Abstract
Application of a modal approach in solving the static stability problem for power systems is examined. It is proposed to use the matrix exponent norm as a generalized transition function of the power system disturbed motion. Based on the concept of a stability radius and the pseudospectrum of Jacobian matrix, the necessary and sufficient conditions for existence of the static margins were determined. The capabilities and advantages of the modal approach in designing centralized or distributed control and the prospects for the analysis of nonlinear oscillations and rendering the dynamic stability are demonstrated.
Similar content being viewed by others
References
V. A. Barinov and S. A. Sovalov, “Modal control of modes of power systems,” Elektrichestvo, No. 8, 1–6 (1986).
E. A. Mikrin, N. E. Zubov, and V. N. Ryabchenko, Matrix Methods in Theory and Practice of Automatic Aircraft Control Systems (Mosk. Gos. Tekh. Univ. im. N. E. Baumana, Moscow, 2016) [in Russian].
W. K. Gawronski, Dynamics and Control of Structures: A Modal Approach (Springer-Verlag, New York, 1998).
V. A. Venikov, Electromechanical Transients in Electric Systems (Vysshaya Shkola, Moscow, 1985) [in Russian].
M. J. Gibbard, P. Pourbeik, and D. J. Vowles, Small-Signal Stability, Control and Dynamic Performance of Power Systems (Univ. of Adelaide Press, Adelaide, 2015).
J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, Oxford, 1965; Fizmatlit, Moscow, 1970).
M. Sh. Misrikhanov and V. N. Ryabchenko, “The quadratic eigenvalue problem in electric power systems,” Autom. Remote Control 67, 698–720 (2006).
Y. Saad, Numerical Methods for Large Eigenvalue Problems (Soc. Ind. Appl. Math., Philadelphia, PA, 2011).
V. V. Voevodin and Yu. A. Kuznetsov, Matrices and Calculations (Fizmatlit, Moscow, 1984) [in Russian].
E. V. Tuzlukova, Candidate’s Dissertation in Engineering (Moscow Power Engineering Univ., Moscow, 2004).
Yu. V. Sharov, “Development of the analysis method of static stability of electric power systems,” Elektrichestvo, No. 1 (2017).
L. N. Trefethen and M. Embree, Spectra and Pseudospectra (Princeton Univ. Press, Princeton, 2005).
M. Sh. Misrikhanov and Yu. V. Sharov, “Estimation of disturbances influence on stability of the electric power system,” Vestn. MEI, No. 5, 42–48 (2009).
P. M. Anderson and A. A. Fouad, Power System Control and Stability (Iowa State Univ. Press, Ames, IA, 1977; Energiya, Moscow, 1980).
M. Sh. Misrikhanov, V. F. Sitnikov, and Yu. V. Sharov, “Modal synthesis of regulators based on FACTS devices,” Elektrotekhnika, No. 10, 22–29 (2007).
M. Sh. Misrikhanov and Yu. V. Sharov, “Modal-optimal control of electric power facilities,” Vestn. IGEU, No. 4, 83–98 (2004).
M. Sh. Misrikhanov, V. F. Sitnikov, and Yu. V. Sharov, “Optimal regulators based on FACTS devices for a decentralized model of united power system,” Vestn. MEI, No. 3, 35–41 (2009).
Yu. V. Sharov, “Nonlinear modal interaction in electric power systems,” Elektrichestvo, No. 12 (2016).
H. M. Shanechi, N. Pariz, and E. Vaahedi, “General nonlinear modal representation of large scale power systems,” IEEE Trans. Power Syst. 18, 1103–1109 (2003).
V. Vittal, N. Bhatia, and A. A. Fouad, “Analysis of the inter-area mode phenomenon in power systems following large disturbances,” IEEE Trans. Power Syst. 6, 1515–1521 (1991).
S. K. Starrett and A. A. Fouad, “Nonlinear measures of mode-machine participation,” IEEE Trans. Power Syst. 13, 389–394 (1994).
V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations (MTsNMO, Moscow, 2012; Springer-Verlag, New York, 2012).
A. P. Seyranian and A. A. Mailybaev, Multiparameter Stability Theory with Mechanical Applications (World Sci., Singapore, 2003).
A. Goriely, “Painlevé analysis and normal forms theory,” Phys. D 152–153, 124–144 (2001).
A. A. Mailybaev and A. P. Seiranyan, “Interaction of eigenvalues under variation of parameters,” Dokl. Math. 68, 466–470 (2003).
A. P. Seyranian, O. N. Kirillov, and A. A. Mailybaev, “Coupling of eigenvalues of complex matrices at diabolic and exceptional points,” J. Phys. A: Math. Gen. 38, 1723–1740 (2005).
I. Dobson, “Strong resonance effects in normal form analysis and subsynchronous resonance,” in Proc. IREP Symp. Bulk Power System Dynamics and Control V — Security and Reliability in a Changing Environment, Onomichi, Japan, Aug. 26–31, 2001 (IREP, 2001).
V. Auvray, I. Dobson, and L. Wehenkel, “Modifying eigenvalue interactions near weak resonance,” in Proc. IEEE Int. Symp. on Circuits and Systems (ISCAS '04), Vancouver, Canada, May 23–26, 2004 (IEEE, New York, 2004).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © J.V. Sharov, 2017, published in Izvestiya Rossiiskoi Akademii Nauk, Energetika.
Rights and permissions
About this article
Cite this article
Sharov, J.V. Application of a Modal Approach in Solving the Static Stability Problem for Electric Power Systems. Therm. Eng. 64, 971–981 (2017). https://doi.org/10.1134/S0040601517130080
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040601517130080
Keywords
- static or dynamic stability
- stability margin
- algebra-differential equations
- linearization
- Jacobi matrix
- eigenvalues
- modal analysis
- generalized transition function
- controllability
- control law synthesis
- centralized or distributed control
- Poincare-Dulac normal form
- nonlinear modal interaction
- strong or weak resonance