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
R. Billinton, R.J. Ringle, A.J. Wood, Power-System Reliability Calculations (MIT Press Classics Series, September, 1973)
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)
P. Kundur, Power System Stability and Control, EPRI Power System Engineering Series. (McGraw-Hill, New York, 1994), p. 328
H. Frank, B. Landstorm, Power factor correction with thyristor-controlled capacitors. ASEA J. 45, 180–184 (1971)
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)
H. Almasoud, Shunt capacitance for a practical 380 kV system. Int. J. Electr. Comput. Sci. (IJECS) 9(10), 23–27 (2009)
G. Ruppeiner, Riemannian geometry in thermodynamic fluctuation theory. Rev. Mod. Phys. 67, 605 (1995); [Erratum 68, 313 (1996)]
G. Ruppeiner, Thermodynamic curvature and phase transitions in Kerr-Newman black holes. Phy. Rev. D 78, 024016 (2008)
G. Ruppeiner, Thermodynamics: a Riemannian geometric model. Phys. Rev. A 20, 1608 (1979)
G. Ruppeiner, Thermodynamic critical fluctuation theory? Phys. Rev. Lett. 50, 287 (1983)
G. Ruppeiner, New thermodynamic fluctuation theory using path integrals. Phys. Rev. A 27, 1116 (1983)
G. Ruppeiner, C. Davis, Thermodynamic curvature of the multicomponent ideal gas. Phys. Rev. A 41, 2200 (1990)
T. Sarkar, G. Sengupta, B.N. Tiwari, On the thermodynamic geometry of BTZ black holes. JHEP 0611, 015 (2006); arXiv:hep-th/0606084v1
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]
B.N. Tiwari, On Generalized Uncertainty Principle (LAP Academic Publication, Germany, 2011); ISBN 978-3-8465-1532-7; arXiv:0801.3402v2 [hep-th]
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]
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]
S. Bellucci, B.N. Tiwari, On the microscopic perspective of black branes thermodynamic geometry. Entropy 12, 2096 (2010); arXiv:0808.3921v1
S. Bellucci, B.N. Tiwari, An exact fluctuating 1/2 BPS configuration. Springer J. High Energy Phys. 05, 23 (2010); arXiv:0910.5314v1
S. Bellucci, B.N. Tiwari, State-space correlations and stabilities. Phys. Rev. D 82, 084008 (2010); arXiv:0910.5309v1 [hep-th]
S. Bellucci, B.N. Tiwari, Thermodynamic geometry and Hawking radiation. JHEP 30, 1011 (2010); arXiv:1009.0633v1 [hep-th]
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]
S. Bellucci, B.N. Tiwari, State-space manifold and rotating black holes. JHEP 118, 1011 (2011); arXiv:1010.1427v1 [hep-th]
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
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
J.E. Aman, I. Bengtsson, N. Pidokrajt, Geometry of black hole thermodynamics. Gen. Rel. Grav. 35,1733 (2003); arXiv:gr-qc/0304015v1
J.E. Aman, N. Pidokrajt, Geometry of higher-dimensional black hole thermodynamics. Phys. Rev. D 73, 024017 (2006); arXiv:hep-th/0510139v3
F. Weinhold, Metric geometry of equilibrium thermodynamics. J. Chem. Phys. 63, 2479 (1975); doi:10.1063/1.431689
F. Weinhold, Metric geometry of equilibrium thermodynamics. II. Scaling, homogeneity, and generalized Gibbs-Duhem relations. J. Chem. Phys. 63, 2482 (1975)
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]
S. Bellucci, V. Chandra, B.N. Tiwari, A geometric approach to correlations and quark number susceptibilities; arXiv:1010.4405v1 [hep-th]
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]
A. Strominger, C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy. Phys. Lett. B 379, 99–104 (1996); arXiv:hep-th/9601029v2
A. Sen, Extremal black holes and elementary string states. Mod. Phys. Lett. A 10, 2081–2094 (1995); arXiv:hep-th/9504147v2
A. Dabholkar, Exact counting of black hole microstates. Phys. Rev. Lett. 94, 241–301 (2005); arXiv:hep-th/0409148v2
L. Andrianopoli, R. D’Auria, S. Ferrara, Flat symplectic bundles of \(N\)-extended supergravities, central charges and black-hole, entropy; arXiv:hep-th/9707203v1
A. Dabholkar, F. Denef, G.W. Moore, B. Pioline, Precision counting of small black holes. JHEP 0510, 096 (2005); arXiv:hep-th/0507014v1
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
A. Sen, Stretching the horizon of a higher dimensional small black hole. JHEP 0507, 073 (2005); arXiv:hep-th/0505122v2
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
J.B. Gutowski, H.S. Reall, General supersymmetric AdS5 black holes. JHEP 0404P, 048 (2004); arXiv:hep-th/0401129v3
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
J.P. Gauntlett, J.B. Gutowski, General concentric black rings. Phys. Rev. D 71, 045002 (2005); arXiv:hep-th/0408122v3
S. Ferrara, R. Kallosh, A. Strominger, \(N=2\) extremal black holes. Phys. Rev. D 52, R5412–R5416 (1995); arXiv:hep-th/9508072v3
A. Strominger, Macroscopic entropy of \(N=2\) extremal black holes. Phys. Lett. B 383, 39–43 (1996); arXiv:hep-th/9602111v3
S. Ferrara, R. Kallosh, Supersymmetry and attractors. Phys. Rev. D 54, 1514–1524 (1996); arXiv:hep-th/9602136
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
S. Bellucci, S. Ferrara, A. Marrani, Attractors in black, Fortsch. Phys. 56, 761 (2008); arXiv:0805.1310
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)
J. Grainger Jr., W. Stevenson, Power System Analysis, 1st edn. (McGraw-Hill Science, Engineering, Math, New York, 1994)
E. Calabi, A construction of non-homogeneous Einstein metrics. Proc. Symp. Pure Math. (AMS, Providence) 27, 17–24 (1975)
J. Li, S.T. Yau, Hermitian Yang-Mills Connections on Non-Kähler Manifolds, Mathematical Aspects of String Theory (World Scientific, Singapore, 1987)
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
P.S. Aspinwall, The Landau-Ginzburg to Calabi-Yau dictionary for \(D\)-branes. J. Math. Phys. 48, 082304 (2007); arXiv:hep-th/0610209v2
A. Ricco, Brane superpotential and local Calabi-Yau manifolds. Int. J. Mod. Phys. A 23, 2187–2189 (2008); arXiv:0805.2738v1 [hep-th]
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
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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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