An Approximate Dynamic Programming Algorithm for the Allocation of High-Voltage Transformer Spares in the Electric Grid

  • Johannes Enders
  • Warren B. Powell
  • David Egan
Part of the Energy Systems book series (ENERGY)


This paper addresses the problem of allocating high-voltage transformer spares (not installed) throughout the electric grid to mitigate the risk of random transformer failures. With this application we investigate the use of approximate dynamic programming (ADP) for solving large scale stochastic facility location problems. The ADP algorithms that we develop consistently obtain near optimal solutions for problems where the optimum is computable and outperform a standard heuristic on more complex problems. Our computational results show that the ADP methodology can be applied to large scale problems that cannot be solved with exact algorithms.


Approximate dynamic programming location analysis spare transformer allocation transformer replacement transformer spares two-stage stochastic optimization 


  1. 1.
    Bertsekas D, Tsitsiklis J (1996) Neuro-dynamic programming. Athena Scientific, BelmontzbMATHGoogle Scholar
  2. 2.
    Powell WB, George A, Bouzaiene-Ayari B Simao H (2005) Approximate dynamic programming for high dimensional resource allocation problems. In: Proceedings of the IJCNN, IEEE Press, New YorkGoogle Scholar
  3. 3.
    Birge J, Louveaux F (1997) Introduction to stochastic programming. Springer, New YorkzbMATHGoogle Scholar
  4. 4.
    Kall P, Wallace S (1994) Stochastic programming. Wiley, New YorkzbMATHGoogle Scholar
  5. 5.
    Sen S (2005) Algorithms for stochastic mixed-integer programming models. In: Aardal K, Nemhauser GL, Weismantel R (eds) Handbooks in operations research and management science: discrete optimization. North Holland, AmsterdamGoogle Scholar
  6. 6.
    Laporte G, Louveaux FV, van Hamme L (1994) Excact solution to a location problem with stochastic demands. Transp Sci 28(2):95–103CrossRefzbMATHGoogle Scholar
  7. 7.
    Louveaux FV, Peeters D (1992) A dual-based procedure for stochastic facility location. Oper Res 40(3):564–573MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ntaimo L, Sen S (2005) The million-variable “march” for stochastic combinatorial optimization. J Global Optim 32(3):385–400MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Powell WB, Ruszczyński A, Topaloglu H (2004) Learning algorithms for separable approximations of stochastic optimization problems. Math Oper Res 29(4):814–836MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Topaloglu H (2001) Dynamic programming approximations for dynamic programming problems. Ph.d. Dissertation, Department of Operations Research and Financial Engineering, Princeton UniversityGoogle Scholar
  11. 11.
    Mirchandani PB (1990) The p-median problem and generalizations. In: Mirchandani PB, Francis RL (eds) Discrete location theory. Wiley, New YorkGoogle Scholar
  12. 12.
    Labbé M, Peeters D, Thisse J-F (1995) Location on networks. In: Ball M, Magnanti TL, Monma CL, Nemhauser GL (eds) Handbooks in operations research and management science: network routing. Elsevier, AmsterdamGoogle Scholar
  13. 13.
    Chowdhury AA, Koval DO (2005) Development of probabilistic models for computing optimal distribution substation spare transformers. IEEE Trans Ind Appl 41(6):1493–1498CrossRefGoogle Scholar
  14. 14.
    Kogan VI, Roeger CJ, Tipton DE (1996) Substation distribution transformers failures and spares. IEEE Trans Power Syst 11(4):1905–1912CrossRefGoogle Scholar
  15. 15.
    Li W, Vaahedi E, Mansour Y (1999) Determining number and timing of substation spare transformers using a probabilistic cost analysis approach. IEEE Trans Power Deliver 14(3):934–939CrossRefGoogle Scholar
  16. 16.
    Godfrey G, Powell WB (2002) An adaptive, dynamic programming algorithm for stochastic resource allocation problems I: single period travel times. Transp Sci 36(1):21–39CrossRefzbMATHGoogle Scholar
  17. 17.
    Topaloglu H, Powell WB (2006) Dynamic programming approximations for stochastic, time-staged integer multicommodity flow problems. Informs J Comput 18(1):31–42MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Powell WB, Topaloglu H (2004) Fleet management. In: Wallace S, Ziemba W (eds) Applications of stochastic programming, SIAM series in optimization. Math Programming Society, PhiladelphiaGoogle Scholar
  19. 19.
    Puterman ML (1994) Markov decision processes. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  20. 20.
    Powell WB (2007) Approximate dynamic programming: solving the curses of dimensionality. Wiley, New YorkCrossRefGoogle Scholar
  21. 21.
    Powell WB, Shapiro JA, Simão HP (2002) An adaptive dynamic programming algorithm for the heterogeneous resource allocation problem. Transp Sci 36(2):231–249CrossRefzbMATHGoogle Scholar
  22. 22.
    Powell WB, Van Roy B (2004) Approximate dynamic programming for high dimensional resource allocation problems. In: Si J, Barto AG, Powell WB, Wunsch D II (eds) Handbook of learning and approximate dynamic programming. IEEE Press, New YorkGoogle Scholar
  23. 23.
    Helmberg C (2000) Semidefinite programming for combinatorial optimization. Technical report, Konrad-Zuse-Zentrum fuer Informationstechnik Berlin, BerlinGoogle Scholar
  24. 24.
    Godfrey GA, Powell WB (2001) An adaptive, distribution-free approximation for the newsvendor problem with censored demands, with applications to inventory and distribution problems. Manage Sci 47(8):1101–1112CrossRefzbMATHGoogle Scholar
  25. 25.
    Topaloglu H, Powell WB (2003) An algorithm for approximating piecewise linear concave functions from sample gradients. Oper Res Lett 31(1):66–76MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Chen QM, Egan DM (2006) A bayesian method for transformer life estimation using perks’ hazard function. IEEE Trans Power Syst 21(4):1954–1965CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Johannes Enders
    • 1
  • Warren B. Powell
    • 1
  • David Egan
    • 2
  1. 1.Department of Operations Research and Financial EngineeringPrinceton UniversityPrincetonUSA
  2. 2.PJM InterconnectionPhiladelphiaUSA

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