Probabilistic Transient Stability Assessment and On-Line Bayes Estimation

  • Elio Chiodo
  • Davide Lauria
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.


  1. 1.
    Kundur P (1993) Power system stability and control. Electric Power System Research, Power System Engineering Series, McGraw-HillGoogle Scholar
  2. 2.
    Amjady N (2004) A framework of reliability assessment with consideration effect of transient and voltage stabilities. IEEE Trans Power Syst 19(2):1005–1014CrossRefGoogle Scholar
  3. 3.
    Billinton R, Kuruganty PRS (1979) An approximate method for probabilistic assessment of transient stability. IEEE Trans Reliab 28(3):255–258CrossRefGoogle Scholar
  4. 4.
    Billinton R, Kuruganty PRS (1980) A probabilistic index for transient stability. IEEE Trans PAS 99(1):195–206Google Scholar
  5. 5.
    Billinton R, Kuruganty PRS (1981) Probabilistic assessment of transient stability. IEEE Trans PAS 100(5):2163–2170Google Scholar
  6. 6.
    Billinton R, Kuruganty PRS (1981) Probabilistic assessment of transient stability in a practical multimachine system. IEEE Trans PAS 100(5):3634–3641Google Scholar
  7. 7.
    Billinton R, Kuruganty PRS (1981) Protection system modelling in a probabilistic assessment of transient stability. IEEE Trans PAS 100(7):3664–3671Google Scholar
  8. 8.
    Anderson PM, Bose A (1983) A probabilistic approach to power system stability analysis. IEEE Trans PAS 102(4):2430–2439Google Scholar
  9. 9.
    Hsu YY, Chang CL (1988) Probabilistic transient stability studies using the conditional probability approach. IEEE Trans PAS 3(4):1565–1572Google Scholar
  10. 10.
    Anders GJ (1990) Probability concepts in electric power systems. Wiley, New YorkGoogle Scholar
  11. 11.
    Chiodo E, Gagliardi F, Lauria D (1994) A probabilistic approach to transient stability evaluation. IEE Proc Generat Transm Distrib 141(5):537–544CrossRefGoogle Scholar
  12. 12.
    Chiodo E, Lauria D (1994) Transient stability evaluation of multimachine power systems: a probabilistic approach based upon the extended equal area criterion. IEE Proc Generat Transm Distrib 141(6):545–553CrossRefGoogle Scholar
  13. 13.
    Allella F, Chiodo E, Lauria D (2003) Analytical evaluation and robustness analysis of power system transient stability probability. Electr Eng Res Rep NR16:1–13Google Scholar
  14. 14.
    Chiodo E, Gagliardi F, La Scala M, Lauria D (1999) Probabilistic on-line transient stability analysis. IEE Proc Generat Transm Distrib 146(2):176–180CrossRefGoogle Scholar
  15. 15.
    Ayasun S, Liang Y, Nwankpa CO (2006) A sensitivity approach for computation of the probability density function of critical clearing time and probability of stability in power system transient stability analysis. Appl Math Comput 176:563–576MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Allella F, Chiodo E, Lauria D (2003) Transient stability probability assessment and statistical estimation. Electric Power Syst Res 67(1):21–33CrossRefGoogle Scholar
  17. 17.
    Pavella M, Murthy PG (1994) Transient stability of power systems. Theory and practice. Wiley, New YorkGoogle Scholar
  18. 18.
    Breipohl AM, Lee FN (1991) A stochastic load model for use in operating reserve evaluation. In: Proceedings of the 3rd international conference on probabilistic methods applied to electric power systems, London, 3–5 July 1991, IEE Publishing, LondonGoogle Scholar
  19. 19.
    Papoulis A (2002) Probability, random variables, stochastic processes. McGraw Hill, New YorkGoogle Scholar
  20. 20.
    Belzer DB, Kellogg MA (1993) Incorporating sources of uncertainty in forecasting peak power loads. A Monte Carlo analysis using the extreme value distribution (with discussion). IEEE Trans Power Syst 8(2):730–737CrossRefGoogle Scholar
  21. 21.
    Crow EL, Shimizu K (1988) Lognormal distributions. Marcel Dekker, New YorkMATHGoogle Scholar
  22. 22.
    Robert CP (2001) The Bayesian choice. Springer Verlag, BerlinMATHGoogle Scholar
  23. 23.
    Press SJ (2002) Subjective and objective Bayesian statistics: principles, models, and applications, 2nd edn. Wiley, New YorkCrossRefGoogle Scholar
  24. 24.
    O′Hagan A (1994) Kendall’s advanced theory of statistics: vol 2b, Bayesian inference. E. Arnold Editor, LondonGoogle Scholar
  25. 25.
    De Finetti B, Galavotti MC, Hosni H, Mura A (eds) (2008) Philosophical lectures on probability. Springer Verlag, BerlinMATHGoogle Scholar
  26. 26.
    Jin M (2009) Estimation of reliability based on zero-failure data. In: Proceedings of the 8th international conference on reliability, maintainability and safety, ICRMS 2009, 20–24 July 2009, pp 308–309Google Scholar
  27. 27.
    Martz HF, Hamm LL, Reed WH, Pan PY (1993) Combining mechanistic best-estimate analysis and level 1 probabilistic risk assessment. Reliab Eng Syst Saf 39:89–108CrossRefGoogle Scholar
  28. 28.
    Rohatgi VK, Saleh AK (2000) An Introduction to Probability and Statistics, 2nd edn. Wiley, New YorkGoogle Scholar
  29. 29.
    West M, Harrison J (1999) Bayesian forecasting and dynamic models. Springer Verlag, BerlinGoogle Scholar
  30. 30.
    Khan UA, Moura JMF (2008) Distributing the Kalman filter for large-scale systems. Part I. IEEE Trans Signal Process 56(10):4919–4935CrossRefMathSciNetGoogle Scholar
  31. 31.
    Robert CP, Casella G (2004) Monte Carlo statistical methods. Springer Verlag, BerlinMATHGoogle Scholar
  32. 32.
    Erto P, Giorgio M (2002) Generalised practical Bayes estimators for the reliability and the shape parameter of the Weibull distribution. In: Proceedings of PMAPS 2002: probabilistic methods applied to power systems, Napoli, ItalyGoogle Scholar
  33. 33.
    Chiodo E, Mazzanti G (2006) Bayesian reliability estimation based on a Weibull stress-strength model for aged power system components subjected to voltage surges. IEEE Trans Dielectric Electric Insulat 13(1):146–159CrossRefGoogle Scholar
  34. 34.
    Kim H, Singh C (2005) Power system probabilistic security assessment using Bayes classifier. Electric Power Syst Res 74:157–165CrossRefGoogle Scholar
  35. 35.
    Robert CP (2007) Bayesian core: a practical approach to computational bayesian statistics. Springer Verlag, BerlinMATHGoogle Scholar
  36. 36.
    Johnson NL, Kotz S, Balakrishnan N (1995) Continuous univariate distributions, 2nd edn. Wiley, New YorkMATHGoogle Scholar
  37. 37.
    Martz HF, Waller RA (1991) Bayesian reliability analysis. Krieger Publishing, Malabar, FLMATHGoogle Scholar
  38. 38.
    Allella F, Chiodo E, Lauria D, Pagano M (2001) Negative log-gamma distribution for data uncertainty modelling in reliability analysis of complex systems: methodology and robustness. Int J Qual Reliab Manag 18(3):307–323CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Electrical Engineering DepartmentUniversity of Naples Federico IINaplesItaly

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