Algorithms for Finding Optimal Flows in Dynamic Networks

  • Maria FonoberovaEmail author
Part of the Energy Systems book series (ENERGY)


This article presents an approach for solving some power systems problems by using optimal dynamic flow problems. The classical optimal flow problems on networks are extended and generalized for the cases of nonlinear cost functions on arcs, multicommodity flows, and time- and flow-dependent transactions on arcs of the network. All parameters of networks are assumed to be dependent on time. The algorithms for solving such kind of problems are developed by using special dynamic programming techniques based on the time-expanded network method together with classical optimization methods.


Dynamic networks Minimum cost flow problem Multicommodity flows Network flows Optimal flows 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ahuja R, Magnati T, Orlin J (1993) Network flows. Prentice-Hall, Englewood CliffsGoogle Scholar
  2. Aronson J (1989) A survey of dynamic network flows. Ann Oper Res 20:1–66zbMATHCrossRefMathSciNetGoogle Scholar
  3. Assad A (1978) Multicommodity network flows: a survey. Networks 8:37–92zbMATHCrossRefMathSciNetGoogle Scholar
  4. Batut J, Renaud A (1992) Daily generation scheduling optimization with transmission constraints: a new class of algorithms. IEEE Trans Power Syst 7(3):982–989CrossRefGoogle Scholar
  5. Bland RG, Jensen DL (1985) On the computational behavior of a polynomial-time network flow algorithm. Technical Report 661, School of Operations Research and Industrial Engineering, Cornell UniversityGoogle Scholar
  6. Cai X, Sha D, Wong CK (2001) Time-varying minimum cost flow problems. Eur J Oper Res 131:352–374zbMATHCrossRefMathSciNetGoogle Scholar
  7. Carey M, Subrahmanian E (2000) An approach to modelling time-varying flows on congested networks. Transport Res B 34:157–183CrossRefGoogle Scholar
  8. Castro J (2000) A specialized interior-point algorithm for multicommodity network flows. Siam J Optim 10(3):852–877zbMATHCrossRefMathSciNetGoogle Scholar
  9. Castro J (2003) Solving difficult multicommodity problems with a specialized interior-point algorithm. Ann Oper Res 124:35–48zbMATHCrossRefMathSciNetGoogle Scholar
  10. Castro J, Nabona N (1996) An implementation of linear and nonlinear multicommodity network flows. Eur J Oper Res Theor Meth 92:37–53zbMATHCrossRefGoogle Scholar
  11. Chambers A, Kerr S (1996) Power industry dictionary. PennWell Books, OKGoogle Scholar
  12. Contreras J, Losi A, Russo M, Wu FF (2002) Simulation and evaluation of optimization problem solutions in distributed energy management systems. IEEE Trans Power Syst 17(1):57–62CrossRefGoogle Scholar
  13. Cook D, Hicks G, Faber V, Marathe M, Srinivasan A, Sussmann Y, Thornquist H (2000) Combinatorial problems arising in deregulated electrical power industry: survey and future directions. In: Panos PM (ed) Approximation and complexity in numerical optimization: continuous and discrete problems. Kluwer, Dordrecht, pp. 138–162Google Scholar
  14. Denny F, Dismukes D (2002) Power system operations and electricity markets. CRC Press, FLGoogle Scholar
  15. Ermoliev I, Melnic I (1968) Extremal problems on graphs. Naukova Dumka, KievGoogle Scholar
  16. Feltenmark S, Lindberg PO (1997) network methods for head-dependent hydro power schedule. In: Panos MP, Donald WH, William WH (eds) Network optimization. Springer, Heidelberg, pp. 249–264Google Scholar
  17. Fleisher L (2000) Approximating multicommodity flow independent of the number of commodities. Siam J Discrete Math 13(4):505–520CrossRefMathSciNetGoogle Scholar
  18. Fleischer L (2001a) Universally maximum flow with piecewise-constant capacities. Networks 38(3):115–125zbMATHCrossRefMathSciNetGoogle Scholar
  19. Fleischer L (2001b) Faster algorithms for the quickest transshipment problem. Siam J Optim 12(1):18–35zbMATHCrossRefMathSciNetGoogle Scholar
  20. Fleisher L, Skutella M (2002) The quickest multicommodity flow problem. Integer programming and combinatorial optimization. Springer, Berlin, pp. 36–53Google Scholar
  21. Fonoberova M, Lozovanu D (2007) Minimum cost multicommodity flows in dynamic networks and algorithms for their finding. Bull Acad Sci Moldova Math 1(53):107–119MathSciNetGoogle Scholar
  22. Ford L, Fulkerson D (1958) Constructing maximal dynamic flows from static flows. Oper Res 6:419–433CrossRefMathSciNetGoogle Scholar
  23. Ford L, Fulkerson D (1962) Flows in networks. Princeton University Press, Princeton, NJzbMATHGoogle Scholar
  24. Glockner G, Nemhauser G (2002) A dynamic network flow problem with uncertain arc capacities: formulation and problem structure. Oper Res 48(2):233–242CrossRefMathSciNetGoogle Scholar
  25. Goldberg AV, Tarjan RE (1987a) Solving minimum-cost flow problems by successive approximation. Proc. 19th ACM STOC, pp. 7–18Google Scholar
  26. Goldberg AV, Tarjan RE (1987b) Finding minimum-cost circulations by canceling negative cycles. Technical Report CS-TR 107-87, Department of Computer Science, Princeton UniversityGoogle Scholar
  27. Hoppe B, Tardos E (2000) The quickest transshipment problem. Math Oper Res 25:36–62zbMATHCrossRefMathSciNetGoogle Scholar
  28. Hu T (1970) Integer programming and network flows. Addison-Wesley Publishing Company, Reading, MAGoogle Scholar
  29. Kersting W (2006) Distribution system modeling and analysis, 2nd edn. CRC, FLGoogle Scholar
  30. Kim BH, Baldick R (1997) Coarse-grained distributed optimal power flow. IEEE Trans Power Syst 12(2):932–939CrossRefGoogle Scholar
  31. Klinz B, Woeginger C (1995) Minimum cost dynamic flows: the series parallel case. Integer programming and combinatorial optimization. Springer, Berlin, pp. 329–343Google Scholar
  32. Klinz B, Woeginger C (1998) One, two, three, many, or: complexity aspects of dynamic network flows with dedicated arcs. Oper Res Lett 22:119–127zbMATHCrossRefMathSciNetGoogle Scholar
  33. Lozovanu D, Fonoberova M (2006) Optimal flows in dynamic networks, Chisinau, CEP USMzbMATHGoogle Scholar
  34. Lozovanu D, Stratila D (2001) The minimum-cost flow problem on dynamic networks and algorithm for its solving. Bull Acad Sci Moldova Math 3:38–56MathSciNetGoogle Scholar
  35. Ma Z, Cui D, Cheng P (2004) Dynamic network flow model for short-term air traffic flow management. IEEE Trans Syst Man Cybern A Syst Hum 34(3):351–358CrossRefGoogle Scholar
  36. McBride R (1998) Progress made in solving the multicommodity flow problems. Siam J Optim 8(4):947–955zbMATHCrossRefMathSciNetGoogle Scholar
  37. McDonald JR, McArthur S, Burt G, Zielinski J (eds) (1997) Intelligent knowledge based systems in electrical power, 1st edn. Springer, HeidelbergGoogle Scholar
  38. Pansini A (2005) Guide to electrical power distribution systems, 6th edn. CRC, FLGoogle Scholar
  39. Papadimitrou C, Steiglitz K (1982) Combinatorial optimization: algorithms and complexity. Prentice-Hall, Englewood Cliffs, NJGoogle Scholar
  40. Pardalos PM, Guisewite G (1991) Global search algorithms for minimum concave cost network flow problem. J Global Optim 1(4):309–330zbMATHCrossRefMathSciNetGoogle Scholar
  41. Powell W, Jaillet P, Odoni A (1995) Stochastic and dynamic networks and routing. In: Ball MO, Magnanti TL, Monma CL, Nemhauser GL (eds) Network routing, vol. 8 of Handbooks in operations research and management science, chapter 3. North Holland, Amsterdam, The Netherlands, pp. 141–295Google Scholar
  42. Rajan GG (2006) Practical energy efficiency optimization. PennWell Books. OKGoogle Scholar
  43. Short T (2003) Electric power distribution handbook, 1st edn. CRC, FLGoogle Scholar
  44. Weber C (2005) Uncertainty in the electric power industry: methods and models for decision support. Springer, HeidelbergzbMATHGoogle Scholar
  45. Weber C, Marechala F, Favrata D (2007) Design and optimization of district energy systems. In: Plesu V, Agachi PS (eds) 17th European Symposium on Computer Aided Process Engineering – ESCAPE17, Elsevier, AmsterdamGoogle Scholar
  46. Willis HL (2004) Power distribution planning reference book. CRC Press, FLGoogle Scholar
  47. Willis HL, Welch G, Schrieber R (2000) Aging power delivery infrastructures. CRC Press, FLGoogle Scholar
  48. Wood A, Wollenberg B (1996) Power generation, operation and control. Wiley, NYGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Aimdyn IncSanta BarbaraUSA

Personalised recommendations