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Intrinsic Geometric Characterization

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Geometrical Methods for Power Network Analysis

Abstract

In this chapter, we offer a concise account of network power flow and stability criteria arising from real intrinsic Riemannian geometry. We begin by considering a brief review of the flow equations and related concepts, for use in the later chapters.

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References

  1. R. Billinton, R.J. Ringle, A.J. Wood, Power-System Reliability Calculations (MIT Press Classics Series, September, 1973)

    Google Scholar 

  2. A. Chakraborty, P. Sen, An analytical investigation of voltage stability of an EHV transmission network based on load flow analysis. J. Inst. Eng. (India) Electr. Eng. Div. 76, (1995)

    Google Scholar 

  3. P. Kundur, Power System Stability and Control, EPRI Power System Engineering Series. (McGraw-Hill, New York, 1994), p. 328

    Google Scholar 

  4. H. Frank, B. Landstorm, Power factor correction with thyristor-controlled capacitors. ASEA J. 45, 180–184 (1971)

    Google Scholar 

  5. R. Rajarman, F. Alvarado, A. Maniaci, R. Camfield, S. Jalali, Determination of location and amount of series compensation to increase power transfer capability. IEEE Trans. Power Syst. 13(2), 294–300 (1998)

    Google Scholar 

  6. H. Almasoud, Shunt capacitance for a practical 380 kV system. Int. J. Electr. Comput. Sci. (IJECS) 9(10), 23–27 (2009)

    Google Scholar 

  7. G. Ruppeiner, Riemannian geometry in thermodynamic fluctuation theory. Rev. Mod. Phys. 67, 605 (1995); [Erratum 68, 313 (1996)]

    Google Scholar 

  8. G. Ruppeiner, Thermodynamic curvature and phase transitions in Kerr-Newman black holes. Phy. Rev. D 78, 024016 (2008)

    Article  MathSciNet  Google Scholar 

  9. G. Ruppeiner, Thermodynamics: a Riemannian geometric model. Phys. Rev. A 20, 1608 (1979)

    Article  Google Scholar 

  10. G. Ruppeiner, Thermodynamic critical fluctuation theory? Phys. Rev. Lett. 50, 287 (1983)

    Article  MathSciNet  Google Scholar 

  11. G. Ruppeiner, New thermodynamic fluctuation theory using path integrals. Phys. Rev. A 27, 1116 (1983)

    Article  Google Scholar 

  12. G. Ruppeiner, C. Davis, Thermodynamic curvature of the multicomponent ideal gas. Phys. Rev. A 41, 2200 (1990)

    Article  Google Scholar 

  13. T. Sarkar, G. Sengupta, B.N. Tiwari, On the thermodynamic geometry of BTZ black holes. JHEP 0611, 015 (2006); arXiv:hep-th/0606084v1

    Google Scholar 

  14. T. Sarkar, G. Sengupta, B.N. Tiwari, Thermodynamic geometry and extremal black holes in string theory. JHEP 0810, 076 (2008); arXiv:0806.3513v1 [hep-th]

    Google Scholar 

  15. B.N. Tiwari, On Generalized Uncertainty Principle (LAP Academic Publication, Germany, 2011); ISBN 978-3-8465-1532-7; arXiv:0801.3402v2 [hep-th]

    Google Scholar 

  16. B.N. Tiwari, Sur les corrections de la géométrie thermodynamique destrous noirs (Éditions Universitaires Européennes, Germany, 2011); ISBN 978-613-1-53539-0; arXiv:0801.4087v2 [hep-th]

    Google Scholar 

  17. B.N. Tiwari, Geometric Perspective of Entropy Function: Embedding, Spectrum and Convexity (LAP Academic Publication, Germany, 2011); ISBN 978-3-8454-3178-9; arXiv:1108.4654v2 [hep-th]

    Google Scholar 

  18. S. Bellucci, B.N. Tiwari, On the microscopic perspective of black branes thermodynamic geometry. Entropy 12, 2096 (2010); arXiv:0808.3921v1

    Google Scholar 

  19. S. Bellucci, B.N. Tiwari, An exact fluctuating 1/2 BPS configuration. Springer J. High Energy Phys. 05, 23 (2010); arXiv:0910.5314v1

    Google Scholar 

  20. S. Bellucci, B.N. Tiwari, State-space correlations and stabilities. Phys. Rev. D 82, 084008 (2010); arXiv:0910.5309v1 [hep-th]

    Google Scholar 

  21. S. Bellucci, B.N. Tiwari, Thermodynamic geometry and Hawking radiation. JHEP 30, 1011 (2010); arXiv:1009.0633v1 [hep-th]

    Google Scholar 

  22. S. Bellucci, B.N. Tiwari, Black strings, black rings and state-space manifold. Int. J. Mod. Phys. A 26(32), 5403–5464 (2011); arXiv:1010.3832v2 [hep-th]

    Google Scholar 

  23. S. Bellucci, B.N. Tiwari, State-space manifold and rotating black holes. JHEP 118, 1011 (2011); arXiv:1010.1427v1 [hep-th]

    Google Scholar 

  24. J.E. Aman, I. Bengtsson, N. Pidokrajt, Flat information geometries in black hole thermodynamics. Gen. Rel. Grav. 38, 1305–1315 (2006); arXiv:gr-qc/0601119v1

    Google Scholar 

  25. J. Shen, R.G. Cai, B. Wang, R.K. Su, Thermodynamic geometry and critical behavior of black holes. Int. J. Mod. Phys. A 22, 11–27 (2007); arXiv:gr-qc/0512035v1

    Google Scholar 

  26. J.E. Aman, I. Bengtsson, N. Pidokrajt, Geometry of black hole thermodynamics. Gen. Rel. Grav. 35,1733 (2003); arXiv:gr-qc/0304015v1

    Google Scholar 

  27. J.E. Aman, N. Pidokrajt, Geometry of higher-dimensional black hole thermodynamics. Phys. Rev. D 73, 024017 (2006); arXiv:hep-th/0510139v3

    Google Scholar 

  28. F. Weinhold, Metric geometry of equilibrium thermodynamics. J. Chem. Phys. 63, 2479 (1975); doi:10.1063/1.431689

    Google Scholar 

  29. F. Weinhold, Metric geometry of equilibrium thermodynamics. II. Scaling, homogeneity, and generalized Gibbs-Duhem relations. J. Chem. Phys. 63, 2482 (1975)

    Google Scholar 

  30. S. Bellucci, V. Chandra, B.N. Tiwari, On the thermodynamic geometry of hot QCD. Int. J. Mod. Phys. A 26, 43–70 (2011); arXiv:0812.3792v1 [hep-th]

    Google Scholar 

  31. S. Bellucci, V. Chandra, B.N. Tiwari, A geometric approach to correlations and quark number susceptibilities; arXiv:1010.4405v1 [hep-th]

    Google Scholar 

  32. S. Bellucci, V. Chandra, B.N. Tiwari, Thermodynamic geometric stability of quarkonia states. Int. J. Mod. Phys. A 26, 2665–2724 (2011); arXiv:1010.4225v2 [hep-th]

    Google Scholar 

  33. A. Strominger, C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy. Phys. Lett. B 379, 99–104 (1996); arXiv:hep-th/9601029v2

    Google Scholar 

  34. A. Sen, Extremal black holes and elementary string states. Mod. Phys. Lett. A 10, 2081–2094 (1995); arXiv:hep-th/9504147v2

    Google Scholar 

  35. A. Dabholkar, Exact counting of black hole microstates. Phys. Rev. Lett. 94, 241–301 (2005); arXiv:hep-th/0409148v2

    Google Scholar 

  36. L. Andrianopoli, R. D’Auria, S. Ferrara, Flat symplectic bundles of \(N\)-extended supergravities, central charges and black-hole, entropy; arXiv:hep-th/9707203v1

    Google Scholar 

  37. A. Dabholkar, F. Denef, G.W. Moore, B. Pioline, Precision counting of small black holes. JHEP 0510, 096 (2005); arXiv:hep-th/0507014v1

    Google Scholar 

  38. A. Dabholkar, F. Denef, G.W. Moore, B. Pioline, Exact and asymptotic degeneracies of small black holes. JHEP 0508, 021 (2005); arXiv:hep-th/0502157v4

    Google Scholar 

  39. A. Sen, Stretching the horizon of a higher dimensional small black hole. JHEP 0507, 073 (2005); arXiv:hep-th/0505122v2

    Google Scholar 

  40. J.P. Gauntlett, J.B. Gutowski, C.M. Hull, S. Pakis, H.S. Reall, All supersymmetric solutions of minimal supergravity in five dimensions. Class. Quant. Grav. 20, 4587–4634 (2003); arXiv:hep-th/0209114v3

    Google Scholar 

  41. J.B. Gutowski, H.S. Reall, General supersymmetric AdS5 black holes. JHEP 0404P, 048 (2004); arXiv:hep-th/0401129v3

    Google Scholar 

  42. I. Bena, N.P. Warner, One ring to rule them all ... and in the darkness bind them?. Adv. Theor. Math. Phys. 9P, 667–701 (2005); arXiv:hep-th/0408106v2

    Google Scholar 

  43. J.P. Gauntlett, J.B. Gutowski, General concentric black rings. Phys. Rev. D 71, 045002 (2005); arXiv:hep-th/0408122v3

    Google Scholar 

  44. S. Ferrara, R. Kallosh, A. Strominger, \(N=2\) extremal black holes. Phys. Rev. D 52, R5412–R5416 (1995); arXiv:hep-th/9508072v3

    Google Scholar 

  45. A. Strominger, Macroscopic entropy of \(N=2\) extremal black holes. Phys. Lett. B 383, 39–43 (1996); arXiv:hep-th/9602111v3

    Google Scholar 

  46. S. Ferrara, R. Kallosh, Supersymmetry and attractors. Phys. Rev. D 54, 1514–1524 (1996); arXiv:hep-th/9602136

    Google Scholar 

  47. S. Ferrara, G.W. Gibbons, R. Kallosh, Black holes and critical points in moduli space. Nucl. Phys. B 500, 75–93 (1997); arXiv:hep-th/9702103

    Google Scholar 

  48. S. Bellucci, S. Ferrara, A. Marrani, Attractors in black, Fortsch. Phys. 56, 761 (2008); arXiv:0805.1310

    Google Scholar 

  49. G. Radman, R.S. Raje, Power flow model/calculation for power system with multiple FACTS controllers. Elsevier Sci. Dir. Electr. Power Syst. Res. 77, 1521–1531 (2007)

    Article  Google Scholar 

  50. J. Grainger Jr., W. Stevenson, Power System Analysis, 1st edn. (McGraw-Hill Science, Engineering, Math, New York, 1994)

    Google Scholar 

  51. E. Calabi, A construction of non-homogeneous Einstein metrics. Proc. Symp. Pure Math. (AMS, Providence) 27, 17–24 (1975)

    Google Scholar 

  52. J. Li, S.T. Yau, Hermitian Yang-Mills Connections on Non-Kähler Manifolds, Mathematical Aspects of String Theory (World Scientific, Singapore, 1987)

    Google Scholar 

  53. A. Klemm, S. Theisen, Considerations of one-modulus Calabi-Yau compactifications: Picard-Fuchs equations, Kähler potentials and mirror maps. Nucl. Phys. B 389, 153–180 (1983); arXiv:hep-th/9205041v1

    Google Scholar 

  54. P.S. Aspinwall, The Landau-Ginzburg to Calabi-Yau dictionary for \(D\)-branes. J. Math. Phys. 48, 082304 (2007); arXiv:hep-th/0610209v2

    Google Scholar 

  55. A. Ricco, Brane superpotential and local Calabi-Yau manifolds. Int. J. Mod. Phys. A 23, 2187–2189 (2008); arXiv:0805.2738v1 [hep-th]

    Google Scholar 

  56. G. Gibbons, R. Kallosh, B. Kol, Moduli, scalar charges, and the first law of black hole thermodynamics. Phys. Rev. Lett. 77, 4992–4995 (1996); arXiv:hep-th/9607108v2

    Google Scholar 

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Bellucci, S., Tiwari, B.N., Gupta, N. (2013). Intrinsic Geometric Characterization. In: Geometrical Methods for Power Network Analysis. SpringerBriefs in Electrical and Computer Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33344-6_3

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  • DOI: https://doi.org/10.1007/978-3-642-33344-6_3

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