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Graphical Models and Belief Propagation Hierarchy for Physics-Constrained Network Flows

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Energy Markets and Responsive Grids

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 162))

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Abstract

We review new ideas and the first results from the application of the graphical models approach, which originated from statistical physics, information theory, computer science, and machine learning, to optimization problems of network flow type with additional constraints related to the physics of the flow. We illustrate the general concepts on a number of enabling examples from power system and natural gas transmission (continental scale) and distribution (district scale) systems.

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Notes

  1. 1.

    In the following we will use the notation {i, j} for the undirected edge and (i, j) for the respective directed edge. When the meaning is clear, we slightly abuse notations, denoting both the set of undirected and directed edges by \(\mathcal {E}\).

References

  1. Ahn S, Park S, Chertkov M, Shin J (2015) Minimum weight perfect matching via blossom belief propagation. In: Neural Information Processing Systems (NIPS) – spotlight presentation

    Google Scholar 

  2. Babonneau F, Nesterov Y, Vial JP (2012) Design and operations of gas transmission networks. Oper Res 60(1):34–47

    Article  MathSciNet  Google Scholar 

  3. Bach F (2015) Submodular functions: from discrete to continous domains. arXiv preprint arXiv:1511.00394

    Google Scholar 

  4. Baran M, Wu F (1989) Network reconfiguration in distribution systems for loss reduction and load balancing. IEEE Trans Power Delivery 4(2):1401–1407. https://doi.org/10.1109/61.25627

    Article  Google Scholar 

  5. Bienstock D (2013) Progress on solving power flow problems. Math Optim Soc Newsl (Optima) 93:1–7. http://www.mathopt.org/Optima-Issues/optima93.pdf

  6. Bienstock D (2015) Electrical transmission system cascades and vulnerability: an operations research viewpoint. Society for Industrial and Applied Mathematics, Philadelphia, PA

    Google Scholar 

  7. Bienstock D, Munoz G (2015) LP approximations to mixed-integer polynomial optimization problems. arXiv preprint arXiv:1501.00288

    Google Scholar 

  8. Bienstock D, Chertkov M, Harnett S (2014) Chance-constrained optimal power flow: risk-aware network control under uncertainty. SIAM Rev 56(3):461–495. https://doi.org/10.1137/130910312

    Article  MathSciNet  Google Scholar 

  9. Bishop CM (2006) Pattern recognition and machine learning, chapter graphical models. Springer, New York, p 359422

    Google Scholar 

  10. Borraz-Sanchez C (2010) Optimization methods for pipeline transportation of natural gas. Ph.D. thesis, Bergen University (Norway)

    Google Scholar 

  11. Chertkov M, Pan F, Stepanov MG (2011) Predicting failures in power grids: the case of static overloads. IEEE Trans Smart Grid 2(1):162–172. https://doi.org/10.1109/TSG.2010.2090912

    Article  Google Scholar 

  12. Chertkov M, Stepanov M, Pan F, Baldick R (2011) Exact and efficient algorithm to discover extreme stochastic events in wind generation over transmission power grids. In: 2011 50th IEEE conference on decision and control and European control conference, pp 2174–2180. https://doi.org/10.1109/CDC.2011.6160669

  13. Cowell RG, Dawid AP, Lauritzen SL, Spiegelhalter DJ (1999) Probabilistic networks and expert systems. Springer, Berlin

    Google Scholar 

  14. District heating: Wikipedia. https://en.wikipedia.org/wiki/District_heating. Accessed: 2017-01-07

  15. Dvijotham K, Turitsyn K (2015) Construction of power flow feasibility sets. arXiv preprint arXiv:1506.07191

    Google Scholar 

  16. Dvijotham K, Chertkov M, Low S (2015) A differential analysis of the power flow equations. 2015 54th IEEE Conference on Decision and Control (CDC), Osaka, pp. 23–30. https://doi.org/10.1109/CDC.2015.7402082

  17. Dvijotham K, Low S, Chertkov M (2015) Convexity of energy-like functions: theoretical results and applications to power system operations. arXiv preprint arXiv:1501.04052

    Google Scholar 

  18. Dvijotham K, Low S, Chertkov M (2015) Solving the power flow equations: a monotone operator approach. arXiv preprint arXiv:1506.08472

    Google Scholar 

  19. Dvijotham K, Vuffray M, Misra S, Chertkov M (2015) Natural gas flow solutions with guarantees: a monotone operator theory approach. arXiv preprint arXiv:1506.06075

    Google Scholar 

  20. Dvijotham K, Chertkov M, Van Hentenryck P, Vuffray M, Misra S (2016) Graphical models for optimal power flow. Constraints 1–26. https://doi.org/10.1007/s10601-016-9253-y

  21. Frolov V, Backhaus S, Chertkov M (2014) Efficient algorithm for locating and sizing series compensation devices in large power transmission grids: I. model implementation. New J Phys 16(10), 105015. http://stacks.iop.org/1367-2630/16/i=10/a=105015

    Article  Google Scholar 

  22. Frolov V, Backhaus S, Chertkov M (2014) Efficient algorithm for locating and sizing series compensation devices in large power transmission grids: II. solutions and applications. New J Phys 16(10):105016. http://stacks.iop.org/1367-2630/16/i=10/a=105016

    Article  Google Scholar 

  23. Frolov V, Thakurta PG, Backhaus S, et al. (2016) Optimal placement and sizing of FACTS devices to delay transmission expansion. arXiv preprint arXiv:1608.04467

    Google Scholar 

  24. Gamarnik D, Shah D, Wei Y (2010) Belief propagation for min-cost network flow: convergence and correctness. In: Proceedings of the twenty-first annual ACM-SIAM symposium on discrete algorithms. Society for Industrial and Applied Mathematics, Philadelphia, PA, pp 279–292. http://dl.acm.org/citation.cfm?id=1873601.1873625

    Chapter  Google Scholar 

  25. Glover JDD, Sarma MS (2001) Power system analysis and design, 3rd edn. Brooks/Cole Publishing, Pacific Grove, CA

    Google Scholar 

  26. Interior point optimizer. https://projects.coin-or.org/Ipopt. Accessed: 2017-07-23

  27. Jabr R (2012) Exploiting sparsity in SDP relaxations of the OPF problem. IEEE Trans Power Syst 27(2):1138–1139. https://doi.org/10.1109/TPWRS.2011.2170772

    Article  Google Scholar 

  28. Jensen F (1996) An introduction to Bayesian networks. Springer, Berlin

    Google Scholar 

  29. Johnson J (2008) Convex relaxation methods for graphical models: lagrangian and maximum entropy approaches. Ph.D. thesis, MIT. http://ssg.mit.edu/~jasonj/johnson_phd.pdf

  30. Kersulis J, Hiskens I, Chertkov M, Backhaus S, Bienstock D (2015) Temperature-based instanton analysis: identifying vulnerability in transmission networks. In: 2015 IEEE Eindhoven PowerTech, pp 1–6. https://doi.org/10.1109/PTC.2015.7232816

  31. Koller D, Friedman N (2009) Probabilistic graphical models. MIT Press, Massachusetts

    MATH  Google Scholar 

  32. Kolmogorov V, Thapper J, Zivny S (2015) The power of linear programming for general-valued CSPs. SIAM J Comput 44(1):1–36

    Article  MathSciNet  Google Scholar 

  33. Kudekar S, Johnson J, Chertkov M (2011) Linear programming based detectors for two-dimensional intersymbol interference channels. In: 2011 IEEE international symposium on information theory proceedings (ISIT), pp 2999–3003. https://doi.org/10.1109/ISIT.2011.6034129

  34. Kudekar S, Johnson J, Chertkov M (2013) Improved linear programming decoding using frustrated cycles. In: 2013 IEEE international symposium on information theory proceedings (ISIT), pp 1496–1500. https://doi.org/10.1109/ISIT.2013.6620476

  35. Lasserre J (2001) Global optimization with polynomials and the problem of moments. SIAM J Optim 11(3):796–817

    Article  MathSciNet  Google Scholar 

  36. Lasserre J (2010) Moments, positive polynomials and their applications. Imperial College Press, London

    MATH  Google Scholar 

  37. Lavaei J, Low S (2012) Zero duality gap in optimal power flow problem. IEEE Trans Power Syst 27(1):92–107. https://doi.org/10.1109/TPWRS.2011.2160974

    Article  Google Scholar 

  38. Lesieutre B, Molzahn D, Borden A, DeMarco C (2011) Examining the limits of the application of semidefinite programming to power flow problems. In: 2011 49th annual Allerton conference on communication, control, and computing (Allerton), pp 1492–1499. https://doi.org/10.1109/Allerton.2011.6120344

  39. Lieu H, Gartner N, Messer CJ, et al. (1999) Traffic flow theory. Public Roads 62:45–47

    Google Scholar 

  40. Lovasz L, Schrijver A (1991) Cones of matrices and set-functions and 0–1 optimization. SIAM J Optim 1(12):166–190

    Article  MathSciNet  Google Scholar 

  41. Low S (2014) Convex relaxation of optimal power flow; part i: formulations and equivalence. IEEE Trans Control Netw Syst 1(1):15–27. https://doi.org/10.1109/TCNS.2014.2309732

    Article  MathSciNet  Google Scholar 

  42. Low S (2014) Convex relaxation of optimal power flow; part ii: exactness. IEEE Trans Control Netw Syst 1(2), 177–189. https://doi.org/10.1109/TCNS.2014.2323634

    Article  MathSciNet  Google Scholar 

  43. Machowski J, Bialek JW, Bumby JR (2008) Power system dynamics: stability and control. Wiley, Chichester. http://opac.inria.fr/record=b1135564. Rev. ed. of: Power system dynamics and stability/Jan Machowski, Janusz W. Bialek, James R. Bumby (1997)

  44. Madani R, Sojoudi S, Lavaei J (2015) Convex relaxation for optimal power flow problem: mesh networks. IEEE Trans Power Syst 30(1):199–211. https://doi.org/10.1109/TPWRS.2014.2322051

    Article  Google Scholar 

  45. Malioutov D, Johnson J, Willsky A (2006) Walk-sums and belief propagation in gaussian graphical models. J Mach Learn Res 7:2031–2064

    MathSciNet  MATH  Google Scholar 

  46. Mezard M, Montanari A (2009) Information, physics, and computation. Oxford graduate texts. Oxford University Press, Oxford

    Book  Google Scholar 

  47. Misra S, Vuffray M, Chertkov M (2015) Maximum throughput problem in dissipative flow networks with application to natural gas systems. arXiv preprint arXiv:1504.02370

    Google Scholar 

  48. Mixture model: Wikipedia. https://en.wikipedia.org/wiki/Mixture_model#Gaussian_mixture_model

  49. Molzahn D, Hiskens I (2014) Moment-based relaxation of the optimal power flow problem. In: Power systems computation conference (PSCC), 2014, pp 1–7. https://doi.org/10.1109/PSCC.2014.7038397

    Google Scholar 

  50. Murphy KP (2012) Machine learning: a probabilistic perspective. MIT Press, Cambridge

    MATH  Google Scholar 

  51. Misra S, Fisher MW, Backhaus S, Bent R, Chertkov M, Pan F (2015) Optimal compression in natural gas networks: a geometric programming approach. IEEE Trans Control Netw Syst 2(1):47–56. https://doi.org/10.1109/TCNS.2014.2367360

    Article  MathSciNet  Google Scholar 

  52. Osiadacz A (1987) Simulation and analysis of gas networks. Gulf Publishing Company, London. http://books.google.com/books?id=cMxTAAAAMAAJ

    MATH  Google Scholar 

  53. Parrilo P (2003) Semidefinite programming relaxations for semialgebraic problems. Math Program Ser B 96(2):293–320

    Article  MathSciNet  Google Scholar 

  54. Pearl J (1988) Probabilistic reasoning in intelligent systems. Morgan Kaufmann, San Mateo, CA

    MATH  Google Scholar 

  55. Rauschenbach T (2016) Modeling, control and optimization of water systems: systems engineering methods for control and decision making tasks. Springer, Berlin

    Book  Google Scholar 

  56. Richardson TJ, Urbanke RL (2008) Modern coding theory. Cambridge University Press, Cambridge

    Book  Google Scholar 

  57. Risteski A (2016) How to calculate partition functions using convex programming hierarchies: provable bounds for variational methods. CoRR abs/1607.03183. http://arxiv.org/abs/1607.03183

  58. Roald L, Misra S, Chertkov M, Andersson G (2016) Optimal power flow with weighted chance constraints and general policies for generation control. In: 2015 54th IEEE conference on decision and control (CDC), pp 6927–6933. IEEE, Piscataway, NJ

    Google Scholar 

  59. Schlesinger MI (1976) Syntatic analysis of two-dimensional visual signals in noisy conditions. Kibernetika [in Russian] 4:113–130

    Google Scholar 

  60. Sherali HD, Adams WP (1990) A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J Discret Math 3(3):411–430. https://doi.org/10.1137/0403036

    Article  MathSciNet  Google Scholar 

  61. Sontag DA (2010) Approximate inference in graphical models using LP relaxations. Ph.D. thesis, MIT. http://www.cs.nyu.edu/~dsontag/papers/sontag_phd_thesis.pdf

  62. Sontag D, Jaakkola T (2008) New outer bounds on the marginal polytope. In: Platt J, Koller D, Singer Y, Roweis S (eds) Advances in neural information processing systems 20, pp 1393–1400. MIT Press, Cambridge, MA

    Google Scholar 

  63. Sontag D, Meltzer T, Globerson A, Weiss Y, Jaakkola T (2008) Tightening LP relaxations for MAP using message-passing. In: 24th Conference in Uncertainty in Artificial Intelligence, pp 503–510. AUAI Press, Corvallis

    Google Scholar 

  64. Sontag D, Choe DK, Li Y (2012) Efficiently searching for frustrated cycles in MAP inference. In: Proceedings of the twenty-eighth conference on uncertainty in artificial intelligence (UAI-12), pp 795–804. AUAI Press, Corvallis

    Google Scholar 

  65. Vuffray M, Misra S, Chertkov M (2015) Monotonicity of dissipative flow networks renders robust maximum profit problem tractable: general analysis and application to natural gas flows. In: 2015 54th IEEE conference on decision and control (CDC), pp 4571–4578. https://doi.org/10.1109/CDC.2015.7402933

  66. Wainwright MJ (2002) Stochastic processes on graphs: geometric and variational approaches. Ph.D. thesis, MIT. http://www.eecs.berkeley.edu/~wainwrig/Papers/Final2_Phd_May30.pdf

  67. Wainwright MJ, Jordan MI (2008) Graphical models, exponential families, and variational inference. Found Trends Mach Learn 1(1):1–305

    MATH  Google Scholar 

  68. Wald Y, Globerson A (2014) Tightness results for local consistency relaxations in continuous MRFs. In: 30th conference in uncertainty in artificial intelligence. AUAI Press, Corvallis

    Google Scholar 

  69. Werner T (2007) A linear programming approach to max-sum problem: a review. IEEE Trans Pattern Anal Mach Intell 29(7):1165–1179. https://doi.org/10.1109/TPAMI.2007.1036

    Article  Google Scholar 

  70. Werner T (2008) High-arity interactions, polyhedral relaxations, and cutting plane algorithm for soft constraint optimisation (map-mrf). In: IEEE conference on computer vision and pattern recognition, 2008 (CVPR 2008), pp 1–8. https://doi.org/10.1109/CVPR.2008.4587355

  71. Werner T (2010) Revisiting the linear programming relaxation approach to Gibbs energy minimization and weighted constraint satisfaction. IEEE Trans Pattern Anal Mach Intell 32(8):1474–1488. https://doi.org/10.1109/TPAMI.2009.134

    Article  Google Scholar 

  72. Wolf DD, Smeers Y (2000) The gas transmission problem solved by an extension of the simplex algorithm. Manag Sci 46(11):1454–1465. http://www.jstor.org/stable/2661661

    Article  Google Scholar 

  73. Wong P, Larson R (1968) Optimization of natural-gas pipeline systems via dynamic programming. IEEE Trans Autom Control 13(5):475–481. https://doi.org/10.1109/TAC.1968.1098990

    Article  Google Scholar 

  74. Wu S, Ros-Mercado R, Boyd E, Scott L (2000) Model relaxations for the fuel cost minimization of steady-state gas pipeline networks. Math Comput Model 31(23):197–220. http://dx.doi.org/10.1016/S0895-7177(99)00232-0. http://www.sciencedirect.com/science/article/pii/S0895717799002320

    Article  MathSciNet  Google Scholar 

  75. Yedidia J, Freeman W, Weiss Y (2005) Constructing free-energy approximations and generalized belief propagation algorithms. IEEE Trans Info Theory 51(7):2282–2312. https://doi.org/10.1109/TIT.2005.850085

    Article  MathSciNet  Google Scholar 

  76. Zinger H (1986) Hydraulic and heating regimes of district heating networks [in Russian]. Energoatomizdat, Moscow

    Google Scholar 

  77. Živný S, Werner T, Průša DA (2014) The power of LP relaxation for MAP inference, pp 19–42. MIT Press, Cambridge

    Google Scholar 

  78. Zlotnik A, Roald L, Backhaus S, Chertkov M, Andersson G (2017) Coordinated scheduling for interdependent electric power and natural gas infrastructures. IEEE Trans Power Syst 32(1):600–610. https://doi.org/10.1109/TPWRS.2016.2545522

    Article  Google Scholar 

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Acknowledgements

The authors are grateful to M. Lubin, N. Ruozzi, and J. B. Lasserre for fruitful discussions and valuable comments. The work at LANL was carried out under the auspices of the National Nuclear Security Administration of the US Department of Energy under Contract No. DE-AC52–06NA25396.

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Authors and Affiliations

  1. Theoretical Division, T-4, CNLS, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Michael Chertkov

  2. Energy System Center, Skoltech, Moscow, 143026, Russia

    Michael Chertkov

  3. Theoretical Division, T-5, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Sidhant Misra

  4. Theoretical Division, T-4, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Marc Vuffray

  5. Pacific Northwest National Laboratory, PO Box 999, Richland, WA, 99352, USA

    Dvijotham Krishnamurthy

  6. University of Michigan, Department of Industrial and Operations Engineering, Ann Arbor, MI, 48109, USA

    Pascal Van Hentenryck

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  1. Michael Chertkov
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  2. Sidhant Misra
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  4. Dvijotham Krishnamurthy
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  5. Pascal Van Hentenryck
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Correspondence to Dvijotham Krishnamurthy .

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Editors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Florida, Gainesville, Florida, USA

    Sean Meyn

  2. Technological Leadership Institute, University of Minnesota, Minneapolis, Minnesota, USA

    Tariq Samad

  3. Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, Michigan, USA

    Ian Hiskens

  4. Department of Electronic Systems, Aalborg University, Aalborg, Denmark

    Jakob Stoustrup

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Chertkov, M., Misra, S., Vuffray, M., Krishnamurthy, D., Hentenryck, P.V. (2018). Graphical Models and Belief Propagation Hierarchy for Physics-Constrained Network Flows. In: Meyn, S., Samad, T., Hiskens, I., Stoustrup, J. (eds) Energy Markets and Responsive Grids. The IMA Volumes in Mathematics and its Applications, vol 162. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-7822-9_10

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