Participation Factors, Actuator Placement, and Some Nonlinear Control Issues in Power Systems

Eyad H. AbedElec. and Comp. Eng. and Inst. Systems Res.

University of Maryland

Presentation at EPRI/NSF Workshopon Global Dynamic Optimization of the Power Grid

April 10-12, 2002, Playcar, Mexico

SummaryIn 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.

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.

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.

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.

Consider a linear time-invariant system

dx/dt = Ax(t),

where x2 Rn, and A is n£ n with n distinct eigenvalues (1,2,…,n).

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.

Basic Background and Original Definition

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 (1,2,…,n), respectively.

Using this criterion, one would say that

the mode associated with i is significantly involved in the state xk if ri

k 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 (1,2,…,n), respectively.

The right and left eigenvectors are taken to satisfy the normalization li rj = ij (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 ri

k

Original logic/motivation for definition…x(t) = eAtx0

where x0 is the initial condition. This yields

x(t)= (li x0) exp(it) ri

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)= (lik ri

k) exp(it)

This formula indicates that pki can be viewed as the relative participation of the i-th mode in the k-th state at t=0.

A similar motivation was given for viewing pki as the relative participation of the k-th state in the i-th mode at t=0.

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:

New Approach and New Definitions

• 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.

Two approaches to handling uncertainty in initial condition:

Take x0 to lie in a connected set S containing the origin: x0 2 SThe case of greatest interest is when S=Rn.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 xk=0, k=1,…,n. That is, for any k2{1,…,n} and z=(z1,…,zk,…,zn)2 Rn, z2 S implies that

(z1,…,-zk,…,zn)2 S.

Set-valued uncertainty in initial condition

Note that the average contribution at time t=0 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 i in state xk with respect to a symmetric uncertainty set S is

pki := avgx0 2 S (li x0) rik / x0

k

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).

This quantity measures the average relative contribution at time t=0 of the i-th mode to state xk. In the definition, the i-th mode is interpreted as the ei 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 t=0.

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 x0

k) rik/x0

k + avgx0j k (lij x0

j) rik/x0

k

= lik ri

k + sx0 2 S j k (lij x0

j) rik/x0

k dx0 / Vol(S)

= lik ri

k + j k lij ri

k sx0 2 S x0j/x0

k dx0 / Vol(S)

= lik ri

k

The last step follows from the observation that, because S is symmetric according to Definition 1,

sx0 2 S x0j/x0

k dx0 =0

for any j 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 isn’t useful.

Assumption 1. The components x0j, j=1,…,n,

of the initial condition vector x0 are independent random variables with probability density functions fX0j(x0

j).

Definition 3 (Participation of modes in states). Suppose Assumption 1 holds. Define pki, the participation at time t=0 of the mode i in

state xk, as the expectation

pki := E { (li x0) rik / x0

k }whenever this expectation exists.

Probabilistic uncertainty in initial condition

In applications of this definition to specific problems, it is useful to rewrite the defining formula as follows:pki = E { (li x0) ri

k / x0k }

= E { j=1n (li

j x0j) ri

k / x0k }

= E { (lik x0

k) rik / x0

k } + E {j k (lij x0

j) rik / x0

k }

= lik ri

k + j k lij ri

k E {x0j / x0

k }. (**)

Note: It is straightforward to verify that this formula for pki is dimensionless and doesn’t depend on the units chosen for state variables.

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

E{g(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.

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.

Example 1. Suppose that the marginal densities

fX0j(x0j) are symmetric with respect to x0

j=0, i.e., that they are even functions of

x0j, for j=1,…,n. With this assumption, Eq. (**) gives

pki = lik ri

k + j k lij ri

k E {x0j / x0

k }

= lik ri

k

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.

Example 2. Suppose that the state variables are restricted to be nonnegative, and, more specifically, that the density functions fX0j(x0

j) are Rayleigh:

fX0j(x)=x/bj exp{ -x2/2bj}

if “x”=x0j ¸ 0, and fX0j(x)=0 otherwise.

The mean of X0j is E{X0

j} = { bj / 2}0.5, j=1,…,n.

After some calculations, pki for this example is found to be

pki = li ri

k + j k (/2) (bj/bk)0.5 li ri

k Thus, this time the new definition does not reduce to the original 1982 Verghese/Perez-Arriaga/Schweppe definition --- note that the uncertainty was nonsymmetric.

Denote by V the matrix of right eigenvectors of A:V=[v1 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 variablesz := V-1 x.Then z follows the dynamicsdz/dt (t) = V-1AVz(t) =: z

where =diag[ 1,…, n]

Participation of states in modes

The new states zi, i=1,…,n evolve according to

zi(t) = ei t z0i,

To define the participation of the original states xk, k=1,…,n in the mode zi, z0

i is written in terms of x0 and then an appropriate expectation is taken:

zi(t) = ei t li x0

= ei t j=1n (li

j x0j).

This equation, which shows the contribution of each component of the initial state x0

j, j=1,…,n, to the i-th mode, motivates the following definition for the participation factor governing participation of states in modes.

Definition 4 (Participation of states in modes). Suppose Assumption 1 holds. Define pki, the participation at time t=0 of the state xk in the mode i, as the expectation

pki = E{ lik x0

k / z0i }

whenever this expectation exists.

After some work, this gives

pki = lik ri

k + j k lij rj

k E {z0j / z0

i }Again, in the case of symmetric uncertainty, we can show that this formula gives the original notion of participation.

Consider the linear control system

dx/dt = Ax(t) + Bu(t)

where u(t) is a vector of controls, uk(t), k=1,…,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.

Controller Placement Implications?

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.

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

A Naïve Calculation:

Suppose B=I, nu=n, and each control uk=-k xk. Thendx/dt = Ax(t) - x(t),

where = diag [1,…,n] .

What is the influence of each gain k on each i?We need the following result from Kato (1982): Let A=A() have distinct e-vals depending smoothly on the k. Then

i()= i+k=1n (li Ak ri / li ri) k + O(||||2)

where

Ak := A(0) / k, k=1,…,n.

For this setting, this quickly leads to

i()= i+k=1n (pki / li ri) k + O(||||2)

= i+[ k=1n pkik ] / li ri + O(||||2)

So the k giving a larger pki ostensibly leads to a greater effectiveness per unit gain in providing eigenvalue mobility.

Notes:(1) Low authority controller setting was needed in

this calculation.(2) This result is related to an interpretation of

participation factors as eigenvalue sensitivities to diagonal entries of state dynamics matrix (Van Ness et al.).

(3) 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.

(4) Other participation factor concepts have been introduced by other groups:

o Fouad, Vittal et al. introduced a concept of higher order participation factors

o Verghese et al. considered participation factors for limit cycles of nonlinear systems

o Control energy concept … etc.

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).

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.

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.

• 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.

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.

Supercritical stationary bifurcation

State x

Bifurcation parameter

Nominalequilibrium

Bifurcatedequilibria

Subcritical stationary bifurcation

State x

Bifurcation parameter

Nominalequilibrium

Bifurcatedequilibria

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.

An Example from Another Field: Nonlinear Active Control

of 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.

Compressor stall and bifurcations

Compressor Stall Control, Cnt’d.

• 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.

Compressor Stall Control, Cnt’d.

• 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.

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