Flow and Elastic Networks on the š-Torus: Geometry, Analysis, and Computation
Abstract
Networks with phase-valued nodal variables are central in modeling several important societal and physical systems, including power grids, biological systems, and coupled oscillator networks. One of the distinctive features of phase-valued networks is the existence of multiple operating conditions corresponding to critical points of an energy function or feasible flows of a balance equation. For networks with phase-valued states it is not yet fully understood how many operating conditions exist, how to characterize them, and how to compute them efficiently. A deeper understanding of feasible operating conditions, including their dependence upon network structures, may lead to more reliable and efficient network systems. This paper introduces flow and elastic network problems on the š-torus and provides a rigorous and comprehensive framework for their study. Based on a monotonicity assumption, this framework localizes the solutions, bounds their number, and leads to an algorithm to compute them. Our analysis is based on a novel winding partition of the š-torus into winding cells, induced by Kirchhoff's angle law for undirected graphs. The winding partition has several useful properties, e.g., each winding cell contains at most one solution. The proposed algorithm is based on a novel contraction mapping and is guaranteed to compute all solutions. Finally, we apply our results to study numerically the active power flow equations in several test cases and estimate the power transmission capacity and congestion of a power network.
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References
1.
R. Abraham, J. E. Marsden, and T. S. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd ed., Appl. Math. Sci. 75, Springer, 1988.
2.
J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc., 30 (1928), pp. 275--306, https://doi.org/10.2307/1989123.
3.
J. Baillieul and C. I. Byrnes, Geometric critical point analysis of lossless power system models, IEEE Trans. Circuits Systems, 29 (1982), pp. 724--737, https://doi.org/10.1109/TCS.1982.1085093.
4.
A. R. Bergen and D. J. Hill, A structure preserving model for power system stability analysis, IEEE Trans. Power Apparatus Systems, 100 (1981), pp. 25--35, https://doi.org/10.1109/TPAS.1981.316883.
5.
N. Biggs, Algebraic potential theory on graphs, Bull. London Math. Soc., 29 (1997), pp. 641--682, https://doi.org/10.1112/S0024609397003305.
6.
J. C. Bronski, T. Carty, and L. DeVille, Configurational stability for the Kuramoto--Sakaguchi model, Chaos, 28 (2018), art. 103109, https://doi.org/10.1063/1.5029397.
7.
F. Bullo, Lectures on Network Systems, Rev. 1.6, Kindle Direct Publishing, 2022, http://motion.me.ucsb.edu/book-lns; with contributions by J. CorteĢs, F. DoĢrfler, and S. MartiĢnez.
8.
N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Automat. Control, 54 (2009), pp. 353--357, https://doi.org/10.1109/TAC.2008.2007884.
9.
T. Coletta, R. Delabays, I. Adagideli, and P. Jacquod, Topologically protected loop flows in high voltage AC power grids, New J. Phys., 18 (2016), art. 103042, https://doi.org/10.1088/1367-2630/18/10/103042.
10.
O. Coss, J. D. Hauenstein, H. Hong, and D. K. Molzahn, Locating and counting equilibria of the Kuramoto model with rank-one coupling, SIAM J. Appl. Algebra Geom., 2 (2018), pp. 45--71, https://doi.org/10.1137/17M1128198.
11.
H. Daido, Quasi-entrainment and slow relaxation in a population of oscillators with random and frustrated interactions, Phys. Rev. Lett., 68 (1992), pp. 1073--1076, https://doi.org/10.1103/PhysRevLett.68.1073.
12.
R. Delabays, T. Coletta, and P. Jacquod, Multistability of phase-locking and topological winding numbers in locally coupled Kuramoto models on single-loop networks, J. Math. Phys., 57 (2016), art. 032701, https://doi.org/10.1063/1.4943296.
13.
M. Desroches, J. Guckenheimer, B. Krauskopf, C. Kuehn, H. M. Osinga, and M. Wechselberger, Mixed-mode oscillations with multiple time scales, SIAM Rev., 54 (2012), pp. 211--288, https://doi.org/10.1137/100791233.
14.
F. DoĢrfler, M. Chertkov, and F. Bullo, Synchronization in complex oscillator networks and smart grids, Proc. Natl. Acad. Sci. USA, 110 (2013), pp. 2005--2010, https://doi.org/10.1073/pnas.1212134110.
15.
M. Elkin, C. Liebchen, and R. Rizzi, New length bounds for cycle bases, Inform. Process. Lett., 104 (2007), pp. 186--193, https://doi.org/10.1016/j.ipl.2007.06.013.
16.
G. B. Ermentrout, The behavior of rings of coupled oscillators, J. Math. Biol., 23 (1985), pp. 55--74, https://doi.org/10.1007/BF00276558.
17.
T. Ferguson, Topological states in the Kuramoto model, SIAM J. Appl. Dyn. Syst., 17 (2018), pp. 484--499, https://doi.org/10.1137/17M112484X.
18.
A. Franci, G. Drion, and R. Sepulchre, Modeling the modulation of neuronal bursting: A singularity theory approach, SIAM J. Appl. Dyn. Syst., 13 (2014), pp. 798--829, https://doi.org/10.1137/13092263X.
19.
C. D. Godsil and G. F. Royle, Algebraic Graph Theory, Springer, 2001.
20.
G. H. Golub and C. F. van Loan, Matrix Computations, 2nd ed., Johns Hopkins University Press, 1989.
21.
C. Grigg et al., The IEEE Reliability Test System-1996. A report prepared by the Reliability Test System Task Force of the Application of Probability Methods Subcommittee, IEEE Trans. Power Systems, 14 (1999), pp. 1010--1020, https://doi.org/10.1109/59.780914.
22.
V. Guillemin and A. Pollack, Differential Topology, American Mathematical Society, 2010.
23.
J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. USA, 79 (1982), pp. 2554--2558, https://doi.org/10.1073/pnas.79.8.2554.
24.
F. C. Hoppensteadt and E. M. Izhikevich, Synchronization of laser oscillators, associative memory, and optical neurocomputing, Phys. Rev. E, 62 (2000), pp. 4010--4013, https://doi.org/10.1103/PhysRevE.62.4010.
25.
R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, 2012.
26.
J. D. Horton, A polynomial-time algorithm to find the shortest cycle basis of a graph, SIAM J. Comput., 16 (1987), pp. 358--366, https://doi.org/10.1137/0216026.
27.
S. Jafarpour and F. Bullo, Synchronization of Kuramoto oscillators via cutset projections, IEEE Trans. Automat. Control, 64 (2019), pp. 2830--2844, https://doi.org/10.1109/TAC.2018.2876786.
28.
N. Janssens and A. Kamagate, Loop flows in a ring AC power system, Internat. J. Electrical Power Energy Systems, 25 (2003), pp. 591--597, https://doi.org/10.1016/S0142-0615(03)00017-6.
29.
T. Kavitha, C. Liebchen, K. Mehlhorn, D. Michail, R. Rizzi, T. Ueckerdt, and K. A. Zweig, Cycle bases in graphs characterization, algorithms, complexity, and applications, Comput. Sci. Rev., 3 (2009), pp. 199--243, https://doi.org/10.1016/j.cosrev.2009.08.001.
30.
A. J. Korsak, On the question of uniqueness of stable load-flow solutions, IEEE Trans. Power Apparatus Systems, 91 (1972), pp. 1093--1100, https://doi.org/10.1109/TPAS.1972.293463.
31.
A. Kumar, R. Mudumbai, S. Dasgupta, M. M. Rahman, D. R. Brown III, U. Madhow, and T. P. Bidigare, A scalable feedback mechanism for distributed nullforming with phase-only adaptation, IEEE Trans. Signal Inform. Process. Networks, 1 (2015), pp. 58--70, https://doi.org/10.1109/TSIPN.2015.2442921.
32.
Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer, 1984.
33.
F. Lazebnik, On systems of linear Diophantine equations, Math. Mag., 69 (1996), pp. 261--266, https://doi.org/10.2307/2690528.
34.
N. E. Leonard, D. Paley, F. Lekien, R. Sepulchre, D. M. Fratantoni, and R. Davis, Collective motion, sensor networks and ocean sampling, Proc. IEEE, 95 (2007), pp. 48--74, https://doi.org/10.1109/JPROC.2006.887295.
35.
B. Lesieutre and D. Wu, An efficient method to locate all the load flow solutions---revisited, in Allerton Conf. on Communications, Control and Computing, Monticello, IL, 2015, pp. 381--388, https://doi.org/10.1109/ALLERTON.2015.7447029.
36.
S. Ling, R. Xu, and A. S. Bandeira, On the landscape of synchronization networks: A perspective from nonconvex optimization, SIAM J. Optim., 29 (2019), pp. 1879--1907, https://doi.org/10.1137/18M1217644.
37.
S. H. Low, Convex relaxation of optimal power flow, Part I: Formulations and equivalence, IEEE Trans. Control Network Systems, 1 (2014), pp. 15--27, https://doi.org/10.1109/TCNS.2014.2309732.
38.
W. Ma and J. S. Thorp, An efficient algorithm to locate all the load flow solutions, IEEE Trans. Power Systems, 8 (1993), pp. 1077--1083, https://doi.org/10.1109/59.260891.
39.
D. Manik, M. Timme, and D. Witthaut, Cycle flows and multistability in oscillatory networks, Chaos, 27 (2017), art. 083123, https://doi.org/10.1063/1.4994177.
40.
M. H. Matheny et al., Exotic states in a simple network of nanoelectromechanical oscillators, Science, 363 (2019), art. eaav7932, https://doi.org/10.1126/science.aav7932.
41.
G. S. Medvedev, Small-world networks of Kuramoto oscillators, Phys. D, 266 (2014), pp. 13--22, https://doi.org/10.1016/j.physd.2013.09.008.
42.
G. S. Medvedev and N. Kopell, Synchronization and transient dynamics in the chains of electrically coupled Fitzhugh--Nagumo oscillators, SIAM J. Appl. Math., 61 (2001), pp. 1762--1801, https://doi.org/10.1137/S0036139900368807.
43.
G. S. Medvedev and X. Tang, Stability of twisted states in the Kuramoto model on Cayley and random graphs, J. Nonlinear Sci., 25 (2015), pp. 1169--1208, https://doi.org/10.1007/s00332-015-9252-y.
44.
K. Mehlhorn and D. Michail, Minimum cycle bases: Faster and simpler, ACM Trans. Algorithms, 6 (2009), art. 8, https://doi.org/10.1145/1644015.1644023.
45.
D. Mehta, N. S. Daleo, F. DoĢrfler, and J. D. Hauenstein, Algebraic geometrization of the Kuramoto model: Equilibria and stability analysis, Chaos, 25 (2015), art. 053103, https://doi.org/10.1063/1.4919696.
46.
D. C. Michaels, E. P. Matyas, and J. Jalife, Mechanisms of sinoatrial pacemaker synchronization: A new hypothesis, Circulation Res., 61 (1987), pp. 704--714, https://doi.org/10.1161/01.RES.61.5.704.
47.
J. W. Milnor, Topology from the Differentiable Viewpoint, Princeton University Press, 1997; reprint of the 1965 edition.
48.
R. E. Mirollo and S. H. Strogatz, The spectrum of the locked state for the Kuramoto model of coupled oscillators, Phys. D, 205 (2005), pp. 249--266, https://doi.org/10.1016/j.physd.2005.01.017.
49.
A. Mirzaei, M. E. Heidari, R. Bagheri, S. Chehrazi, and A. A. Abidi, The quadrature LC oscillator: A complete portrait based on injection locking, IEEE J. Solid-State Circuits, 42 (2007), pp. 1916--1932, https://doi.org/10.1109/JSSC.2007.903047.
50.
D. K. Molzahn, F. DoĢrfler, H. Sandberg, S. H. Low, S. Chakrabarti, R. Baldick, and J. Lavaei, A survey of distributed optimization and control algorithms for electric power systems, IEEE Trans. Smart Grid, 8 (2017), pp. 2941--2962, https://doi.org/10.1109/TSG.2017.2720471.
51.
D. K. Molzahn, B. C. Lesieutre, and H. Chen, Counterexample to a continuation-based algorithm for finding all power flow solutions, IEEE Trans. Power Systems, 28 (2013), pp. 564--565, https://doi.org/10.1109/TPWRS.2012.2202205.
52.
J. A. Momoh, R. Adapa, and M. E. El-Hawary, A review of selected optimal power flow literature to 1993. I. Nonlinear and quadratic programming approaches, IEEE Trans. Power Systems, 14 (1999), pp. 96--104, https://doi.org/10.1109/59.744492.
53.
J. A. Momoh, M. E. El-Hawary, and R. Adapa, A review of selected optimal power flow literature to 1993. II. Newton, linear programming and interior point methods, IEEE Trans. Power Systems, 14 (1999), pp. 105--111, https://doi.org/10.1109/59.744495.
54.
J. Munkres, Topology, 2nd ed., Pearson, 2000.
55.
NERC Steering Group, Technical Analysis of the August 14, 2003, Blackout: What Happened, Why, and What Did We Learn?, Tech. report, North American Electric Reliability Council, Princeton, NJ, 2004.
56.
J. C. Neu, Coupled chemical oscillators, SIAM J. Appl. Math., 37 (1979), pp. 307--315, https://doi.org/10.1137/0137022.
57.
T. Nishikawa, Y.-C. Lai, and F. C. Hoppensteadt, Capacity of oscillatory associative-memory networks with error-free retrieval, Phys. Rev. Lett., 92 (2004), art. 108101, https://doi.org/10.1103/PhysRevLett.92.108101.
58.
J. Ochab and P. F. GoĢra, Synchronization of coupled oscillators in a local one-dimensional Kuramoto model, Acta Phys. Polonica Ser. B Proc. Ser., 3 (2010), pp. 453--462, https://www.actaphys.uj.edu.pl/S/3/2/453.
59.
D. A. Paley, N. E. Leonard, R. Sepulchre, D. Grunbaum, and J. K. Parrish, Oscillator models and collective motion, IEEE Control Systems, 27 (2007), pp. 89--105, https://doi.org/10.1109/MCS.2007.384123.
60.
A. Pluchino, V. Latora, and A. Rapisarda, Changing opinions in a changing world: A new perspective in sociophysics, Internat. J. Modern Phys. C, 16 (2005), pp. 515--531, https://doi.org/10.1142/S0129183105007261.
61.
S. Rao, Y. Feng, D. J. Tylavsky, and J. K. Subramanian, The holomorphic embedding method applied to the power-flow problem, IEEE Trans. Power Systems, 31 (2016), pp. 3816--3828, https://doi.org/10.1109/TPWRS.2015.2503423.
62.
J. A. Rogge and D. Aeyels, Stability of phase locking in a ring of unidirectionally coupled oscillators, J. Phys. A, 37 (2004), pp. 11135--11148, https://doi.org/10.1088/0305-4470/37/46/004.
63.
R. Sepulchre, D. A. Paley, and N. E. Leonard, Stabilization of planar collective motion: All-to-all communication, IEEE Trans. Automat. Control, 52 (2007), pp. 811--824, https://doi.org/10.1109/TAC.2007.898077.
64.
J. W. Simpson-Porco, A theory of solvability for lossless power flow equations---Part I: Fixed-point power flow, IEEE Trans. Control Network Systems, 5 (2018), pp. 1361--1372, https://doi.org/10.1109/TCNS.2017.2711433.
65.
J. W. Simpson-Porco, F. DoĢrfler, and F. Bullo, Synchronization and power sharing for droop-controlled inverters in islanded microgrids, Automatica J. IFAC, 49 (2013), pp. 2603--2611, https://doi.org/10.1016/j.automatica.2013.05.018.
66.
K. D. Smith, S. Jafarpour, and F. Bullo, Transient stability of droop-controlled inverter networks with operating constraints, IEEE Trans. Automat. Control, to appear, https://doi.org/10.1109/TAC.2021.3053552.
67.
P. A. Tass, A model of desynchronizing deep brain stimulation with a demand-controlled coordinated reset of neural subpopulations, Biol. Cybernet., 89 (2003), pp. 81--88, https://doi.org/10.1007/s00422-003-0425-7.
68.
R. Taylor, There is no non-zero stable fixed point for dense networks in the homogeneous Kuramoto model, J. Phys. A, 45 (2012), pp. 1--15, https://doi.org/10.1088/1751-8113/45/5/055102.
69.
R. Taylor, Finding Non-zero Stable Fixed Points of the Weighted Kuramoto Model Is NP-Hard, preprint, https://arxiv.org/pdf/1502.06688, 2015.
70.
A. Trias, The holomorphic embedding load flow method, in IEEE Power & Energy Society General Meeting, 2012, pp. 1--8, https://doi.org/10.1109/PESGM.2012.6344759.
71.
F. Varela, J. P. Lachaux, E. Rodriguez, and J. Martinerie, The brainweb: Phase synchronization and large-scale integration, Nature Rev. Neurosci., 2 (2001), pp. 229--239, https://doi.org/10.1038/35067550.
72.
T. Vicsek, A. CziroĢk, E. Ben-Jacob, I. Cohen, and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), pp. 1226--1229, https://doi.org/10.1103/PhysRevLett.75.1226.
73.
K. Wiesenfeld, P. Colet, and S. H. Strogatz, Frequency locking in Josephson arrays: Connection with the Kuramoto model, Phys. Rev. E, 57 (1998), pp. 1563--1569, https://doi.org/10.1103/PhysRevE.57.1563.
74.
D. A. Wiley, S. H. Strogatz, and M. Girvan, The size of the sync basin, Chaos, 16 (2006), art. 015103, https://doi.org/10.1063/1.2165594.
75.
R. D. Zimmerman, C. E. Murillo-SaĢnchez, and R. J. Thomas, MATPOWER: Steady-state operations, planning, and analysis tools for power systems research and education, IEEE Trans. Power Systems, 26 (2011), pp. 12--19, https://doi.org/10.1109/TPWRS.2010.2051168.
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Submitted: 31 January 2019
Accepted: 30 March 2021
Published online: 3 February 2022
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Defense Threat Reduction Agency https://doi.org/10.13039/100000774 : HDTRA1-19-1-0017
U.S. Department of Energy https://doi.org/10.13039/100000015 : DE-EE0000-1583
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