Analyzing stable regimes of electrical power systems and tropical geometry of power balance equations over complex multifields (original) (raw)

References

  1. Litvinov, G.L. and Maslov, V.P., Idempotent Mathematics: The Correspondence Principle and Its Computer Applications, Usp. Mat. Nauk, 1996, vol. 51, no. 6, pp. 209–210.
    Article MathSciNet Google Scholar
  2. Litvinov, G.L. and Maslov, V.P., The Correspondence Principle for Idempotent Calculus and Some Computer Applications, in Idempotency, Gunawardena, J., Ed., Cambridge: Cambridge Univ. Press, 1998, pp. 420–443, e-print math. GM/0101021 (http://arXiv.org).
    Chapter Google Scholar
  3. Litvinov, G.L., The Maslov Dequantization, Idempotent and Tropical Mathematics: A Brief Introduction, J. Math. Sci., 2007, vol. 140, no. 3, pp. 426–444, e-print math. GM/0507014 (http://arXiv.org).
    Article MathSciNet Google Scholar
  4. Noell, V., Grigoriev, D., Vakulenko, S., et al., Tropical Geometries and Dynamics of Biochemical Networks. Application to Hybrid Cell Cycle Models, e-print arXiv:1109.4085v2 [q-bio.MN], 2011 (http://arXiv.org).
    Google Scholar
  5. Del Moral, P. and Duazi, M., On Applications of Maslov Optimization Theory, Mat. Zametki, 2001, vol. 69, no. 2, pp. 262–276.
    Article Google Scholar
  6. Viro, O., Dequantization of Real Algebraic Geometry on a Logarithmic Paper, Proc. Eur. Congr. Math., July 10–14, 2000, vol. 2, pp. 135–146.
    Google Scholar
  7. Fomin, S. and Mikhalkin, G., Labeled Floor Diagrams for Plane Curves, J. Eur. Math. Soc., 2010, vol. 12, pp. 1453–1496.
    Article MathSciNet MATH Google Scholar
  8. Itenberg, I. and Mikhalkin, G., Geometry in the Tropical Limit, e-print arXiv:1108.3011 [math.AG], 2011 (http://arXiv.org).
    Google Scholar
  9. Kapranov, M., Thermodynamics and the Moment Map, e-print arXiv: 1108.3472v1 [math.QA], 2011 (http://arXiv.org).
    Google Scholar
  10. Fortuin, C.M. and Kasteleyn, P.W., On the Random Cluster Model: I. Introduction and Relations to Other Models, Physica, 1972, vol. 57, no. 4, pp. 536–564.
    Article MathSciNet Google Scholar
  11. Foster, R., The Average Impedance of an Electrical Network, in Contributions to Applied Mechanics (Reissner Anniversary Volume), Ann Arbor: J.W. Edwards, pp. 333–340.
  12. Gel’fand, A.M. and Kirshtein, B.Kh., Idempotent Systems of Nonlinear Equations and Problems of Computing Electric Power Nets, in Proc. Int. Conf. “Idempotent and Tropical Mathematical and Problems of Mathematical Physics,” Moscow, 2007, vol. 2, pp. 84–87, e-print arXiv:0709.4119 (http://arXiv.org).
    Google Scholar
  13. Viro, O.Ya., On Basic Notions of Tropical Geometry, Tr. MIAN, 2011, vol. 273, pp. 271–303.
    MathSciNet Google Scholar
  14. Kontorovich, A.M. and Kryukov, A.V., Predel’nye rezhimy energosistem. Osnovy teorii i metody raschetov (Limit Modes of Power Systems. Theoretical Foundations and Computational Methods), Irkutsk: Irkut. Gos. Univ., 1985.
    Google Scholar
  15. Arzhannikov, S.G., Zakharkin, O.V., and Petrov, A.M., Estimating the Stability Margin of an Stable Regime and Choosing Controlling Influences to Bring It to the Admissible Region, Novoe Ross. Elektroenerget., 2005, no. 5, pp. 6–19.
    Google Scholar
  16. Tarasov, V.I., Teoreticheskie osnovy analiza ustanovivshikhsya rezhimov elektroenergeticheskikh sistem (Theoretical Foundations for the Analysis of Stable Regimes in Electric Power Systems), Novosibirsk: Nauka, 2002.
    Google Scholar
  17. Gelfand, I.M., Kapranov, M.M., and Zelevinsky, A., Discriminants, Resultants, and Multidimensional Determinants, Boston: Birkhauser, 1994.
    Book MATH Google Scholar
  18. Passare, M. and Rullgard, H., Amoebas, Monge-Ampere Measures and Triangulations of the Newton Polytope, Duke Math. J., 2004, vol. 121, no. 3, pp. 481–507.
    Article MathSciNet MATH Google Scholar
  19. Passare, M. and Tsikh, A., Amoebas: Their Spines and Their Contours, in Contemporary Math., Providence: AMS, 2005, vol. 377, pp. 275–288.
    Google Scholar
  20. Baker, M. and Faber, X., Metric Properties of the Tropical Abel-Jacobi Map, J. Algebr. Combinat., 2011, vol. 33, no. 3, pp. 349–381.
    Article MathSciNet MATH Google Scholar
  21. Biggs, N.I., Algebraic Potential Theory on Graphs, Bull. London Math. Soc., 1997, vol. 29, no. 6, pp. 641–682.
    Article MathSciNet Google Scholar
  22. Wagner, D.G., Combinatorics of Electrical Networks, A Series of Lectures Prepared for the Undergraduate Summer Research Assistants, Waterloo: Univ. of Waterloo, 2009.
    Google Scholar

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