• number,
• placement, and
• type
• of actuators and sensors
• A nice review of techniques on input/output
  selection is given by van de Wal and de Jager
  (Automatica 2001).

35
The Input/Output Selection Problem, Cntd.

• In the electric power field, we need a careful
  re-examination of sensor/actuator placement
  issues in light of related activities in other
  fields.
• The computational complexity requires innovative
  techniques.
• This is a hard problem.

36
Border of Stability Issues/Control

• Nonlinear systems are pervasive in engineering
  and natural sciences.
• Nonlinear systems exhibit amazingly rich behavior
  compared to linear systems which cannot even
  exhibit robust periodic orbits. Interest in
  nonlinear systems has been steadily growing
  because of the success in modeling a variety of
  phenomena across many disciplines.
• Nonlinear systems show changes in steady state
  behavior as parameters vary. This is known as
  bifurcation.

37

• Technically, bifurcation is a change in the
  number and/or nature of steady state solutions as
  a parameter is slowly changed.
• Bifurcations are triggered by a loss of
  linearized stability.
• Bifurcations are possible in both smooth and
  nonsmooth systems.
• Chaos is an intriguing dynamical behavior -
  seemingly random behavior for a deterministic
  system. Bifurcation and chaos are linked since
  chaos often arises through a cascading sequence
  of bifurcations.

38

• Important to remember
• Existence of bifurcation can be checked by linear
  analysis around nominal operating condition.
  However,
• Severity of a bifurcation in terms of
  implications for system operation depends
  strongly on system nonlinearities.

39
Supercritical stationary bifurcation
Bifurcated equilibria
State x
Nominal equilibrium
Bifurcation parameter m
40
Subcritical stationary bifurcation
Bifurcated equilibria
State x
Nominal equilibrium
Bifurcation parameter m
41

• Notes
• Subcritical bifurcations generally lead to
  departure from nominal operation and possibly
  hysteresis or chaotic or hunting motions. Also
  called dangerous or hard bifurcations.
• Supercritical bifurcations are less severe. Also
  called safe or soft bifurcations.
• Local bifurcation control can entail using
  feedback to render supercritical an originally
  subcritical bifurcation.

42
An Example from Another Field Nonlinear Active
Controlof Jet Engine Stall

• The drive toward lighter jet engines is
  motivating studies into operability of the
  engine's axial flow compressor close to its
  maximum pressure rise.
• The compressor is only marginally stable in this
  condition, and can stall under small or moderate
  flow disturbances.

43
Compressor stall and bifurcations
44
Compressor Stall Control, Cntd.

• Active control laws were designed for stabilizing
  jet engine stall, by applying bifurcation control
  to nonlinear models of axial flow compressors.
• The controllers employ one dimensional actuation
  in the form of throttle feedback.

45
Compressor Stall Control, Cntd.

• The project has shown the superiority of
  nonlinear bifurcation-theory based methods for
  design of simple one-dimensional throttle
  controllers. 
• The controls allow operation up to and past the
  stall stability limit ("stall line" in
  schematic).
• The control designs have been been expanded on
  and implemented at university and industry
  laboratories.

46
Detection of Impending Instability using Noise
Bifurcation Precursors

• Some observations
• Noisy precursors give a robust nonparametric
  indicator of impending instability.
• Resonant perturbations delay or advance
  bifurcations
• 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