Probability theory applied to reservoir storage

  • N. T. Kottegoda


Stochastic models applicable to hydrologic time series and extreme events are investigated in earlier chapters. Such models are important for the simulation of complex water resource systems and for flood estimation purposes. However, in the design of individual reservoirs where one of the main criteria is the probability of failure, a more direct approach can be adopted if the inflow data are independent or have a Markov type of dependence. The choice between alternatives is usually made so that this probability does not exceed a stipulated value, which depends on the purposes served. Also important is the average number of times a reservoir will spill or empty during a given period. Another interesting outcome is the probability of first time emptiness.


Markov Chain Reservoir Storage Steady State Probability Unconditional Probability Multipurpose Reservoir 
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  1. Benjamin, J. R., and Cornell, C. A. (1970). Probability, Statistics and Decision for Civil Engineers, McGraw-Hill, New YorkGoogle Scholar
  2. Billingsley, P. (1961). Statistical Inference for Markov Processes, University of Chicago Press, Chicago, IllinoisGoogle Scholar
  3. Brooks, C. E. P., and Carruthers, N. C. (1953). Handbook of Statistical Methods in Meteorology, H. M. Stationary Office, London, pp. 281–339Google Scholar
  4. Chin, E. H. (1977). Modelling daily precipitation occurrence process with Markov chain. Water Resour. Res., 13, 949–56CrossRefGoogle Scholar
  5. Chung, K. L. (1967). Markov Chains with Stationary Transition Probabilities, 2nd edn, Springer, BerlinGoogle Scholar
  6. Cole, J. A. (1966). Application of two season statistics to reservoir yield calculations. Proceedings of the International Association of Scientific Hydrology Symposium, Garda, Italy, pp. 590–1Google Scholar
  7. Cox, D. R., and Miller, H. D. (1965). The Theory of Stochastic Processes, Methuen, LondonGoogle Scholar
  8. Cox, D. R., and Smith, W. L. (1961). Queues, Monographs on Applied Probability and Statistics, Methuen, LondonGoogle Scholar
  9. Crovelli, R. A. (1973). Stochastic models for precipitation. Proceedings of the International Symposium on Uncertainties in Hydrologic and Water Resource Systems, vol. 1, 11–14 December 1972, University of Arizona, Tucson, pp. 284–98Google Scholar
  10. David, F. N., and Barton, D. E. (1962). Combinatorial Chance, Griffin, LondonGoogle Scholar
  11. Doran, D. G. (1975). An efficient transition definition for discrete state reservoir analysis: the divided interval technique. Water Resour. Res., 11, 867–73CrossRefGoogle Scholar
  12. Dyck, S., and Schramm, M. (1966). Queuing theory and multipurpose reservoir design. Proceedings of the International Association of Scientific Hydrology Symposium, Garda, Italy, paper 2 (71), pp. 707–10Google Scholar
  13. Encyclopaedia Britannica (1977), vol. 14, 15th edn, William Benton, Chicago, Illinois, pp. 1113–15Google Scholar
  14. Farmer, E. E., and Homeyer, J. W. (1974). The probability of consecutive rainless days. Water Resour. Bull., 10, 914–24CrossRefGoogle Scholar
  15. Feller, W. (1968). Introduction to Probability Theory and its Applications, vol. 1, 3rd edn, Wiley, New YorkGoogle Scholar
  16. Fiering, M. B. (1962). Queuing theory and simulation in reservoir design. Trans. Am. Soc. Civ. Eng., 127, part I, paper 3367, 1114–44Google Scholar
  17. Gabriel, K. R., and Neumann, J. (1962). A Markov chain model for daily rainfall occurrence at Tel Aviv. Q. J. R. Meteorol. Soc., 88, 90–5CrossRefGoogle Scholar
  18. Gani, J. (1957). Problems in the probability theory of storage systems. J. R. Statist. Soc. B, 19, 181–206Google Scholar
  19. Gani, J. (1965). Flooding models. Proceedings of the Reservoir Yield Symposium,21–23 September 1965, Water Research Association, Medmenham, paper 4, pp. 4–1–4–16Google Scholar
  20. Gillett, B. E. (1976). Introduction to Operations Research, A Computer Oriented Algorithmic Approach, McGraw-Hill, New YorkGoogle Scholar
  21. Gnedenko, B. V. (1968). The Theory of Probability, 4th edn (translated from Russian by B. D. Seckler), Chelsea, New YorkGoogle Scholar
  22. Gould, B. W. (1961). Statistical methods for estimating the design capacity of dams. J. Inst. Eng. Aust., 33, 405–16Google Scholar
  23. Gould, B. W. (1966). Communication on `Probability of reservoir yield failure using Moran’s steady state method and Gould’s probability writing method’ by R. A. Harris. J. Inst. Water Eng., 20, 141–6Google Scholar
  24. Green, J. R. (1964). A model for rainfall occurrence. J. R. Statist. Soc. B., 26, 345–53Google Scholar
  25. Green, J. R. (1970). A generalized probability model for sequences of wet and dry days. Mon. Weath. Rev., 98, 238–41CrossRefGoogle Scholar
  26. Gross, D., and Harris, C. M. (1974). Fundamentals of Queuing Theory, Wiley, New YorkGoogle Scholar
  27. Harris, R. A. (1965). A probability of reservoir yield failure using Moran’s steady state probability method and Gould’s probability routing method. J. Inst. Water Eng., 19, 302–28Google Scholar
  28. Harris, R. A., Dearlove, R. E., and Morgan, M. (1965). Reservoir yield, 2, serially correlated inflows and subsequent attainment of steady state probabilities, Water Res. Assoc., Medmenham, Tech. Paper, No. 45Google Scholar
  29. Hershfield. D. M. (1971). Parameter estimation for wet-dry sequences. Water Resour. Bull., 7, 441–6CrossRefGoogle Scholar
  30. Hillier, F. S., and Lieberman, G. J. (1974). Operations Research, 2nd edn, Holden Day, San Francisco, CaliforniaGoogle Scholar
  31. Isaacson, D. L., and Madsen, R. W. (1976). Markov Chains—Theory and Applications, Wiley, New YorkGoogle Scholar
  32. Jarvis, C. L. (1964). An Application of Moran’s theory of dams to the Ord River project, western Australia. J. Hydrol., 2, 232–47CrossRefGoogle Scholar
  33. Joy, C. S., and McMahon, T. A. (1972). Reservoir-yield estimation procedures. Civ. Eng. Trans., Inst. Eng. Aust., 14, 28–36Google Scholar
  34. Kemeny, J. G., and Snell, J. L. (1960). Finite Markov Chains, Van Nostrand, Princeton, New JerseyGoogle Scholar
  35. Kendall, D. G. (1957). Some problems in the theory of dams. J. R. Statist. Soc. B., 19, 207–33Google Scholar
  36. Kiemeš, V. (1970). A two-step probabilistic model of storage reservoir with correlated inputs. Water Resour. Res., 6, 756–67CrossRefGoogle Scholar
  37. Kiemeš, V. (1971). On one difference between the Gould and Moran storage models. Water Resour. Res., 7, 410–4CrossRefGoogle Scholar
  38. Kiemeš, V. (1974). Probability distribution of outflow from a linear reservoir. J. Hydrol., 21, 305–14CrossRefGoogle Scholar
  39. Kiemeš, V. (1977). Discrete representation of storage for stochastic reservoir optimization. Water Resour. Res., 13, 149–58CrossRefGoogle Scholar
  40. Langbein, W. B. (1958). Queuing Theory and Water Storage. J. Hydraul. Div., Am. Soc. Civ. Eng., 84 (HY5), 1811–1–1811–24Google Scholar
  41. Lloyd, E. H. (1963). A probability theory of reservoirs with serially correlated inputs, J. Hydrol., 1, 99–128CrossRefGoogle Scholar
  42. Lloyd, E. H. (1967). Stochastic reservoir theory. Adv. Hydrosci., 4, 281–339CrossRefGoogle Scholar
  43. Lloyd, E. H. (1974). What is, and what is not, a Markov chain. J. Hydrol., 22, 1–28CrossRefGoogle Scholar
  44. Lloyd, E. H., and Odoom, S. (1964). Probability theory of reservoirs with seasdnal input. J. Hydrol., 2, 1–10CrossRefGoogle Scholar
  45. Lowry, W. P., and Guthrie, D. (1968). Markov chains of order greater than one. Mon. Weath. Rev., 96, 798–801CrossRefGoogle Scholar
  46. Moran, P. A. P. (1954). A probability theory of dams and storage systems. Aust. J. Appl. Sci., 5, 116–24Google Scholar
  47. Moran, P. A. P. (1955). A probability theory of dams and storage systems, modifications of the release rule. Aust. J. Appl. Sci., 6, 117–30Google Scholar
  48. Moran, P. A. P. (1959). The Theory of Storage, Methuen, LondonGoogle Scholar
  49. Mosteller, F. (1965). Fifty Challenging Problems in Probability, Addison-Wesley, Reading, MassachusettsGoogle Scholar
  50. National Bureau of Standards (1959). Tables of the bivariate normal distribution function and related functions. U.S. Dep. Comm., Washington, D.C., Appl. Math. Ser., No. 50Google Scholar
  51. Parzen, E. (1962). Stochastic Processes, Holden Day, San Francisco, CaliforniaGoogle Scholar
  52. Pegram, G. G. S. (1974). Factors affecting draft from a Lloyd reservoir. Water Resour. Res., 10, 63–6CrossRefGoogle Scholar
  53. Phatarfod, R. M. (1976). Some aspects of stochastic reservoir theory. J. Hydrol., 30, 199–217CrossRefGoogle Scholar
  54. Prabhu, N. U. (1958). Some exact results for the finite dam. Ann. Math. Statist., 29, 1234–43CrossRefGoogle Scholar
  55. Prabhu, N. U. (1964). Time dependent results in storage theory. J. Appl. Prob.,1, 1–64Google Scholar
  56. Reardon, T. J. (1970). Storage yield and probability from an engineer’s viewpoint (with discussion). Civ. Eng. Trans., Inst. Eng. Aust., 12, 119–24, 168, 169Google Scholar
  57. Revuz, D. (1975). Markov Chains, North-Holland, AmsterdamGoogle Scholar
  58. Ross, S. M. (1972). Introduction to Probability Models, Academic Press, New YorkGoogle Scholar
  59. Saverenskiy, A. D. (1940). Metod rascheta regulirovaniya stoka. Gidrotekhnicheskoe Stroitel’stvo, No. 2, 24–8Google Scholar
  60. Shamblin, J. E., and Stevens, G. T., Jr. (1974). Operations Research, a Fundamental Approach, McGraw-Hill, New YorkGoogle Scholar
  61. Strang, G. (1976). Linear Algebra and its Applications, Academic Press, New YorkGoogle Scholar
  62. Venetis, C. (1969a). Conditional probabilities of failures in reservoir operation for the Moran–Gould model. Water Resour. Res., 5, 514–18.CrossRefGoogle Scholar
  63. Venetis, C.(1970). Comments. Water Resour. Res., 6, 1427–32Google Scholar
  64. Venetis, C. (1969b). On the distribution of the frequency of reservoir deficit, J. Hydrol., 8, 341–6.CrossRefGoogle Scholar
  65. Venetis, C. (1970). Correction. J. Hydrol., 10, 103–4. Discussion. J. Hydrol.,10, 199–201CrossRefGoogle Scholar
  66. Venetis, C. (1969c). A stochastic model of monthly reservoir storage. Water Resour. Res., 5, 729–34.CrossRefGoogle Scholar
  67. Venetis, C. (1970). Correction. Water Resour. Res. 6, 351CrossRefGoogle Scholar
  68. Weesakul, B. (1961). First emptiness of a finite dam. J. R. Statist. Soc. B, 23, 343–51Google Scholar
  69. Weiss, L. L. (1964). Sequences of wet or dry days described by a Markov chain probability model. Mon. Weath. Rev., 92, 169–76CrossRefGoogle Scholar
  70. White, J. B. (1965). Probability of emptiness, II, a variable–season model. Proceedings of the Reservoir Yield Symposium, 21–23 September 1965 Water Research Association, Medmenham, paper 6, pp. 6–1–6–11Google Scholar

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© N. T. Kottegoda 1980

Authors and Affiliations

  • N. T. Kottegoda
    • 1
  1. 1.Department of Civil EngineeringUniversity of BirminghamUK

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