Skip to main content
SpringerLink
Log in

Bifurcation and Manifold Based Approach for Voltage and Oscillatory Stability Assessment and Control

  • Chapter
Applied Mathematics for Restructured Electric Power Systems

Part of the book series: Power Electronics and Power Systems ((PEPS))

  • 2882 Accesses

Abstract

This paper presents a framework based on a bifurcation and differential manifold approach that combines identification and tracing of both saddle node and Hopf bifurcation margin boundaries. The bifurcation related margin boundary could be traced along any control scenario in a multi-control parameter space combined with any given loading scenario. This is achieved by moving from one boundary point to the next without re-tracing the entire PV curve. In addition, to take into account damping, an integration-based approach to trace the critical eigenvalue near the imaginary axis is proposed. Indices are developed to identify the critical eigenvalue to be traced for further analysis. They take into account the rate of change as well as the direction of the movement. This method in combination with maximum real part calculation provides reliable information related to margins with respect to oscillatory stability and minimum damping requirements. Eigenvalue and eigenvector sensitivities (with respect to any explicit/implicit parameters) are by products of this approach. This approach can lead to cost-based fast monitoring and control for voltage and damping related margins.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. IEEE Power System Stability Subcommittee. Voltage Stability Assessment: Concepts, Practices, and Tools. IEEE/PES Publication SP101PSS, ISBN: 0780378695, 2002.

    Google Scholar 

  2. IEEE System Dynamic Performance Subcommittee. Voltage Stability of Power Systems: Concepts, Analytical Tools, and Industry Experience. IEEE/PES Publication 90TH0358-2-PWR, 1990.

    Google Scholar 

  3. Y. Mansour, ed. Suggested Techniques for Voltage Stability Analysis. IEEE PES Working Group on Voltage Stability, Publication 93TH0620-5-PWR, 1993.

    Google Scholar 

  4. C.W. Taylor. Power System Voltage Stability. McGraw-Hill, New York, 1994.

    Google Scholar 

  5. P. Kundur. Power System Stability and Control. McGraw-Hill, New York, 1994.

    Google Scholar 

  6. T. Van Cutsem and C. Vournas. Voltage Stability of Electric Power Systems. Kluwer Academic Publishers, Norwell, MA, 1990.

    Google Scholar 

  7. M. Ilic and J. Zaborszky. Dynamics and Control of Large Electric Power Systems. John Wiley & Sons, New York, 2000.

    Google Scholar 

  8. Proceedings of Bulk Power System Voltage Phenomena III— Seminar on Voltage Stability, Security, and Control. Davos, Switzerland, August 22–26, 1994.

    Google Scholar 

  9. Proceedings of the Symposium on Bulk Power System Dynamics and Control IV—Restructuring. Santorini, Greece, August 24–28, 1998.

    Google Scholar 

  10. Proceedings of Bulk Power System Dynamics and Control V— Security and Reliability in a Changing Environment. Onomichi, Japan, August 26–31, 2001.

    Google Scholar 

  11. V. Ajjarapu and B. Lee. Bibliography on Voltage Stability. IEEE Transactionz on Power Systems, 13:115–125, 1998.

    Article  Google Scholar 

  12. V. Ajjarapu. Iowa State University’s Web Based Voltage Stability Search Engine. Accessible at http://design-2.ee.iastate.edu/biblio/.

    Google Scholar 

  13. B.D. Hassard, N.D. Kazarinoff, and Y.-H. Wan. Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge, 1981.

    Google Scholar 

  14. I. Dobson, F. Alvarado, and C.L DeMarco. Sensitivity of Hopf Bifurcations to Power System Parameters. Proceedings of the 31StIEEE Conference on Decision and Control, 3:2928–2933, 1992.

    Article  Google Scholar 

  15. C.D. Vournas, M.A. Pai, and P.W. Sauer. The Effect of Automatic Voltage Regulation on the Bifurcation Evolution in Power Systems. IEEE Transactions on Power Systems, 11:1683–1688, 1996.

    Article  Google Scholar 

  16. W. Zhu, R.R. Mohler, R. Spee, W.A. Mittelstadt, and D. Maratukulam. Hopf Bifurcations in a SMIB Power System with SSR. IEEE Transactions on Power Systems, 11: 1579–1584, 1996.

    Article  Google Scholar 

  17. V. Ajjarapu and B. Lee. Bifurcation Theory and its Application to Nonlinear Dynamical Phenomena in an Electrical Power System. IEEE Transactions on Power Systems, 7:424–431, 1992.

    Article  Google Scholar 

  18. K. Kim, H. Schattler, V. Venkatasubramanian, J. Zaborszky, and P. Hirsch. Methods for Calculating Oscillations in Large Power Systems. IEEE Transactions on Power Systems, 12:1639–1648, 1997.

    Article  Google Scholar 

  19. F. Howell and V. Venkatasubramanian. Transient Stability Assessment with Unstable Limit Cycle Approximation. IEEE Transactions on Power Systems, 14:667–677, 1999.

    Article  Google Scholar 

  20. V. Ajjarapu and C. Christy. The Continuation Power Flow: a Tool for Steady State Voltage Stability Analysis. IEEE Transactions on Power Systems, 7:416–423, 1992.

    Article  Google Scholar 

  21. Z. Feng, V. Ajjarapu, and B. Long. Identification of Voltage Collapse through Direct Equilibrium Tracing. IEEE Transactions on Power Systems, 15:342–349, 2000.

    Article  Google Scholar 

  22. C.A. Canizares, A.C.Z. De Souza, and V.H. Quintana. Comparison of Performance Indices for Detection of Proximity to Voltage Collapse. IEEE Transactions on Power Systems, 11:1441–1450, 1996.

    Article  Google Scholar 

  23. C.A. Canizares and F.L. Alvarado. Point of Collapse and Continuation Methods for Large AC/DC Systems. IEEE Transactions on Power Systems, 8:1–8, 1993.

    Article  Google Scholar 

  24. A.C.Z. De Souza, C.A. Canizares, and V.H. Quintana. New Techniques to Speed Up Voltage Collapse Computations using Tangent Vectors. IEEE Transactions on Power Systems, 12:1380–1387, 1997.

    Article  Google Scholar 

  25. A.J. Flueck and J.R. Dondeti. A New Continuation Power Flow Tool for Investigating the Nonlinear Effects of Transmission Branch Parameter Variations. IEEE Transactions on Power Systems, 15:223–227, 2000.

    Article  Google Scholar 

  26. H.D. Chiang, A.J. Flueck, K.S. Shah, and N. Balu. CPFLOW: A Practical Tool for Tracing Power System Steady-State Stationary Behavior due to Load and Generation Variations. IEEE Transactions on Power Systems, 10:623–634, 1995.

    Article  Google Scholar 

  27. I. Dobson and L. Lu. Computing an Optimal Direction in Control Space to Avoid Saddle Bifurcation and Voltage Collapse in Electrical Power Systems. IEEE Transactions on Automatic Control, 37:1616–1620, 1992.

    Article  MathSciNet  Google Scholar 

  28. S. Greene, I. Dobson, and F.L. Alvarado. Contingency Ranking for Voltage Collapse via Sensitivities from a Single Nose Curve. IEEE Transactions Power Systems, 14:232–240, 1999.

    Article  Google Scholar 

  29. S. Greene, I. Dobson, and F.L. Alvarado. Sensitivity of the Loading Margin to Voltage Collapse with respect to Arbitrary Parameters. IEEE Transactions on Power Systems, 12:262–272, 1997.

    Article  Google Scholar 

  30. B. Long and V. Ajjarapu. The Sparse Formulation of ISPS and its Application to Voltage Stability Margin Sensitivity and Estimation. IEEE Transactions on Power Systems, 14:944–957, 1999.

    Article  Google Scholar 

  31. A. Berizzi, P. Finazzi, D. Dosi, P. Marannino, and S. Corsi. First and Second Order Methods for Voltage Collapse Assessment and Security Enhancement. IEEE/PES Publication PE-422-PWRS-0-01-1997, IEEE PES Winter Meeting, New York, 1997.

    Google Scholar 

  32. J. Barauin, T. Gomez, and F.L. Pagola. Estimating the Loading Limit Margin Taking into Account Voltage Collapse Areas. IEEE Transactions on Power Systems, 10:1952–1962, 1995.

    Article  Google Scholar 

  33. W.C. Rheinboldt. Numerical Analysis of Parameterized Nonlinear Equations. John Wiley & Sons, New York, 1986.

    Google Scholar 

  34. R. Dai and W.C. Rheinboldt. On the Computation of Manifolds of Fold Points for Parameter-Dependent Problems. SIAM Journal of Numerical Analysis, 27:437–446, 1990.

    Article  MathSciNet  Google Scholar 

  35. Y. Zhou and V. Ajjarapu. Local Parameterization Approach for Unified Time Domain Simulation of Power System Dynamics. Proceedings of IEEE Winter Power Meeting, New York, January 2002.

    Google Scholar 

  36. T. Van Cutsem. An Approach to Corrective Control of Voltage Instability using Simulation and Sensitivity. IEEE Transactions on Power Systems, 10:616–622, 1995.

    Article  Google Scholar 

  37. T. Van Cutsem and C.D. Vournas. Voltage Stability Analysis in Transient and Mid-Term Time Scales. IEEE Transactions on Power Systems, 11:146–154, 1996.

    Article  Google Scholar 

  38. T. Van Cutsem. Voltage Instability: Phenomena, Countermeasures, and Analysis Methods. Proceedings of the IEEE, 88:208–227, 2000.

    Article  Google Scholar 

  39. T. Van Cutsem, Y. Jacquemart, J.N. Marquet, and P. Pruvot. A Comprehensive Analysis of Mid-Term Voltage Stability. IEEE Transactions on Power Systems, 10:1173–1182, 1995.

    Article  Google Scholar 

  40. D. Kosterev, B. Mittelstadt, M. Viles, B. Tuck, J. Burns, M. Kwok, J. Jardim, and G. Garnett. Model Validation and Analysis of WSCC System Oscillations following Alberta Separation on August 4, 2000. Final Report, January 2001.

    Google Scholar 

  41. G. Angelidis and A. Semlyen. Improved Methodologies for the Calculation of Critical Eigenvalues in Small Signal Stability Analysis. IEEE Transactions on Power System, 11: 1209–1217, 1996.

    Article  Google Scholar 

  42. G. Angelidis and A. Semlyen. Efficient Calculation of Critical Eigenvalue Clusters in the Small Signal Stability Analysis of Large Power Systems. IEEE Transactions on Power System, 10:427–432, 1995.

    Article  Google Scholar 

  43. L.T.G. Lima, L.H. Bezerra, C. Tomei, and N. Martins. New Method for Fast Small-Signal Stability Assessment of Large Scale Power Systems. IEEE Transactions on Power Systems, 10:1979–1985, 1995.

    Article  Google Scholar 

  44. N. Martins, L.T.G. Lima, and H.J.C.P. Pinto. Computing Dominant Poles of Power System Transfer Function. IEEE Transactions on Power Systems, 11:162–170, 1996.

    Article  Google Scholar 

  45. N. Martins. The Dominant Pole Spectrum Eigensolver. IEEE Transactions on Power Systems, 12:245–254, 1997.

    Article  MathSciNet  Google Scholar 

  46. J.M. Campagnolo, N. Martins, J.L.R. Pereira, L.T.G. Lima, H.J.C.P. Pinto, and D.M. Falcao. Fast Small-Signal Stability Assessment using Parallel Processing. IEEE Transactions on Power Systems, 9:949–956, 1994.

    Article  Google Scholar 

  47. J.M. Campagnolo, N. Martins, and D.M. Falcao. Refactored Bi-Iteration: A High Performance Eigensolution Method for Large Power System Matrices. IEEE Transactions on Power Systems, 11:1228–1235, 1996.

    Article  Google Scholar 

  48. J.M. Campagnolo, N. Martins, and D.M. Falcao. An Efficient and Robust Eigenvalue Method for Small-Signal Stability Assessment in Parallel Computers. IEEE Transactions on Power Systems, 10:506–511, 1995.

    Article  Google Scholar 

  49. L. Wang and A. Semlyen. Application of Sparse Eigenvalue Techniques to the Small Signal Stability Analysis of Large Power Systems. IEEE Transactions on Power Systems, 5:635–642, 1990.

    Article  Google Scholar 

  50. M. Kubicek and M. Marek. Computational Methods in Bifurcation Theory and Dissipative Structures. Springer Series in Computational Physics, Springer-Verlag, New York, 1983.

    Google Scholar 

  51. Y.V. Makarov, Z.Y. Dong, and D.J. Hill. A General Method for Small Signal Stability Analysis. IEEE Transactions on Power System, 13:979–985, 1998.

    Article  Google Scholar 

  52. D.J. Hill and Z.Y. Dong. Nonlinear Computation and Control for Small Disturbance Stability. Panel Session on Recent Applications of Small-Signal Stability Techniques, Proceedings of IEEE Summer Power Meeting, Seattle, July 2000.

    Google Scholar 

  53. S. Gomes Jr., N. Martins, and C. Portela. Computing Small-Signal Stability Boundaries for Large-Scale Power Systems. IEEE Transactions on Power Systems, 18:747–752, 2003.

    Article  Google Scholar 

  54. Y. Zhou and V. Ajjarapu. Identification and Tracing of Voltage and Oscillatory Stability Margins. Proceedings of IEEE PES Winter Meeting, 1:250–255, 2002.

    Article  Google Scholar 

  55. R.T.H. Alden and F.A. Qureshy. Eigenvalue Tracking due to Parameter Variation. IEEE Transactions on Automatic Control, 30:923–925, 1985.

    Article  MathSciNet  Google Scholar 

  56. R. Kalaba, K. Spingarn, and L. Tesfatsion. Variational Equations for the Eigenvalues and Eigenvectors of Nonsymmetric Matrices. Journal of Optimization Theory and Applications, 33:1–8, 1981.

    Article  MathSciNet  Google Scholar 

  57. R. Kalaba, K. Spingarn, and L. Tesfatsion. Individual Tracking of an Eigenvalue and Eigenvector of Parameterized Matrix. Nonlinear Analysis: Theory, Methods and Application, 54:337–340, 1981.

    Article  MathSciNet  Google Scholar 

  58. Y. Zhou and V. Ajjarapu. Local Parameterization Based Tracing for Stability Boundary, Optimal Margin Control and Time Domain Simulation. Proceedings of Bulk Power System Dynamics and Control V, Onomichi, Japan, August 2001.

    Google Scholar 

  59. X. Wen and V. Ajjarapu. Critical Eigenvalue Trajectory Tracing for Power System Oscillatory Stability Assessment. Proceedings of IEEE PES General Meeting, Denver, June 2004.

    Google Scholar 

  60. Y. Zhou, X. Wen, and V. Ajjarapu. Identification and Tracing of Oscillatory Stability Margin Boundaries. Presented at IEEE Panel Session on Recent Applications of Linear Analysis Techniques, IEEE PES General Meeting, Toronto, Canada, 2003.

    Google Scholar 

  61. Z. Zhou and V. Ajjarapu. Optimal Margin Boundary Tracing With Continuation Optimal Power Flow. Proceedings of the 34thNorth American Power Symposium, Tempe, AZ, 2002.

    Google Scholar 

  62. Z. Zhou and V. Ajjarapu. A Novel Approach for Cost Based Optimization with Variable Voltage Stability Margins. Proceedings of the 35thNorth American Power Symposium, Rolla, MO, 2003.

    Google Scholar 

  63. Z. Zhou and V. Ajjarapu. A Novel Approach for Cost Based Optimization With Variable Oscillatory Stability Limits. Proceedings of the 36thNorth American Power Symposium, Moscow, Idaho, 2004.

    Google Scholar 

  64. Z. Zhou and V. Ajjarapu. A Novel Approach to Trace Time Domain Trajectories of Power Systems in Multiple Time Scales. Accepted for publication in IEEE Transactions on Power Systems, 2004.

    Google Scholar 

  65. J. Huang, S.S. Venkata, V. Ajjarapu, and Z. Zhou. Adaptive Wide Area Protection to Mitigate Voltage Collapse. Proceedings of the 35thNorth American Power Symposium, Rolla, MO, 2003.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Iowa State University, USA

    Venkataramana Ajjarapu

Authors
  1. Venkataramana Ajjarapu
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dept. of Electrical, Computer, & Systems Engineering, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY, 12180

    Joe H. Chow

  2. Dept. of Electrical and Electronic Engineering, Hong Kong University, Pokfulam Road, Hong Kong

    Felix F. Wu

  3. Center for Energy Systems and Control, Howard University, 2300 Sixth Street NW, Washington, DC, 20059

    James Momoh

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer Science+Business Media, Inc.

About this chapter

Cite this chapter

Ajjarapu, V. (2005). Bifurcation and Manifold Based Approach for Voltage and Oscillatory Stability Assessment and Control. In: Chow, J.H., Wu, F.F., Momoh, J. (eds) Applied Mathematics for Restructured Electric Power Systems. Power Electronics and Power Systems. Springer, Boston, MA. https://doi.org/10.1007/0-387-23471-3_4

Download citation

  • DOI: https://doi.org/10.1007/0-387-23471-3_4

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-0-387-23470-0

  • Online ISBN: 978-0-387-23471-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation