Signal Processing Methods for Estimating Small-Signal Dynamic Properties from Measured Responses

  • Daniel Trudnowski
  • John Pierre
Part of the Power Electronics and Power Systems book series (PEPS)


Power system small-signal electromechanical dynamic properties are often described using linear system concepts. The underlying hypothesis is that small motions of the system can be described by a set of ordinary differential equations. Modal analysis of these governing equations provides considerable insight into the stability properties of the system. Over the past two decades, many signal processing techniques have been developed to conduct modal analysis using only time-synchronized actual system measurements. Some techniques are appropriate for transient signals, others are for ambient signal conditions, and some are for conditions where a known probing signal is exciting the system. In this chapter, an overview of many of the more successful analysis techniques is presented. The theoretical basis for these methods is described as well as application properties and performance. Examples include computer simulations and actual system experiments from the western North American power system. Analysis goals center on estimating the modal properties of the system including modal frequency, damping, and shape.


Power System Power Spectral Density Mode Shape Ambient Noise Mode Estimate 
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.



The authors wish to acknowledge the contribution of the many graduate students over the years. Also, the technical leadership of Dr. John Hauer of Pacific Northwest National Laboratory (retired) and Mr. Bill Mittelstadt of the Bonneville Power Administration (retired) are acknowledged. Much of this work was supported by the US Department of Energy; the authors wish to thank Mr. Phil Overholt for his support.


  1. 1.
    DSIToolbox. Posted as freeware at Scholar
  2. 2.
    M. Parashar, J. Mo,“Real Time Dynamics Monitoring System (RTDMS™): Phasor applications for the control room”, 42nd Annual Hawaii International Conference on System Sciences (HICSS-42), Big Island, Hawaii, January 5–8, 2009 (accepted).Google Scholar
  3. 3.
    D. J. Trudnowski, J. E. Dagle, “Effects of generator and static-load nonlinearities on electromechanical oscillations,” IEEE Transactions on Power Systems, vol. 12, no. 3, pp. 1283–1289, Aug. 1997.CrossRefGoogle Scholar
  4. 4.
    G. Rogers, Power System Oscillations, Kluwer Academic Publishers, Boston, 2000.Google Scholar
  5. 5.
    P. Kundur, Power System Stability and Control, New York: McGraw-Hill, Inc., 1994.Google Scholar
  6. 6.
    J. F. Hauer, R. L. Cresap, “Measurement and modeling of pacific AC intertie response to random load switchings,” IEEE Transactions on PAS, vol. PAS-100, no. 1, Jan. 1981.Google Scholar
  7. 7.
    J. Hauer, D. Trudnowski, J. DeSteese, “A perspective on WAMS analysis tools for tracking of oscillatory dynamics,” IEEE Power Engineering Society General Meeting, paper no. PESGM2007-001391, June 2007.Google Scholar
  8. 8.
    J. F. Hauer, C. J. Demeure, L. L. Scharf, ”Initial results in Prony analysis of power system response signals,” IEEE Transactions on Power Systems, vol. 5, no. 1, pp. 80–89, Feb. 1990.CrossRefGoogle Scholar
  9. 9.
    L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis, Addison Wesley Publishing Company, Inc., 1991.Google Scholar
  10. 10.
    D. A. Pierre, D. J. Trudnowski, J. F. Hauer, “Identifying reduced-order models for large nonlinear systems with arbitrary initial conditions and multiple outputs using Prony signal analysis,” Proceedings of the 1990 American Control Conference, vol. 1, pp. 149–154, May 1990.Google Scholar
  11. 11.
    J. F. Hauer, “Application of Prony analysis to the determination of modal content and equivalent models for measured power system response,” IEEE Transactions on Power Systems, vol. 6, no. 3, pp. 1062–1068, Aug. 1991.CrossRefGoogle Scholar
  12. 12.
    D. A. Pierre, D. J. Trudnowski, J. F. Hauer, “Identifying linear reduced-order models for systems with arbitrary initial conditions using Prony signal analysis,” IEEE Transactions on Automatic Control, vol. 37, no. 6, pp. 831–835, June 1992.CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    D. J. Trudnowski, M. K. Donnelly, J. F. Hauer, “Advances in the identification of transfer function models using Prony analysis,” Proceedings of the 1993 American Control Conference, Vol. 2, pp. 1561–1562, June 1993.Google Scholar
  14. 14.
    J. R. Smith, F. Fatehi, C. S. Woods, J. F. Hauer, D. J. Trudnowski, “Transfer function identification in power system applications,” IEEE Transactions on Power Systems, vol. 8, no. 3, pp. 1282–1290, Aug. 1993.CrossRefGoogle Scholar
  15. 15.
    D. J. Trudnowski, “Order reduction of large-scale linear oscillatory system models,” IEEE Transactions on Power Systems, vol. 9, no. 1, pp. 451–458, Feb. 1994.CrossRefGoogle Scholar
  16. 16.
    D. Trudnowski, J. Johnson, J. Hauer, “SIMO system identification from measured ringdowns,” Proceedings of the 1998 American Control Conference, pp. 2968–2972, June 1998.Google Scholar
  17. 17.
    D. J. Trudnowski, J. M. Johnson, J. F. Hauer, “Making Prony analysis more accurate using multiple signals,” IEEE Transactions on Power Systems, vol. 14, no.1, pp. 226–231, Feb. 1999.CrossRefGoogle Scholar
  18. 18.
    I. Kamwa, R. Grondin, E. J. Dickinson, S. Fortin, “A minimal realization approach to reduced-order modelling and modal analysis for power system response signals,” IEEE Transactions on Power Systems, vol. 8, no. 3, pp. 1020–1029, Aug. 1993.CrossRefGoogle Scholar
  19. 19.
    J. Sanchez-Gasca, J. Chow, “Computation of power system low-order models from time domain simulations using a Hankel matrix,” IEEE Transactions on Power Systems, vol. 12, no. 4, pp. 1461–1467, Nov. 1997.CrossRefGoogle Scholar
  20. 20.
    L. Guoping, J. Quintero, V. Venkatasubramaniar, “Oscillation monitoring system based on wide area synchrophasors in power systems,” Bulk Power System Dynamics and Control – VII. Revitalizing Operational Reliability, 2007 iREP Symposium, pp. 1-13, IEEE Digital Object Identifier 10.1145/1176254.1176261, Aug. 19–24, 2007.Google Scholar
  21. 21.
    J. Sanchez-Gasca, J. Chow, “Performance comparison of three identification methods for the analysis of electromechanical oscillations,” IEEE Transactions on Power Systems, vol. 14, no. 3, pp. 995–1002, Aug. 1999.CrossRefGoogle Scholar
  22. 22.
    J. W. Pierre, R. F. Kubichek, “Spectral Analysis: Analyzing a Signal Spectrum,” Tektronix Application Note,, 2002.
  23. 23.
    J. G. Proakis, D. G. Manolakis, Digital Signal Processing Principles, Algorithms, and Applications, 4th ed., Prentice Hall, 2007.Google Scholar
  24. 24.
    J. W. Pierre, D. J. Trudnowski, M. K. Donnelly, “Initial results in electromechanical mode identification from ambient data,” IEEE Transactions on Power Systems, vol. 12, no. 3, pp. 1245–1251, Aug. 1997.CrossRefGoogle Scholar
  25. 25.
    R. W. Wies, J. W. Pierre, D. J. Trudnowski, “Use of ARMA block processing for estimating stationary low-frequency electromechanical modes of power systems,” IEEE Transactions on Power Systems, vol. 18, no. 1, pp. 167–173, Feb. 2003.CrossRefGoogle Scholar
  26. 26.
    M. G. Anderson, N. Zhou, J. W. Pierre, R. W. Wies, “Boostrap-based confidence interval estimates for electromechanical modes from multiple output analysis of measured ambient data,” IEEE Transactions on Power Systems, vol. 20, no. 2, pp. 943–950, May 2005.CrossRefGoogle Scholar
  27. 27.
    N. Zhou, J. W. Pierre, R. W. Wies, “Estimation of low-frequency electromechanical modes of power systems from ambient measurements using a subspace method,” Proceedings of the North American Power Symposium, Oct. 2003.Google Scholar
  28. 28.
    D. Trudnowski, J. Pierre, N. Zhou, J. Hauer, M. Parashar, “Performance of three mode-meter block-processing algorithms for automated dynamic stability assessment,” IEEE Transactions on Power Systems, vol. 23, no. 2, pp. 680–690, May 2008.CrossRefGoogle Scholar
  29. 29.
    L. Guoping, V. Venkatasubramaniar, “Oscillation monitoring from ambient PMU measurements by frequency domain decomposition,” Proceedings of IEEE International Symposium on Circuits and Systems, pp. 2821–2824, May 2008.Google Scholar
  30. 30.
    R. W. Wies, J. W. Pierre, D. J. Trudnowski, “Use of least-mean squares (LMS) adaptive filtering technique for estimating low-frequency electromechanical modes in power systems,” Proceedings of the IEEE Power Engineering Society General Meeting, vol. 2, pp. 1863–1870, June 2004.Google Scholar
  31. 31.
    N. Zhou, J. W. Pierre, D. J. Trudnowski, R. T. Guttromson, “Robust RLS methods for online estimation of power system electromechanical modes,” IEEE Transactions on Power Systems, vol. 22, no. 3, pp. 1240–1249, Aug. 2007.CrossRefGoogle Scholar
  32. 32.
    N. Zhou, D. Trudnowski, J. Pierre, W. Mittelstadt, “Electromechanical mode on-line estimation using regularized robust RLS methods,” IEEE Transactions on Power Systems, vol. 23, no. 4, pp.1670–1680, Nov. 2008.Google Scholar
  33. 33.
    J. F. Hauer, W. A. Mittelstadt, K. E. Martin, J. W. Burns, H. Lee, In association with the Disturbance Monitoring Work Group of the Western Electricity Coordinating Council, “Integrated dynamic information for the western power system: WAMS analysis in 2005,” Chapter 14 in the Power System Stability and Control volume of The Electric Power Engineering Handbook, edition 2, L. L. Grigsby ed., CRC Press, Boca Raton, FL, 2007.Google Scholar
  34. 34.
    N. Zhou, J. W. Pierre, and J. F. Hauer, “Initial results in power system identification from injected probing signals using a subspace method,” IEEE Transactions on Power Systems, vol. 21, no. 3, pp. 1296–1302, Aug. 2006.CrossRefGoogle Scholar
  35. 35.
    P. Stoica and R. Moses, Introduction to Spectral Analysis, Prentice Hall, New Jersey, 1997.MATHGoogle Scholar
  36. 36.
    L. Ljung, System Identification Theory for the User, 2nd Ed., Prentice Hall, Upper Saddle River, NJ, 1999.Google Scholar
  37. 37.
    D. J. Trudnowski, J. R. Smith, T. A. Short, and D. A. Pierre, “An application of Prony methods in PSS design for multimachine systems,” IEEE Transactions on Power Systems, vol. 6, no. 2, pp. 118–126, Feb. 1991.CrossRefGoogle Scholar
  38. 38.
    D. Trudnowski, M. Donnelly, and E. Lightner, “Power-system frequency and stability control using decentralized intelligent loads,” Proceedings of the 2005/2006 IEEE PES T&D Conference and Exposition, pp. 1453–1459, May 2006.Google Scholar
  39. 39.
    D. J. Trudnowski, J. W. Pierre, and N. Zhou, “Performance and properties of ambient-data swing-mode estimation algorithms, version 1.0,” Report no. ENGR 2006-1, Engineering Dept., Montana Tech of the University of Montana, Butte, MT, USA, 2006.Google Scholar
  40. 40.
    F.K. Tuffner, “Computationally efficient weighted updating of statistical parameter estimates for time varying signals with application to power system identification,” Ph.D. dissertation, Department of Electrical and Computer Engineering, University of Wyoming, Laramie, WY, USA, 2008.Google Scholar
  41. 41.
    B. Efron, R. Tibshirani, “Bootstrap methods: another look at the jackknife,” The Annals of Statistics, vol. 7, no. 1, pp. 1–26, 1979.CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    D. Trudnowski, “Estimating electromechanical mode shape from synchrophasor measurements,” IEEE Transactions on Power Systems, vol. 23, no. 3, pp. 1188–1195, Aug. 2008.CrossRefGoogle Scholar
  43. 43.
    L. Dosiek, D. Trudnowski, J. Pierre, “New algorithms for mode shape estimation using measured data,” IEEE Power & Energy Society General Meeting, paper no. PESGM2008-001014, July 2008.Google Scholar
  44. 44.
    J. S. Bendat, A. G. Piersol, Engineering Applications of Correlation and Spectral Analysis, 2nd Ed., John Wiley & Sons, New York, 1993.MATHGoogle Scholar
  45. 45.
    D. Trudnowski, J. Hauer, J. Pierre, W. Litzenberger, D. Maratukulam, “Using the coherency function to detect large-scale dynamic system modal observability,” Proceedings of the 1999 American Control Conference, pp. 2886–2890, June 1999.Google Scholar

Copyright information

© Springer Science+Business Media,LLC 2009

Authors and Affiliations

  1. 1.Electrical Engineering DepartmentMontana Tech of the University of MontanaButteUSA

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