Probabilistic Transient Stability Assessment and On-Line Bayes Estimation

Part of the Springer Series in Reliability Engineering book series (RELIABILITY)


It is a well-known fact that the increase in energy demand and the advent of the deregulated market mean that system stability limits must be considered in modern power systems reliability analysis. In this chapter, a general analytical method for the probabilistic evaluation of power system transient stability is discussed, and some of the basic contributes available in the relevant literature and previous results of the authors are reviewed. The first part of the chapter is devoted to a review of the basic methods for defining transient stability probability in terms of appropriate random variables (RVs) (e.g. system load, fault clearing time and critical clearing time) and analytical or numerical calculation. It also shows that ignoring uncertainty in the above parameters may lead to a serious underestimation of instability probability (IP). A Bayesian statistical inference approach is then proposed for probabilistic transient stability assessment; in particular, both point and interval estimation of the transient IP of a given system is discussed. The need for estimation is based on the observation that the parameters affecting transient stability probability (e.g. mean value and variances of the above RVs) are not generally known but have to be estimated. Resorting to “dynamic” Bayes estimation is based upon the availability of well-established system models for the description of load evolution in time. In the second part, the new aspect of on-line statistical estimation of transient IP is investigated in order to predict transient stability based on a typical dynamic linear model for the stochastic evolution of the system load. Then, a new Bayesian approach is proposed in order to perform this estimation: such an approach seems to be very appropriate for on-line dynamic security assessment, which is illustrated in the last part of this article, based on recursive Bayes estimation or Kalman filtering. Reported numerical application confirms that the proposed estimation technique constitutes a very fast and efficient method for “tracking” the transient stability versus time. In particular, the high relative efficiency of this method compared with traditional maximum likelihood estimation is confirmed by means of a large series of numerical simulations performed assuming typical system parameter values. The above results could be very important in a modern liberalized market in which fast and large variations are expected to have a significant effect on transient stability probability. Finally, some results on the robustness of the estimation procedure are also briefly discussed in order to demonstrate that the methodology efficiency holds irrespective of the basic probabilistic assumptions made for the system parameter distributions.


Stability Margin Load Demand Load Forecast Transient Stability Clearing Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of main symbols and acronyms


Bayesian confidence interval


Cumulative distribution function


Complementary standard Gaussian distribution function


Critical clearing time (T cr or T x )


Coefficient of variation


Set of observed data used for inference


Dynamic linear model


Expectation (or “mean value”) of the RV R


Extreme value distribution


Fault clearing time


Generic cdf


Generic pdf

g(ω), g(ω|D)

Prior and posterior pdf of a generic parameter ω

G(r, ϕ)

Gamma distribution with parameters (r, ϕ)


s-Independent and identically distributed (random variables)


Instability probability


Logarithm of the CCT

LF, L(D|β)

Likelihood function, conditional to given parameter β

L, L(t)

Load (at time t)


Maximum likelihood


Mean square error


Log-Normal distribution with parameters α and β


Normal (Gaussian) distribution with mean μ and SD σ


Probability density function


Random variable


Standard deviation of measurement errors in the DLM of the LCCT


Standard deviation of measurement errors in the DLM of the load

SD, σ

Standard deviation


Statistically independent


Standard deviation of the RV Y


Stability margin (i.e. the quantity u defined below), for a given fault


Denotes a variance

Tcr or Tx

Critical clearing time

Tcl or Ty

Fault clearing time


x  − α y )/(β x 2  + β y 2 )1/2


CV values of the CCT and FCT, respectively


Very short time


Variance of the RV R


Standard deviation of system equation error in the DLM of the LCCT


Standard deviation of system equation error in the DLM of the load


Peak value of the load L(t), over a given time interval


White Gaussian noise


Logarithm of the CCT


Logarithm of the FCT










Bayes estimate of a generic parameter ζ


ML estimate of a generic parameter ζ


Denotes a mean value (expectation)

\( \hat{\mu }_{k} \)

Bayes estimate of a “dynamic” parameter μ at time k


Euler–Gamma function


Expectation of the generic RV R


Standard normal cdf


1 − Φ(z) (Complementary standard Gaussian distribution function)


Standard normal pdf

R ∽ N(α,β)

The RV R has a Gaussian distribution N(α, β) (and similarly for the LN model, etc.)



The authors wish to thank sincerely prof. Francesco Gagliardi, of the University of Naples Federico II, Italy, for encouraging them in undertaking the researches which constituted the foundations of this study.


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Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Electrical Engineering DepartmentUniversity of Naples Federico IINaplesItaly

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