Instability Monitoring and Control of Power Systems

1. 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 abed@umd.edu

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

9. Closed-loop precursor-based monitoring systems – Illustration for closeness to zero eigenvalue

10. 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.

11. 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.

12. 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.

13. 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.

14. 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.

15. 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.

16. 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 statesand 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.

17. 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

18. 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.

19. 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.

20. 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?

21. 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 problemsbefore 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.