Synchronous solutions of power systems: existence, uniqueness and stability
Author:
Sherwin J. Skar
Journal:
Quart. Appl. Math. 38 (1980), 331-342
MSC:
Primary 93D05; Secondary 34D05
DOI:
https://doi.org/10.1090/qam/592200
MathSciNet review:
592200
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Abstract: This paper contains a study of the asymptotic stability and uniqueness of equilibrium solutions of multi-dimensional Hamiltonian-like systems. The results are applied to the swing equations, the classical model for power systems. By developing some results in matrix theory, it is shown that asymptotically stable equilibrium solutions may exist even though most rotor angle pairs are more than ${90^ \circ }$, some even ${180^ \circ }$, out of phase. In contrast to the numerical criteria usually used, an analytic criterion for the existence of asymptotically stable equilibrium solutions of the swing equations is given.
P. M. Anderson and A. A. Fouad, Power system control and stability, Iowa State University Press, Ames, 1977
T. M. Apostal, Mathematical analysis, second edition, Addison-Wesley, Reading, Mass., 1975
M. Araki, M-matrices (matrices with nonpositive off-diagonal elements and positive principal minors), Publication 74/19, Imperial College of Science and Technology, London, 1974
F. Brauer and J. A. Nohel, The qualitative theory of ordinary differential equations, W. A. Benjamin, New York, 1969
O. G. C. Dahl, Electric power circuits: theory and applications, Vol. II, Power system stability, McGraw-Hill, New York, 1938
M. Fiedler and V. Pták, On matrices with nonpositive off-diagonal elements and positive principal minors, Czech. Math. J., 12, 382–400 (1962)
A. A. Fouad, Stability theory—criteria for transient stability, ERDA Report, Engineering Research Institute, Iowa State University, Ames, 1975
F. R. Gantmacher, The theory of matrices, Vol. I, Chelsea, New York, 1959
- Wolfgang Hahn, Stability of motion, Die Grundlehren der mathematischen Wissenschaften, Band 138, Springer-Verlag New York, Inc., New York, 1967. Translated from the German manuscript by Arne P. Baartz. MR 0223668
C. J. Tavora and O. J. M. Smith, Equilibrium analysis of power systems, IEEE Trans. Power Apparatus and Systems, 91, 1131–1137 (1972)
- Jacques L. Willems, Direct methods for transient stability studies in power system analysis, IEEE Trans. Automatic Control AC-16 (1971), 332–341. MR 0289174, DOI https://doi.org/10.1109/tac.1971.1099743
P. M. Anderson and A. A. Fouad, Power system control and stability, Iowa State University Press, Ames, 1977
T. M. Apostal, Mathematical analysis, second edition, Addison-Wesley, Reading, Mass., 1975
M. Araki, M-matrices (matrices with nonpositive off-diagonal elements and positive principal minors), Publication 74/19, Imperial College of Science and Technology, London, 1974
F. Brauer and J. A. Nohel, The qualitative theory of ordinary differential equations, W. A. Benjamin, New York, 1969
O. G. C. Dahl, Electric power circuits: theory and applications, Vol. II, Power system stability, McGraw-Hill, New York, 1938
M. Fiedler and V. Pták, On matrices with nonpositive off-diagonal elements and positive principal minors, Czech. Math. J., 12, 382–400 (1962)
A. A. Fouad, Stability theory—criteria for transient stability, ERDA Report, Engineering Research Institute, Iowa State University, Ames, 1975
F. R. Gantmacher, The theory of matrices, Vol. I, Chelsea, New York, 1959
W. Hahn, Stability of motion, Springer-Verlag, New York, 1967
C. J. Tavora and O. J. M. Smith, Equilibrium analysis of power systems, IEEE Trans. Power Apparatus and Systems, 91, 1131–1137 (1972)
J. L. Willems, Direct methods for transient stability studies in power systems analysis, IEEE Trans. Automat. Contr., 16, 332–341 (1971)
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Article copyright:
© Copyright 1980
American Mathematical Society